Geometry for the Artist 0367628236, 9780367628239

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Geometry for the Artist Geometry for the Artist is based on a course of the same name which started in the 1980s at Maharishi International University. It is aimed both at artists willing to dive deeper into geometry and at mathematicians open to learning about applications of mathematics in art. The book includes topics such as perspective, symmetry, topology, fractals, curves, surfaces, and more. A key part of the book’s approach is the analysis of art from a geometric point of view—looking at examples of how artists use each new topic. In addition, exercises encourage students to experiment in their own work with the new ideas presented in each chapter. This book is an exceptional resource for students in a general-education mathematics course or teachereducation geometry course, and since many assignments involve writing about art, this text is ideal for a writing-intensive course. Moreover, this book will be enjoyed by anyone with an interest in connections between mathematics and art. Features • Abundant examples of artwork displayed in full color. • Suitable as a textbook for a general-education mathematics course or teacher-education geometry course. • Designed to be enjoyed by both artists and mathematicians. Catherine A. Gorini is a professor of mathematics at Maharishi International University (MIU) in Fairfield, Iowa, where she has taught for over forty years and served as dean of faculty and dean of the College of Arts and Sciences. Her interests include geometry and connections between geometry and art, as well as mathematics education, connections between mathematics and consciousness, and liberal arts education. She started developing a geometry course for art majors over thirty years ago; this course is now popular with students of all majors. Her numerous teaching awards include the Outstanding Teacher of the Year award by MIU students in 2021 and the Award for Distinguished College or University Teaching of Mathematics given by the Mathematical Association of America in 2001. She received the 2019 Wege Award for Research from MIU. Gorini edited Geometry at Work, published by the Mathematical Association of America, wrote the Facts on File Geometry Handbook, and has published journal articles in geometry, mathematics education, connections between mathematics and art, and the relationship between consciousness and mathematics. She is the executive editor of the International Journal of Mathematics and Consciousness and holds a PhD in mathematics from the University of Virginia in algebraic topology.

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Geometry for the Artist

Catherine A. Gorini Maharishi International University United States of America

Designed cover image: Gregory Latta First edition published 2023 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2023 Catherine A. Gorini Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Gorini, Catherine A., author. Title: Geometry for the artist / Catherine A. Gorini, Maharishi International University, United States of America. Description: First edition. | Boca Raton : C&H/CRC Press, 2023. | Includes bibliographical references and index. Identifiers: LCCN 2022054960 (print) | LCCN 2022054961 (ebook) | ISBN 9780367628239 (hardback) | ISBN 9780367628253 (paperback) | ISBN 9781003110972 (ebook) Subjects: LCSH: Art--Mathematics. | Geometry. Classification: LCC N72.M3 G67 2023 (print) | LCC N72.M3 (ebook) | DDC 700/.45--dc23/eng/20230112 LC record available at https://lccn.loc.gov/2022054960 LC ebook record available at https://lccn.loc.gov/2022054961

ISBN: 978-0-367-62823-9 (hbk) ISBN: 978-0-367-62825-3 (pbk) ISBN: 978-1-003-11097-2 (ebk) DOI: 10.1201/9781003110972 Typeset in Latin Modern font by KnowledgeWorks Global Ltd. Publisher’s note: This book has been prepared from camera-ready copy provided by the author.

With deepest appreciation to Maharishi Mahesh Yogi

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Contents Preface

xiii

Acknowledgments

xv

Section I

Introduction

Chapter

1  Introduction

3

1.1

OVERVIEW

3

1.2

WAYS TO ANALYZE ART

5

1.3

HOW TO USE THIS BOOK

6

Section II Symmetry Chapter

2  Symmetry

11

2.1

THE NATURE OF SYMMETRY

11

2.2

OVERVIEW OF SYMMETRY

12

Chapter

3  Finite Designs

15

3.1

SYMMETRIC DESIGNS

15

3.2

SYMMETRY TRANSFORMATIONS

15

3.3

SYMMETRIES OF A SQUARE

16

3.4

SYMMETRIES OF A PINWHEEL

17

3.5

LEONARDO’S THEOREM

18

3.6

SIGNATURE OF A SYMMETRIC DESIGN

19

3.7

IMPORTANT EXAMPLES OF SYMMETRIC DESIGNS

19

3.8

COMPOUND OR LAYERED DESIGNS

24

3.9

SYMBOLS AND LOGOS

24

3.10 SACRED GEOMETRY

25

vii

viii  Contents

3.11 TIPS FOR DETERMINING THE SYMMETRIES OF A DESIGN

25

3.12 CREATING SYMMETRIC DESIGNS

26

Chapter

4  Band Ornaments

31

4.1

OVERVIEW

31

4.2

SYMMETRIES OF A BAND

31

4.3

THE SEVEN TYPES OF BAND ORNAMENTS

32

4.4

THE SEVEN TYPES OF SYMMETRIES OF BAND ORNAMENTS

36

4.5

MAKING A BAND ORNAMENT OF YOUR OWN

38

Chapter

5  The Regular Tilings

43

5.1

OVERVIEW

43

5.2

TRANSLATIONS OF A TILING

44

5.3

A TILING BY SQUARES

44

5.4

A TILING BY EQUILATERAL TRIANGLES

45

5.5

A TILING BY REGULAR HEXAGONS

46

5.6

WHAT WE HAVE LEARNED ABOUT REGULAR TILINGS

46

5.7

NEW TILINGS FROM OLD

47

Chapter

6  Tilings

51

6.1

OVERVIEW

51

6.2

TILINGS WITHOUT ROTATIONS—FOUR TILINGS

52

6.3

TILINGS WITH GYRATIONS ONLY—SIX TILINGS

53

6.4

TILINGS WITH KALEIDOSCOPES—FIVE NEW TILINGS PLUS TWO REGULAR TILINGS

56

6.5

TIPS FOR DETERMINING THE SIGNATURE OF A TILING PATTERN

58

6.6

SUMMARY OF TILINGS

60

Chapter

7  Symmetry in the Work of MC Escher

65

7.1

MC ESCHER’S TILINGS

65

7.2

HOW TO CREATE ESCHER-TYPE TILINGS

66

7.3

DAY AND NIGHT

69

7.4

CYCLE

70

Contents  ix

Section III Perspective Chapter

8  Introduction to Linear Perspective

75

8.1

OVERVIEW

75

8.2

INTRODUCTION TO LINEAR PERSPECTIVE

76

8.3

THE GEOMETRY OF LINEAR PERSPECTIVE

77

8.4

THE CENTRAL VANISHING POINT

79

8.5

ANALYSIS OF A PERSPECTIVE PICTURE

81

8.6

HOW TO DRAW A PICTURE IN ONE-POINT PERSPECTIVE

84

8.7

AERIAL OR ATMOSPHERIC PERSPECTIVE

85

Chapter

9  Drawing Grids in Perspective

91

9.1

OVERVIEW

91

9.2

DRAWING A SQUARE GRID IN PERSPECTIVE

92

9.3

WHERE IS THE DIAGONAL VANISHING POINT?

92

9.4

FINDING THE DIAGONAL VANISHING POINT IN A PAINTING

94

9.5

FORESHORTENING DISTANCES IN A PERSPECTIVE PAINTING

95

9.6

USING GRIDS TO DRAW SHAPES IN PERSPECTIVE

96

Chapter 10  The Conic Sections

99

10.1 OVERVIEW

99

10.2 THE CONIC SECTIONS

99

10.3 THE CIRCLE

100

10.4 THE ELLIPSE: A CIRCLE IN PERSPECTIVE

101

10.5 THE PARABOLA: A FOUNTAIN OF WATER

102

10.6 THE HYPERBOLA: A SHADOW ON A WALL

103

10.7 DRAWING A CIRCLE IN PERSPECTIVE

103

Chapter 11  Two-Point and Three-Point Perspective

109

11.1 OVERVIEW

109

11.2 TWO-POINT PERSPECTIVE

111

11.3 ESHER’S CYCLE —AN EXAMPLE OF TWO-POINT PERSPECTIVE

111

11.4 THREE-POINT PERSPECTIVE

112

11.5 USING TWO-POINT AND THREE-POINT PERSPECTIVE

113

x  Contents

Chapter 12  Perspective in the Work of MC Escher

119

12.1 OVERVIEW

119

12.2 RELATIVITY

120

12.3 BELVEDERE

120

12.4 ASCENDING AND DESCENDING

123

Section IV Fractals Chapter 13  Proportion and Similarity

131

13.1 OVERVIEW

131

13.2 CONGRUENT FIGURES

131

13.3 SIMILAR FIGURES

133

13.4 CONGRUENCE AND SIMILARITY IN ART

134

13.5 IMPORTANT EXAMPLES OF CONGRUENCE AND SIMILARITY

135

Chapter 14  Fractals

145

14.1 OVERVIEW

145

14.2 CONSTRUCTING GEOMETRIC FRACTALS

146

14.3 NATURAL FRACTALS

148

14.4 FRACTAL DIMENSION

149

14.5 APPLICATIONS OF FRACTALS

151

Chapter 15  Dynamical Systems and Chaos

155

15.1 OVERVIEW

155

15.2 DYNAMICAL SYSTEMS

155

15.3 MATHEMATICAL DYNAMICAL SYSTEMS

157

15.4 CHAOS

160

15.5 DYNAMICAL SYSTEMS AND CHAOS IN ART

161

15.6 CHAOS AND CHANCE

161

Chapter 16  The Mandelbrot Set

165

16.1 OVERVIEW

165

16.2 THE COMPLEX NUMBERS

165

Contents  xi

16.3 GRAPHING COMPLEX NUMBERS

166

16.4 CONSTRUCTING THE MANDELBROT SET

167

16.5 PROPERTIES OF THE MANDELBROT SET

169

Section V Curves, Spaces, and Geometries Chapter 17  Lines and Curves

175

17.1 OVERVIEW

175

17.2 STRAIGHT LINES AND THEIR SLOPES

176

17.3 CURVES AND CURVATURE

178

17.4 SPECIAL CURVES

181

17.5 HOW ARTISTS USE LINES AND CURVES

182

Chapter 18  Surfaces

187

18.1 OVERVIEW

187

18.2 SURFACES

188

18.3 EUCLIDEAN GEOMETRY

188

18.4 ELLIPTIC GEOMETRY

188

18.5 HYPERBOLIC GEOMETRY

190

18.6 THE THREE GEOMETRIES AND CURVATURE

190

18.7 THE THREE GEOMETRIES AROUND YOU

192

Chapter 19  Euclidean and Non-Euclidean Geometries

195

19.1 OVERVIEW

195

19.2 EUCLID’S POSTULATES

195

19.3 THE VALIDITY OF NON-EUCLIDEAN GEOMETRIES

197

Chapter 20  Topology

201

20.1 OVERVIEW

201

¨ 20.2 THE KONIGSBERG BRIDGE PROBLEM

202

20.3 TOPOLOGY

203

20.4 TOPOLOGY IN ART

206

xii  Contents

Chapter 21  Pictorial Composition

213

21.1 OVERVIEW

213

21.2 GEOMETRIC PRINCIPLES OF PICTORIAL COMPOSITION

214

21.3 PORTRAITS

218

21.4 HOW THE EYE MOVES AROUND A PICTURE

219

21.5 UNDERSTANDING PICTORIAL COMPOSTION

220

Bibliography

223

Appendix

225

A  Suggestions for Final Projects

A.1

ART RESEARCH PROJECT

225

A.2

ART RESEARCH TEAM PROJECT

225

A.3

CREATIVE PROJECT

225

Appendix

Index

B  Answers to Selected Exercises

227

233

Preface I first learned about the connections between geometry and art when I read Mathematics in Western Culture by Morris Kline and then later when I saw a collection of Japanese family crests. I was delighted to see that the geometry I loved for its austere beauty was used by artists to create beauty that could be appreciated by everyone. As I continued to study mathematics, I saw more and more such connections between mathematics and art, and I started to include those connections in my teaching.

One [science] is the systematic knowledge of life while the other [art] is the systematic expression of it; systematic expression and systematic knowledge go hand-in-hand. Maharishi Mahesh Yogi The Unmanifest Canvas

This textbook is based on the course Geometry for the Artist, which started in the 1980s when Michael Cain, the chairperson of the Art Department at Maharishi International University (MIU), asked for a geometry course for art students. Art majors took the first offerings of the course, somewhat reluctantly, as their mathematics requirement, but they seemed to enjoy connecting geometry to their artwork. Then mathematics majors wanted to take the course, and today majors from every department take the course. The topics initially included perspective (inspired by Mathematics in Western Culture) and symmetry (inspired by the Japanese crests), but other topics were added over time. I wanted students to see the topological magic of the M¨obius band. The use of topology in art is pervasive—from El Greco’s elongated figures to Salvador Dalí’s melted clocks, to delightful animated characters like Mister Fantastic. As fractals became popular, students wanted to understand them. Conversations with Dale Divoky of the MIU Art Department about art and nature inspired lessons on curves and surfaces since students wanted to know about the wide variety of curves and surfaces that they see in nature. The final lesson on pictorial composition brings all of these topics together—in the context of a single work of art. This textbook covers only two-dimensional art for several reasons. The first is that a picture in a book of a three-dimensional work of art or a still from a film cannot do justice to the whole work. Also, there is no time in a one-semester course to cover both two-dimensional and three-dimensional art. And finally, if you understand the xiii

xiv  Preface

geometric principles that apply to one-dimensional art, you can apply those principles to three-dimensional art and film. A key part of the course is each student’s analysis of works of art from a geometric point of view, looking to see if and how artists use each new topic. Students have found this to be rewarding. One student analyzed painted buffalo hides, an important part of her cultural heritage. At first, she felt that such an analysis would be at odds with the spiritual values of the works. But later, as she saw the highly developed use of geometric principles supporting the spiritual meanings, she felt that she had a much greater appreciation and respect for the traditional artists. Mathematical analysis of a work of art does not make it dry and lifeless, but rather gives new life to the work, enriching our insight and appreciation. It is not enough to see how others use geometry, so this course offers opportunities in every lesson for students to create works of their own using the new concepts and techniques that they have just learned. This book was written for the artist who is open to learning about geometry and the mathematician who wants to know more about applications in art. I encourage you to take a journey into the mathematics of art and see for yourself all that mathematics has to offer to the artist.

Acknowledgments This work would not have been possible without the contributions of many people— including the hundreds of students who have taken the course Geometry for the Artist during the past thirty years. I have learned so much from them about art, about artists, and about how art and geometry are connected. I can’t thank everyone of them individually, and I know that I have left out too many, but I do want to make special mention of many of those who have an impact on my understanding of art and geometry, teaching about art and geometry, or who have helped in other ways. My heartfelt thanks go to Corina Acosta, Toni Alazraki, Andy Averill, Judy Bales, Thomas Banchoff, Matthew Beaufort, James Billingsly, Tom Brooks, Charlotte Cain, Michael Cain, Paul Calter, Jim Casey, Anna Chepourkova, Fran Clark, Annalisa Crannell, Meg Custer, Marina D’Angiolillo, Michael Deitzel, Thérèse Dignard, Susie Dillbeck, Dale Divoky, Anne Dow, Connie Eyberg, Jim Fairchild, Rodney Franz, Paul Glossop, Helen Gorini, John Gorini, Margaret Gorini, Sarah Greenwald, Taniya Hallman, Eric Hart, Vicki Alexander Herriot, David Henderson, Betsy Henry, Peter Huestis, Nick Jackiw, Shiraz Januwalla, Noah Johnson, Thomas Kadar, Jay Kappraff, Janet Kernis, Carolyn King, Frances Knight, Shashi Kumar, Roy Lane, Gregory Latta, Carl Lee, Gurdy Leete, Isabelle Levi, Viktoria Luhaste, Susan Metrican, Walter Meyer, Ken Morrissey, Joe O’Rourke, Lynne Osborn, Rachel Osborn, Mary Platt, Marina Pokrovskaya, John Price, Claudia Rodrigues, Ann Robertson, Susan Runkle, Sanaa Sayani, Doris Schattschneider, Daniel Scher, Jane SchmidtWilk, Behnaz Shahidi, Lawrence Sheaff, Jim Shrosbree, Richard Sorensen, Daina Taimin, a, Joe Tarver, Mary Jo Toles, Narcissa Vanderlip, Rouzanna Vardanyan, Ted Wallace, Laura Wege, Michael Weinless, Kim Williams, Emi Wherry, Elinor Wolfe, Kris Wood, and the MIU Mathematics and Consciousness Study Group. M.C. Escher’s pictures Relativity, Ascending and Descending, Belvedere, Day and c 2022 The M.C. Escher Night. and Cycle are courtesy of the M.C. Escher Company, Company-The Netherlands. All rights reserved. www.mcescher.com I want to give very special thanks to Dr. Tony Nader, Riaz Ahmed, Veda Govender, and Jonathan Looney for significant contributions to this textbook. And finally, it is a pleasure to express my greatest appreciation and admiration for my editors, Callum Fraser and Mansi Kabra, for their advice, continuous support, encouraging feedback, and kind patience.

xv

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I Introduction

Susan Chrysler White. My Own Private Ecosystem, 2021.

1

Figure 1.1

Pieter Bruegel. Hunters in the Snow, 1565.

Figure 1.2

Wassily Kandinsky. Yellow-Red-Blue, 1925.

CHAPTER

1

Introduction

EXPLORATIONS • Look at the picture Hunters in the Snow by Pieter Bruegel, shown in Figure 1.1. What is portrayed? What impressions do you get from the picture? What geometrical shapes, either distinct or implied, do you see? • Look at the picture Yellow-Red-Blue by Wassily Kandinsky, shown in Figure 1.2. Does the picture portray anything familiar to you? What impressions do you get from the picture? What geometrical shapes, either distinct or implied, do you see? • Name some ways in which the two pictures are different. Do you see any ways in which the pictures are alike or similar?

1.1

OVERVIEW

To fully appreciate mathematics, we must know how it can be used in art and other areas, and to fully appreciate art, we must know how artists use mathematics in their work. The full range of mathematics is from the very small to the very large, ranging from the number 0 to the infinity of all possible numbers in algebra and from a point to lines and planes that extend infinitely in geometry. Art also has an infinite range, encompassing all human emotions and all possible ways to express those emotions.

Geometry exists as an innate phenomenon in our consciousness. In the external world a perfectly formed snow crystal would never exist. But in our consciousness lies the glittering and flawless knowledge of perfect ice. Peter Høeg Smilla’s Sense of Snow

A work of art has many details. Somehow, the artist combines a multitude of separate details into a wholeness that is the work of art, a wholeness that is more DOI: 10.1201/9781003110972-1

3

4  Geometry for the Artist

than the sum of the parts. In the process of creating art, artists incorporate many tools from geometry. In this book, we look at some important aspects of geometry; we will see how artists have used them and you will learn how to apply them to the creation of art. You will see richness and variety in geometry and in the unbounded creativity of artists who use this geometry in their work. You will also have opportunities to create your own work using the geometry that you are learning. The themes of our study are symmetry, perspective, fractals, Euclidean and nonEuclidean geometries, and topology. We discuss these in the next sections.

Ultimately, everyone says, there is no art without geometry. Frank Stella

Symmetry

Symmetric patterns convey a feeling of orderliness, harmony, and balance. Geometrical techniques give ways to characterize regular patterns. We’ll study individual symmetric designs, frieze patterns or band ornaments, and tilings or wallpaper patterns, like those shown in Figure 2.1; our tools will be symmetry transformations like rotations, reflections, and translations. We’ll also learn how to create intricate tilings like those of MC Escher. Perspective

Perspective gives geometric techniques that can represent three-dimensional objects on a two-dimensional surface or picture plane. Artists use perspective to bring the viewer into a scene and can make the scene intimate and personal or dramatic and imposing. Different types of perspective, each appropriate for a particular purpose, include one-point, two-point, and three-point perspective. Understanding these kinds of perspective will help us understand how MC Escher used perspective to create his “impossible worlds.” Euclidean geometry

Euclidean geometry uses measurement to study points, lines, planes, and their relationships in a flat, unbounded plane and is the foundation and inspiration for all other geometries. Topics from Euclidean geometry that we will study include circles and conic sections, lines and curves, and similarity and proportion. We’ll see many examples of Euclidean geometry in art. Fractal geometry

A fractal is a self-similar shape, as we see in a fern, a coastline, clouds, or lightning. Since many natural shapes are fractals, fractal geometry has been called the

Introduction  5

“geometry of nature.” We’ll see how dynamical systems—systematic processes that are repeated over and over—create fractal shapes, and we’ll see applications of fractals in art. Non-Euclidean Geometry

Non-Euclidean geometry goes beyond Euclidean geometry, which studies flat space, to the study of curved surfaces. Elliptic geometry is geometry on a sphere, and hyperbolic geometry is geometry on a saddle or ruffled surface. We’ll see many examples of both elliptic and hyperbolic geometries in art and nature. Topology

Topology is the most abstract area of geometry that we will study; it has been called “rubber sheet geometry,” because it studies the properties and relationships of a shape that stay the same even when the shape is bent, twisted, shrunk, or stretched, as if it were made of a super-stretchy kind of rubber. It is surprising how many artists use topology in their work.

I want the painter, as far as he is able, to be learned in all the liberal arts, but I wish him above all to have a good knowledge of geometry. Leon Battista Alberti (1404–1472) On Painting

1.2 WAYS TO ANALYZE ART Each work of art is a unique whole, created by an artist for us to enjoy. Whether from the distant past or by a contemporary artist, a work of art expresses and communicates the deep personal feelings and individuality of the artist using different artistic methods, techniques, and materials. As the viewer of a work of art, we connect with the artist, expanding our experience of life. We can make this experience richer by learning more about the artist and the work. There are many ways to analyze a work of art, each of which can uncover valuable insights about the work. One could look at the place this work has in the body of the work of the artist. Or one could look at the historical environment of the work— the influences of other artists on the artist, the cultural or political environment in which the artist produced the work, and the art movements surrounding the work. One might look at the role that the work had in society—the reception of the work by the public, the influence of the work on other artists, and the response of critics to the work. A chemist could study the chemical composition of the paint on a painting to determine the nature of pigments and where they came from or use carbon dating to

6  Geometry for the Artist

determine the age of ancient works of art. A curator will study the provenance of a work of art—who has owned it in the past and where it has been shown. In this book, you will learn a different approach to analyzing art—by using geometry. You will find that understanding the geometrical features of a work of art will support and reinforce what you know about the work of art from other ways of analysis. To appreciate a work of art and gain personal value from it, you must look at it, not once but many times. For this reason, many of the activities in this book will give you opportunities to study different works of art repeatedly, using different aspects of geometry each time. By the end of this course, you will have learned new ways to analyze, understand, and appreciate art and you will have a deeper appreciation of geometry.

The value of art is that it has its boundaries, and within those boundaries it is the full expression of life, yet it tells the story of the beyond. It speaks in silence; it speaks of the unboundedness of life, and this is the glory of it. Maharishi Mahesh Yogi The Unmanifest Canvas

1.3 HOW TO USE THIS BOOK While each part of the book is mainly independent of the other parts, it is helpful to study the chapters in each part in sequence. To get the most out of the book, you should work through some of the exercises in each chapter. The exercises include those that will reinforce your understanding of the material and many that allow you to be creative in different ways using the material you have just learned. The “Two by Two” and “Three by One” exercises in each chapter encourage you to look at selected works of art in depth, from many different perspectives. While not all geometric topics are relevant to any one picture, you will be surprised to find that every picture uses a variety of geometric concepts in unique ways. Please enjoy this journey into geometry and see for yourself why mathematicians consider mathematics to be an art.

EXERCISES Apply

1.1 Three Pictures by One Artist: 3by1. Choose three paintings, by one artist or, for work that is not attributed to a single artist, from one time period or style. These are the pictures you will use for all 3by1 exercises in the following chapters. It

Introduction  7

is all right to choose pictures that you are very familiar with since further study will give you new insights and deeper appreciation of the pictures. Give the names of the pictures and the name of the artist and sketches or copies of the paintings or links to them. Spend some time looking at your paintings and reflecting on your feelings about them. You may want to do a historical study of your artist and paintings, or you might want to make your own copies of the paintings. Then spend some time free writing about the pictures to put down how you respond to them. Here are some prompts to help you get started with the free write. a. Describe the paintings. What does each painting show? Is there a story or is the painting abstract? b. Does the title tell you something about the painting? c. What shapes are used? What colors are used? How are the shapes and colors arranged? d. Write about why you like the paintings or explain why you chose them. e. Do the paintings remind you of anything? How do you feel when you look at them? What do you think was the intention of the artist in creating each picture? f. Do the paintings show spatial depth or do they feel flat? g. How does your eye move around the paintings? h. Is there a feeling of balance or imbalance? What gives it that feeling? i. Is there silence in the paintings? Is there dynamism? Does the painting as a whole convey predominantly silence or predominantly dynamism or is there a balance? j. Do the paintings express the infinite in any way? Do they convey a feeling of infinity? Explain. k. Describe any geometric aspects of the pictures that strike you. 1.2 Two Pictures by Two Artists: 2by2. In this exercise, you will be comparing two different paintings by different artists. Comparing different pictures leads to a greater understanding of the pictures individually. Choose two pictures by different artists that you like. Give the name of each painting, the artist (if known), and a sketch of the painting or a link to the painting. These are the pictures you will use for all 2by2 exercises in the following chapters. For each of the two paintings, answer the following questions: a. b. c. d.

Outline some of the important shapes in the painting. How are those shapes arranged or related to one another? Is there a feeling of balance or imbalance? What gives it that feeling? Is there silence in the paintings? Is there dynamism? Does the painting as a whole convey predominantly silence or predominantly dynamism or is there a balance? e. What do you think was the intention of the artist?

8  Geometry for the Artist

f. Is there anything in the pictures that conveys infinity? Explain. Discuss one significant difference between the two paintings and one significant way in which they are similar. You may also do a free write about the two paintings using the prompts in Exercise 1.1. Create

1.3 Geometry You Remember. Draw some of the shapes that you remember from a previous mathematics or geometry course. Create a picture or design using some of the shapes that you remember. 1.4 Geometry around You. Look around your home or classroom. Sketch or photograph shapes that you feel have a geometric interest or that remind you of shapes that you have studied in previous courses. Create a picture or design using some of the shapes that you find. 1.5 Geometry in the Environment. Go to an interesting place near you. Sketch or photograph shapes that you feel have a geometric interest or that remind you of shapes that you have studied in previous courses. Create a picture or collage using some of the shapes that you find. Review

1.6 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 1.7 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

READINGS AND RESOURCES • Mathematics + Art: A Cultural History by Lynn Gamwell • Manifold Mirrors: The Crossing Paths of the Arts and Mathematics by Felipe Cucker • Squaring the Circle by Paul Calter • The Geometer’s Sketchpad, interactive geometry software, https://www.ke ycurriculum.com • Web Sketchpad, interactive geometry web app, https://geometricfunct ions.org/fc/tools/#welcome • GeoGebra, interactive mathematics software and web app, https://www.ge ogebra.org

II Symmetry

Handwoven Navajo Klagetoh rug.

Persian rug.

9

Designs and patterns from Oceania. From The Grammar of Ornament by Owen Jones.

Figure 2.1

CHAPTER

2

Symmetry

EXPLORATIONS Figure 2.1 shows many different designs and patterns from Oceania. Choose one design or pattern that appeals to you. What about the design appeals to you? See if you can describe it in words. Copy the design or create a similar design of your own—grid paper is very useful for copying or creating symmetric designs. Where might you make use of the design?

2.1

THE NATURE OF SYMMETRY

Symmetry is important in both mathematics and art. The word symmetry connotes beauty, harmony, and grace. We see symmetry all around us in patterns and designs from cultures all over the world as well as in natural shapes—flowers, butterflies, snowflakes, crystals, and even our own bodies. Artists and artisans use designs and patterns in many different ways to enhance or decorate other objects—emblems, jewelry, borders, textiles, walls, floors, and so on. Mathematicians look at the abstract structure of a pattern rather than the details. So, for example, in Figure 2.2, we see two different patterns that could extend to cover the plane. Even though the designs are different, a mathematician would say that the two designs have the same type of symmetry. One design has a diagonal grid with crosses inside the squares. The other design has a horizontal-vertical grid with crosses inside some squares and dots inside other squares. But nevertheless, a mathematician can see the same structure underlying the two different designs.

Figure 2.2

Two patterns from Oceania. From The Grammar of Ornament by Owen

Jones.

DOI: 10.1201/9781003110972-2

11

12  Geometry for the Artist

Looking at the different patterns of Figure 2.1, we see that elements of each design are repeated over and over. It may be a curved line, a cross, a red or black square, a diagonal line, or another geometric shape. The repetition of shapes is the key to the mathematical quantification of symmetry. In the next five chapters, we will see how mathematicians use symmetry transformations to connect repeated elements of a design, to classify designs, and to create new designs. We’ll see that there are many different types of symmetry, and we’ll also see that for any one type of symmetry, artists have created many different designs.

2.2

OVERVIEW OF SYMMETRY

In this part, we analyze three types of designs: finite designs or rosettes, band ornaments or frieze patterns, and tilings or wallpaper patterns. We start in the next chapter with finite designs and their symmetry transformations—rotations and reflections. Then, in the following chapter, we look at band ornaments, also called strip patterns or frieze patterns—designs that go on forever left and right, back and forth. We add two new symmetry transformations to our toolbox, translations and glide reflections. Following that, we study tilings—patterns like those seen in brick pavements, printed fabric, or wallpaper, which can go on forever to cover an infinite plane. These are more complex and take up the next two chapters. Finally, for some fun, we look at the symmetric patterns that the Dutch artist MC Escher used in unusual and imaginative ways. You will learn how to create your own patterns using his techniques.

And, in fact, the sense of the symmetrical is an instinct which may be depended on with an almost blindfold reliance. It is the poetical essence of the Universe—of the Universe which, in the supremeness of its symmetry, is the most sublime of poems. Edgar Allan Poe (1809–1849)

Eureka

Taylor & Francis Taylor & Francis Group

http://taylorandfrancis.com

Japanese designs used on cloisonné enamels, from L’ornement Polychrome by Albert Racinet.

Figure 3.1

CHAPTER

3

Finite Designs

EXPLORATIONS • Look at the Japanese designs shown in Figure 3.1. Choose some designs that appear symmetric to you and copy them. What are the features of the designs that make you think that they are symmetric? Choose some designs that do not appear to be symmetric. What are the features of these designs that make you think that they are not symmetric? Create some designs of your own based on these designs.

3.1

SYMMETRIC DESIGNS

The individual designs shown in Figure 3.1 are examples of finite designs or rosettes. In this chapter, we study the kinds of symmetries that finite designs can have and we present a technique for creating symmetric designs of your own. The key feature of a symmetrical design is that some parts of the design are exactly like other parts. In Figure 3.1, you can see that most of the smaller individual designs are made up of parts that are the same. It is this repetition of parts that creates a feeling of harmony, balance, and unity in a design—and is what we will capture in the idea of a symmetry transformation.

3.2

SYMMETRY TRANSFORMATIONS

Consider the familiar designs shown in Figure 3.2—a circle, the letter S, a heart, a triangle, and a square.

Figure 3.2

Five finite designs.

The circle looks the same if we rotate it by any amount or flip it in any direction. In the triangle, every edge looks the same and every corner looks the same; we can say the same for the square. The top and bottom of the letter S both look the same DOI: 10.1201/9781003110972-3

15

16  Geometry for the Artist

although they are facing in different directions, and the two sides of the heart look the same even though they are also facing in different directions. You can rotate the letter S by one-half turn, and it looks the same; flip the heart over, and it looks the same. To capture the idea of parts that look the same mathematically, we use a symmetry transformation, or symmetry for short, a motion or transformation that moves a shape onto itself so that it appears unchanged. If you don’t look while someone performs a symmetry transformation, you would think that nothing happened. A symmetry transformation must take corners to corners and edges to edges— otherwise, you would see a change. A line must go to a line of the same length, and an angle must go to an angle of the same measurement. This means that lengths and angle measurements stay the same under a symmetry transformation—otherwise, you could detect the change. A symmetry transformation is a rigid motion, a motion that changes only the position or orientation of a shape. In the next section, we look at the symmetry transformations of a square.

3.3

SYMMETRIES OF A SQUARE

To study the symmetries of a square, start by labeling the corners of the square A, B, C, and D, as shown in Figure 3.3. The labels aren’t part of the design, but they help us describe the transformations. Rotations

If we rotate the square through an angle of 90◦ about the point at the center of the square, each corner will land on the next corner and each edge will land on the next edge. The labels will change, but the square itself looks exactly the same—and this is what makes the 90◦ rotation a symmetry transformation. Rotating the square clockwise by 90◦ takes point A to point B, point B to point C, and so on, but the square itself looks the same. If we rotate by 90◦ again and again, it will still look the same. In fact, there are four different clockwise rotations, through angles of 90◦ , 180◦ , 270◦ , and 360◦ or 0◦ ; so point A can move to point B, point C, point D, or stay at point A. Figure 3.3 The The center of the square remains fixed by all of the rosquare ABCD and tations. This is a feature of any rotation—there is a central its center point. point that doesn’t move. This point is called the center of rotation or the rotocenter. We say that the rotocenter of the square has period 4, since there are four different rotations (0◦ , 90◦ , 180◦ , and 270◦ ) that leave the square unchanged. It may seem strange to include a rotation of 0◦ , which does nothing, but it means that we end up with four rotations in all, matching the fact that the square has four corners and four edges. To describe a rotation of a design, it is enough to give the rotocenter and either the smallest non-zero angle that leaves the design unchanged or the period of the rotation. Depending on the context, you may find the angle or the period more

Finite Designs  17

convenient. These numbers are related to each other by a formula: (smallest non-zero angle of rotation) × (period of rotation) = 360◦ This formula is valid because the period is the number of times we rotate the design by the smallest non-zero angle of rotation to get back to the starting point, and this gives the full circle 360◦ rotation. For the square, this formula is 90◦ × 4 = 360◦ Reflections of a Square

There is another kind of symmetry that a finite design can have, a reflection. Flip a square over; it is still a square. However, the flip or reflection will be a symmetry only in certain cases, just as only certain rotations were symmetries. In fact, there are just four ways to reflect a square so that it looks exactly the same. To describe a reflection, we give a specific line, called the mirror line, that we reflect across. Points on opposite sides of this mirror line are interchanged, while the points on the mirror line itself remain fixed. The square in Figure 3.4 has four mirror lines that give symmetry transformations. Two of the mirror lines go through the center of opposite edges, while the other two go through opposite corners or vertices. If the square is reflected across the horizontal line, vertices A and D are interchanged and vertices B and C are interchanged. Figure 3.4 The square Similarly, if the square is reflected across the diagonal line ABCD and its mirror from A to C, vertices B and D are interchanged while ver- lines. tices A and C remain fixed. For each of these reflections, the square remains unchanged. The mirror lines of the square go through the center of the square. This will be true for every mirror line that is a symmetry transformation of a design—it goes through the center of the design. We’ll call a design like a square that has both reflections and rotations a kaleidoscope, since a kaleidoscope toy shows designs with many reflections. The rotocenter of a kaleidoscope is called a kaleidoscopic point.

3.4

SYMMETRIES OF A PINWHEEL

The pinwheel design shown in Figure 3.5 is based on a square, but its symmetry is different. The four triangles create a feeling of direction, like a real-life pinwheel that spins if you blow on it. If you reflect the pinwheel, as shown in Figure 3.6, it looks different. The pinwheel looks the same under 90◦ rotations, but not under reflections. Like the square, the pinwheel has four rotational symmetries, through angles of 0◦ , 90◦ , 180◦ , and 270◦ . But unlike the square, these four rotations are the only symmetries of the pinwheel.

18  Geometry for the Artist

A pinwheel with its rotocenter in red. Figure 3.5

Figure 3.6

A pinwheel and its re-

flection.

A design like a pinwheel that has only rotations is called a gyration, and we call its rotocenter a gyration point.

3.5

LEONARDO’S THEOREM

In his study of architectural design, Leonardo da Vinci (1452–1519) observed that symmetric designs have very special properties. He determined that it is no coincidence that the number of mirror lines of the square is the same as the number of rotation symmetries that it has. He saw that this holds for any design that has both rotation symmetries and reflection symmetries. Let’s take a look at this. The square has four corners, four edges, four different rotational symmetries, and four mirror lines. A rotation takes a corner to a corner, so there are four different rotations—vertex A can go to vertex A, B, C, or D. Similarly, the rotation can take an edge to itself or any of the other three edges. A mirror line must go through the center of an edge or through a vertex, and the square has only four such lines. What is important is that if a design has reflection symmetries, it must also have rotation symmetries. To get a feel for this, think about what happens when you reflect the square twice. Reflecting across the same line twice is the same as doing nothing. Reflecting a square across the horizontal mirror line and then the vertical mirror line is shown in Figure 3.7. You can see that this is the same as a rotation through an angle of 180◦ , so the fact that we have these two reflection symmetries means that we must also have a rotation symmetry.

Reflecting a square across a horizontal line and then across a vertical line is the same as rotating the square clockwise by an angle of 180◦ . Figure 3.7

Leonardo da Vinci was the first to discover this fact, now called Leonardo’s theorem, that a finite design has either only rotations or else both rotations and reflections. If a design has reflections and rotations, the number of rotations is the same as the

Finite Designs  19

number of reflections. He also noticed that designs of each type can exist for any integer starting at 1. Summing up Leonardo’s theorem about symmetric designs: • Some designs have only rotational symmetries. Such a design is called a gyration. • If a design has reflections, it must also have rotations. Such a design is called a kaleidoscope. • If a design has reflections and rotations, then the number of rotations is the same as the number of reflections. • There are gyration designs and kaleidoscopic designs with periods equal to every whole number starting at 1.

3.6

SIGNATURE OF A SYMMETRIC DESIGN

It is convenient to assign a code or signature to a design in order to describe its symmetries. In this section, we show how to assign a signature to a finite design. Considering the square and the pinwheel as examples, it is clear that we must give two pieces of information to describe the symmetry of a design: the period of the rotations that the design has, and whether or not the design has any mirror lines. A design with rotational symmetry has a fixed point in the center, which we indicate by a dot (•). We also need to give the smallest non-zero angle of rotation or the period of the rotation; it is easier to give the period as it is a smaller number and can usually be determined just by looking at the design. For designs without reflections, the signature is the period of the rotation followed by a dot. Thus, the signature of a pinwheel is 4 • and completely describes its symmetries—there is a rotocenter • and rotations of period 4. If a design has mirror symmetry, we use the star symbol ∗ since a star also has refection symmetry. Thus, the signature of a square is ∗ 4 •. The star tells us that the square has reflections. The numeral 4 tells us that there are exactly four reflections and four rotations. The dot (•) at the end tells us that there is exactly one fixed point, the center of the square. If a design has no symmetry—no reflections and no rotations other than 0◦ – we say that it is asymmetric or that it has trivial symmetry. The signature is 1 •. Examples of designs with trivial symmetry are the letters F, G, J, L, P, and Q. Table 3.1 gives examples of simple designs with small period along with their signatures. In the following section, we look at examples that are frequently found in art and design.

3.7

IMPORTANT EXAMPLES OF SYMMETRIC DESIGNS

We now look at common examples of symmetry—including point symmetry and bilateral symmetry. We begin with regular polygons; a regular polygon is a polygon, all of whose edges have equal length and all of whose angles have the same measure.

20  Geometry for the Artist

TABLE 3.1

Examples of gyrations and kaleidoscopes with small periods. Gyrations

Kaleidoscopes

1•

∗1•

2•

∗2•

3•

∗3•

4•

∗4•

5•

∗5•

6•

∗6•

8•

∗8•

The Equilateral Triangle

The equilateral triangle—a regular triangle—has three edges equal in length and three angles, each measuring 60◦ . It is very easy to construct a regular triangle using a compass, as shown in Figure 3.8. The signature of an equilateral triangle is ∗ 3 •.

Finite Designs  21

To construct an equilateral triangle, draw a straight line segment. At each endpoint, construct a circle with radius equal to the length of the segment. Connect the two endpoints to one of the intersections of the two circles, as shown. Figure 3.8

The Square

The square has four edges equal in length and four angles, each measuring 90◦ . It is very easy to construct a square from a piece of paper, as shown in Figure 3.9. The signature of a square is ∗ 4 •.

To construct a square from a rectangular piece of paper, fold one of the corners until it lands on the opposite edge. Cut and unfold to get a square. Figure 3.9

The Regular Pentagon

The regular pentagon has five edges equal in length and five angles, each measuring 108◦ . An easy paper-folding construction of the regular pentagon is shown in Figure 3.10. Probably more familiar than the regular pentagon is the pentagram, the star formed by the diagonals of the regular pentagon, as shown in Table 3.1 next to the regular pentagon. The signature of both the regular pentagon and the pentagram is ∗ 5 •.

To construct a regular pentagon, tie a strip of paper in an overhand knot, not allowing the paper to wrinkle. Tighten the knot carefully. Cut off the ends as shown to get a regular pentagon. Figure 3.10

22  Geometry for the Artist

The Regular Hexagon

The regular hexagon has six edges equal in length and six angles, each measuring 120◦ . It is very easy to construct a regular hexagon from a circle, as shown in Figure 3.11. The cross section of a honeycomb contains many hexagons. The signature of a regular hexagon is ∗ 6 •.

To construct a regular hexagon, start with a circle; construct another circle of the same radius with its center on the circumference of the original circle, as shown at far left in the diagram. Construct two more circles of the same radius with centers at the points of intersection of the first two circles. Continue in this way until there are six circles and connect the intersections of the six new circles with the original circle, as shown in the center of the diagram. Erase the construction lines to get a regular hexagon, far right. Figure 3.11

The Regular Polygons

For every integer n greater than 2, there is a regular polygon, with all edges of equal length and each angle of size (n − 2)180◦ . n The most convenient way to construct regular polygons is with computer software such as The Geometer’s Sketchpad. The signature of a regular polygon with n edges and n angles is ∗ n •. Designs with Point Symmetry

The letter S and the yin-yang symbol are both gyrations with rotations of only 180◦ and 0◦ . This type of symmetry has a special name, point symmetry. If you trace designs with point symmetry and then rotate the tracing paper, the original design and the traced design will match only when you have rotated 180◦ or 0◦ . These designs do not have any mirror lines because they look different after flipping them over. Four designs with point symmetry are shown in Figure 3.12. Each of these designs has signature 2 •. A common shape with point symmetry is the rhombus, which has four edges equal in length. Unlike the square, only opposite angles have equal measure. The rhombus

Finite Designs  23

Figure 3.12

Four designs with signature 2 •.

has point symmetry ∗ 2 •. Examples are shown in Figure 3.13. Other shapes with point symmetry are parallelograms and the letters Z and N.

Figure 3.13

Four rhombuses of different sizes and shapes.

One way to create a design with point symmetry is to choose any finite design and choose a rotocenter, which can be on, near, or even inside the design. Then rotate a copy of the design by an angle of 180◦ around the rotocenter that you chose. The resulting design, as shown in Figure 3.14, has point symmetry.

To create a design with point symmetry, start with any design (far left), rotate the design 180◦ around a point (center), and then erase the rotocenter (far right). Figure 3.14

Designs with Bilateral Symmetry

A design with exactly one mirror line has just one rotation of period 1; its signature is ∗ 1 •. Such a design is said to have bilateral symmetry. The word “bilateral” means two sided, which refers to the two parts of the design that are reflections of one another. The mirror line can be oriented in any direction, but the most common directions are vertical and horizontal. Common geometric bilateral designs include the heart and the letters A, D, E, and M. Shapes with bilateral symmetry are abundant in nature: the human body, most other mammals, butterflies and moths, many birds and fish, and many leaves. A simple method for creating a design with bilateral symmetry is shown in Figure 3.15.

24  Geometry for the Artist

To create a design with bilateral symmetry, start with any asymmetric design (left), reflect the design across a mirror line (center), and then erase the mirror line (right). Figure 3.15

The Circle

The circle has very special symmetry: a circle looks the same if you rotate it through any angle about its center or reflect it across any line through its center. A design with this symmetry is said to have radial symmetry.

3.8

COMPOUND OR LAYERED DESIGNS

You must be careful with designs like those in Figure 3.16. They are constructed from an overlay of designs of different periods. For a rotation to leave the design unchanged, it must be a symmetry of each layer of the design. Tracing the design and then rotating or reflecting the design will help you to determine the period of the symmetry. The first design of Figure 3.16 has a triangle inside a hexagon. The hexagon has a rotocenter of period 6, while the triangle has a rotocenter of period 3; the design as a whole has signature ∗ 3 •. The second design consists of a rectangle inside a square and has signature ∗ 2 •. The third design consists of a square inside a hexagon; only two of the mirror lines of each layer coincide, and the design as a whole has signature ∗ 2 •. To determine the period of a complex design, you can trace and rotate, counting the number of rotations that leave the design unchanged, until you get back to where you started. Reflect your tracing to see if there are any mirror lines.

Figure 3.16

Compound designs made up of simpler designs with different symme-

tries.

3.9

SYMBOLS AND LOGOS

Many national symbols, corporate logos, and emoticons have symmetry. The recycling symbol, shown in Figure 3.17, has gyrational symmetry, referring to the cyclic nature

Finite Designs  25

of the activity of recycling. The flag of the Isle of Man, Figure 3.18, has a symbol with period 3 gyration that is based on an ancient symbol.

The recycling symbol, with signature 3 •. Figure 3.17

3.10

Flag of the Isle of Man. The symbol has signature 3 •. Figure 3.18

3.19 A Japanese emblem with signature 6 •. Figure

3.20 A Celtic design with signature 3 •. Figure

SACRED GEOMETRY

Sacred geometry refers to the geometry used by religious and spiritual traditions for symbols and architecture. Many of the shapes in sacred geometry, such as the Christian cross, the Star of David, the Star of Lakshmi, the Wheel of Dharma, the yinyang symbol, and the mandala, use the balancing and harmonious effect of symmetry.

3.11

TIPS FOR DETERMINING THE SYMMETRIES OF A DESIGN

Trace the design on tracing paper or clear plastic. Reflect the tracing and see if you can get it to match up with the original design. If you can’t get it to match up, the design is a gyration. If you can get it to match up, it is a kaleidoscope. If your design is a gyration, • Match your tracing to the original design. Rotate the tracing and see how many times it matches the design until you get back to where you started. The number of times it matches, including the starting match, gives the period of the rotocenter. • Count the number of times a distinctive feature—a “petal” or an “arm” or other distinctive shape—is repeated around the center of the design. This number will be the period of the rotocenter of the design or a multiple of the period in case of a compound design. If your design is a kaleidoscope, • Match your tracing to the original design. Rotate the tracing and see how many times it matches the design until you get back to where you started. The number of times it matches, counting the starting match, gives the period of the rotocenter and the number of mirror lines. Mirror lines always go through the rotocenter of the design. To find the mirror lines, look for features such

26  Geometry for the Artist

as petals or other bilateral shapes; mirror lines will go through the center or between such features. If the rotocenter has odd period, all mirror lines will look the same; if the rotocenter has even period, there will be mirrors of two types, alternating with one another. • If each petal or arm has bilateral symmetry on its own, there may be mirror lines through the petals or arms or between petals or arms. • If there are an even number of petals with bilateral symmetry, there may be mirror lines through petals and between petals. • If there are an odd number of petals, the mirror lines may pass through the middle of one petal and between two opposite petals. • Draw all the mirror lines and count them. Be careful to count each mirror line once—even though it appears on two sides of the design. Artists have created a wide variety of designs, but the signature of any design will always be either ∗ n • or n •. The signature is ∗ n • if there are n mirror lines and n rotations. The signature is n • if there are n rotations but no mirror lines.

3.12

CREATING SYMMETRIC DESIGNS

To create a design of your own, you must first make two decisions: will your design have mirror lines or not—in other words, will it be a kaleidoscope or gyration? What period will the rotations have? If your design has mirror lines, the number of mirror lines will be the same as the period of the rotation. Your design will have signature ∗ n • if you chose to create a kaleidoscope of period n and will have signature n • if you chose to create a gyration of period n. Once you have chosen the period n, construct a regular polygon with n vertices. You can do this with a compass and protractor or a free app like The Geometer’s Sketchpad, WebSketchpad, or GeoGebra.

Figure 3.21

Construction of designs with signatures 5 • and ∗ 5 •.

Figure 3.21 illustrates the construction of a design with period 5. Start with a regular pentagon and connect the center point to each vertex. To get a design with signature 5 •, put an asymmetric design in one of the five regions and rotate the design four times. To get a design with signature ∗ 5 •, divide one of the five regions in half as shown. Draw an asymmetric design in one of these smaller regions, reflect it across the radius, and then rotate this bilateral design four times, getting a design with signature ∗ 5 •.

Finite Designs  27

EXERCISES Understand

3.1 Symmetric Designs. Give the signature of each design shown in Figure 3.22.

Figure 3.22

Eight finite designs.

3.2 Corporate Logos. Give the signature of each corporate logo shown in Figure 3.23.

Figure 3.23

Eight corporate logos.

3.3 Parallelograms. Find or draw a parallelogram and trace it on tracing paper. Flip the tracing paper over. Can you get the flipped parallelogram to match the original parallelogram? What is the signature of your parallelogram? 3.4 Compound Designs. Give the signature of each of the compound designs shown in Figure 3.24.

Figure 3.24

Seven Compound Designs.

3.5 Flipping a Square. What rotation of the square would be the result of reflecting first across the vertical mirror line and then across the horizontal mirror line? What rotation of the square would be the result of doing the rotations in reverse order: reflecting first across the horizontal mirror line and then across the vertical mirror line?

28  Geometry for the Artist

3.6 Flipping a Square Again. What rotation of the square would be the result of reflecting across the diagonal mirror line through B and D of the square shown in Figure 3.4 and then across the vertical mirror line? What rotation of the square would be the result of doing these rotations in reverse order—reflect across the vertical mirror line and then across the diagonal mirror line through B and D? Compare your results. 3.7 Comparing Rotations. Compare the results of Exercise 3.5 with the results of Exercise 3.6. What do you notice? Apply

3.8 Designs around You. Find some interesting designs in your environment and photograph or draw them. Classify them by giving their signatures. 3.9 Logos. Find two logos, one with kaleidoscopic symmetry and one with gyrational symmetry. Do the symmetries fit the purpose or use of the logos? Explain your answer. 3.10 2by2. Are there any symmetric designs in the two paintings you chose by two artists? If so, copy and classify them. 3.11 3by1. Are there any symmetric designs in the three paintings you chose by one artist? If so, copy and classify them. Create

3.12 Finite Designs. Create a table, like that in Figure 3.1, using your own designs with signatures 1 •, ∗ 1 •, 2 •, ∗ 2 •, 3 •, ∗ 3 •, 4 •, ∗ 4 •, 5 •, ∗ 5 •, 6 •, and ∗ 6 •. 3.13 Using Designs around You. Find some interesting designs in your environment and photograph or draw them. Create your own designs based on these designs. Classify them by giving their signatures. 3.14 Quilt Square. Find some symmetric shapes in the Pictorial Quilt by Harriet Powers on page 214 and classify them. This quilt uses the method of appliqué, where a small piece of cloth cut in a particular shape is sewn to the background cloth. Note that the symmetries are only approximate, due to the handcrafting of the shapes. Design a quilt square of your own that incorporates symmetric designs. 3.15 Rug Patterns. Find some symmetric shapes in the Navajo rug or the Persian rug on page 9 and classify them. Design a symmetric design of your own that could be used on a rug. 3.16 Create Your Own. Create four designs with different signatures. Give the signatures of each. 3.17 Create Your Own Logo. Create your own symmetric logo or emoticon. What kind of symmetry did you choose? Why did you choose it?

Finite Designs  29

Review

3.18 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 3.19 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

READINGS • Symmetry: A Unifying Concept by Istv´an Hargittai and Magdolna Hargittai • Symmetry: The Ordering Principle by David Wade

ARTISTS • Lawrence Sheaff (b. 1940), British painter; symmetric designs • Japanese crests • Indian rangoli designs • Sona designs from Angola

NOTE The classification scheme used here was developed by William Thurston, and the associated signature notation was developed by John Conway. Other notations are also used, notably the one developed by the International Union of Crystallography.

Figure 4.1

Five band ornaments with the same type of symmetry.

Figure 4.2

Seven band ornaments with seven different types of symmetry.

CHAPTER

4

Band Ornaments

EXPLORATIONS • Look at the band ornaments in Figure 4.1. They all belong to the same symmetry classification. How are they alike? How are they different? • Look at the band ornaments in Figure 4.2. They all have different symmetry classifications. Can you see the differences? Which one do you think has the same symmetry classification as all four bands in Figure 4.1?

4.1

OVERVIEW

A band ornament or frieze pattern is simply a row of repeated finite designs, as in Figures 4.1 and 4.2. Our understanding of finite designs will help us study band ornaments. To classify band ornaments, we will use rotations and reflections along with two new symmetry transformations: translation, which just moves a band to the left or the right, and glide reflection, which is a combination of a translation (“glide”) and a reflection.

4.2

SYMMETRIES OF A BAND

Mathematicians like to think of a band ornament extending infinitely left and right—in other words, the designs keep on repeating infinitely to the left and to the right. Thinking of a band ornament in this way makes it easier to understand the symmetries of band ornaments; it also means that once we understand the pattern of a band ornament, we can extend the band ornament indefinitely, as far as we like, to the left or to the right. We think of the symmetries of the band ornament moving this infinite band in different ways. Another way is to think of a band ornament as a band Figure 4.3 Large around the equator of a very large sphere, as shown in Figsphere with band. ure 4.3. This way of thinking about the band helps us classify its symmetries. First, in this section, we’ll find the symme-

DOI: 10.1201/9781003110972-4

31

32  Geometry for the Artist

tries of the underlying band, and then we’ll look for symmetries of different band ornaments. The plain band going around the equator of a large sphere has the following symmetries: • Rotation by any amount around the vertical axis that goes through the north and south poles • Reflection across the equator, the midline of the band • Reflection across any longitude line—a curve along the sphere connecting the north pole and the south pole • Rotation around any point on the midline of the band Of course, any combination of these symmetry transformations will be another symmetry of the band. The infinite strip has the following closely related symmetries: • Moving by any amount, left or right (remember that the strip is infinite in both directions)—translation • Reflection across the midline of the band—horizontal reflection • Reflection across any vertical line through the strip—vertical reflection • Rotation by 0◦ or 180◦ about any point that lies on the horizontal line through the middle of the strip—rotation Again, any combination of these symmetry transformations will be another symmetry of the band.

4.3

THE SEVEN TYPES OF BAND ORNAMENTS

Once we put designs or motifs on the band, everything is changed. We can only translate by an amount that will assure that one motif lands exactly on another. The location of mirror lines and rotocenters will depend on the type and location of the motifs. These restrictions allow only seven different symmetry types of band ornaments, as mathematicians are able to demonstrate. We now look at these seven types. A Band with Only Translations

Put an asymmetrical flower at regular intervals along the band; this gives us the top band ornament in Figure 4.2. We call this a periodic pattern because it repeats at regular intervals of space. This design has translations that will preserve symmetry, but only if the band is moved through certain distances left or right. The distance must be such that each flower lands exactly on top of another flower.

Band Ornaments  33

The rectangle highlighted in the figure is called the unit cell of the band ornament. By translating the unit cell over and over by an amount equal to the width—or any multiple of the width— Figure 4.4 A band ornament that has of the unit cell, we get the whole band only translations. The unit cell is shaded. ornament. This band reminds us of rotation around the sphere. But in this case, there is no fixed point that lies on the band, so we won’t have a dot (•) as we did for a finite design. The signature will be ∞ ∞ to remind us of the two as-if-infinite rotocenters at the north and south poles. A Band with Rotations and Translations

Look at the fourth band in Figure 4.2. The unit cell includes one S-shaped design. We know that each S on its own has rotational symmetry of period 2 in the center of the S, but not reflection symmetry. Looking at two S’s together, Figure 4.5 A band ornament with rotaSS, as if it were a finite design, we see a tions and translations. The unit cell is rotocenter of period 2 between the two shaded, and two rotocenters are marked. S’s that interchanges them. If you trace the band, you can see that there is a rotocenter of the whole infinite band on each S and between each pair of neighboring S’s. The signature for this band ornament must indicate that we have two different rotocenters of period 2 and no mirror lines. To indicate that there are no mirror lines is easy—we don’t have a star (∗). There are two different rotocenters of period 2, so we will need two 2’s, 2 2. We still have the rotocenters at the north and south poles, but any of the rotocenters on an S or between two S’s will interchange the poles, so we regard them as being the same. This means we will have only one infinity symbol ∞. We will adopt a convention that we start with smaller numbers and put the infinity symbol ∞ at the end, so for the band of all S’s, we have the signature 2 2 ∞. Figure 4.1 shows five band ornaments with the same signature, 2 2 ∞. See if you can find the two different types of rotocenters in each band. A Band with Vertical Mirror Lines and Translations

Now let’s see what happens when we repeat a design or motif that has bilateral symmetry, as shown in the second row of Figure 4.2. Each motif has a vertical mirror line, but between two adjacent mo- Figure 4.6 A band ornament with vertitifs, there is also a vertical mirror line, cal mirror lines and translations. The unit cell is shaded, and mirror lines are marked. as shown in Figure 4.6. As before, since the band ornament has translations, the poles are rotocenters with very large period. Following the reasoning in the previous chapter, we put (∗)

34  Geometry for the Artist

in front of a rotocenter that lies on a mirror line, and the signature of this band is ∗ ∞ ∞. A Band with a Horizontal Mirror Line and Translations

The third band in Figure 4.2 has a motif with bilateral symmetry repeated over and over. This is different from the second band because the mirror line of the motif is horizontal, whereas in the second band it is vertical. The motif on its own has one horizontal mirror line that matches perfectly with the mirror line of the band on its Figure 4.7 A band ornament with hor- own. Since there is a mirror line, the sigizontal mirror line and translations. The nature must have a star (∗). The rotounit cell is shaded, and the mirror line is centers at the north and south poles of the sphere are interchanged by this remarked. flection, so there is just one infinity sign and it must be written before the star (∗) since the rotocenter is not on the mirror line. The signature is thus ∞ ∗. A Band with Both Horizontal and Vertical Mirror Lines along with Translations

When we repeat a design like a cross that has both vertical and horizontal mirror lines, we expect the band to also have vertical and horizontal mirror lines, as Figure 4.8 A band ornament with a hor- shown in Figure 4.8. For such a band, izontal mirror line, vertical mirror lines, there are two vertical mirror lines: one and translations. The unit cell is shaded, on the cross and one between two neighand mirror lines are marked. boring crosses. This means the signature must have a star (∗). The horizontal mirror line interchanges the north and south poles, so we will have just one infinity symbol (∞). The north and south poles land on the vertical mirror lines, so the infinity symbol goes after the star. The cross also has rotational symmetry, and we see that the band also has rotational symmetry, with two different rotocenters: one in the center of each cross and one between every pair of neighboring crosses. These rotocenters are on mirror lines, so they go after the star (∗) as well. In keeping with our convention that numbers are listed in increasing order, the signature of this design is ∗ 2 2 ∞. A Band with Glide Reflection plus Translations—A Miracle

Footprints in snow or in sand form a band ornament, as shown in Figure 4.9. A unit cell, containing both a left and right footprint, is shaded. This band has a very special type of symmetry. Translations take one unit cell, which contains two footprints, to another unit cell. This band doesn’t have any reflections or rotations, as you can see by tracing. But there is symmetry, even though there are no reflections of the band as

Band Ornaments  35

a whole; after all, a left footprint is the mirror image of a right footprint. We capture that special relationship with a new symmetry, the glide reflection. If we reflect the band across a horizontal mirror line, it doesn’t match, but it does look the same and will actually Figure 4.9 A band ornament with a glide reflecmatch up perfectly if we also tion and translations. A unit cell is shaded. translate the band, but only if we translate half the length of the unit cell. Neither the horizontal reflection nor the translation by half the unit cell is a symmetry transformation on its own, but taken together they do give a symmetry transformation. This is called a glide reflection. The word “glide” refers to the translation part of the symmetry. The key feature of a glide reflection is that translation by half the unit cell and horizontal reflection are not symmetries of the band on their own, but their combination, in either order, gives a symmetry of the whole band. This symmetry is called a miracle because it is a symmetry that is made up of transformations that are not symmetries on their own; a cross (×) represents the miracle. A glide reflection interchanges the north and south poles of the sphere that the band is on, so the signature has only one infinity symbol (∞). The miracle is special, so we put the cross at the end of the signature and get ∞ × for the footprint band. A Band with Vertical Mirror Lines and Glide Reflection plus Translations

A band consisting of motifs with bilateral symmetry that alternate between upward and downward, as shown in Fig- Figure 4.10 A band ornament with vertical mirror ure 4.10, also has a glide reflec- lines and a glide reflection. The unit cell is shaded, tion. We won’t call this a mir- and mirror lines and rotocenters are marked. acle because there is a vertical mirror line that crosses the glide reflection. This band also has a unit cell that stretches over two squares. There are vertical mirror lines through the center of the motifs, but no horizontal mirror line. There are two rotocenters as shown; neither one of them is on a mirror line. This band has mirror lines, so we have a star (∗) in the signature. We see two rotocenters of period 2, but they are interchanged by the vertical mirror lines, so we treat them as the same and have just one 2 in the signature. They are not on the mirror lines, so the 2 goes before the star (∗). The rotocenters and the glide reflection both interchange the north and south poles, so there is only one infinity symbol (∞) in the signature. The glide reflection comes along automatically with the other symmetry transformations and crosses a mirror line, so we don’t include it in the signature. This means that the signature is 2 ∗ ∞.

36  Geometry for the Artist

4.4

THE SEVEN TYPES OF SYMMETRIES OF BAND ORNAMENTS

We have now seen all seven mathematical classifications of band ornaments, shown in Table 4.1. Every band ornament you see will always be of one of these seven types— that’s what the mathematics says—but artists do not let these classifications limit their creativity and you will see an unending variety of band ornament designs. Here are the kinds of symmetries a band ornament can have: • Translation, left or right of a distance equal to the width of the unit cell (remember that the strip is infinite in both directions) • Reflection across a vertical line through the band • Reflection across the horizontal midline of the band • Rotation of 180◦ around a point on the midline of the band • Glide reflection, combining a reflection across the midline of the band and a translation of half the width of the unit cell Let’s summarize the important properties of band ornaments. Then, we give tips on finding the signature of a band ornament and show how to create your own band ornament. Properties of Signatures of Band Ornaments

1. There is either one or two infinity symbols (∞) in the signature of a band ornament. A rotation (gyration or kaleidoscope) or a horizontal mirror line will interchange the north and south poles of the infinite sphere, so there is one infinity symbol (∞). Otherwise, there are two infinity symbols (∞ ∞). 2. Gyrations are given first. A gyration with period 2 is indicated by the number 2; if there are two different gyration points, put two 2’s. 3. If there is a mirror line, there is a star (∗) in the signature. There is a star (∗) before the infinity symbol (∞) if the band ornament has vertical mirror lines and a star (∗) after the infinity symbol (∞) if the band ornament has a horizontal mirror line but no vertical mirror lines. 4. If there are two different kaleidoscopic points, there are two 2’s after the star. 5. A miracle is given by a cross (×) at the end of the signature. Because a band ornament with a miracle will never have a mirror line, there will never be a cross and star together in the signature of a band ornament. Note: we will see a cross and star together when we look at tilings in Chapter 6. 6. A glide reflection is trivial if the translation by itself and the reflection by itself are both symmetries of the design.

Band Ornaments  37

TABLE 4.1 Signature

∞∞

∞∗

∗∞∞ 22∞ ∗22 ∞

Signatures for all seven band ornaments. Mirrors

Rotations

Glide Reflections

None

None

None

Horizontal

None

Trivial

Vertical

None

None

None

Gyration

None

Vertical and

Kaleidoscope

Trivial

Vertical

Gyration

Non-trivial

None

None

Miracle

Example

horizontal 2∗∞ ∞×

7. The special feature of the glide reflections in the checkerboard and the footprints (Figures 4.9 and 4.10) is that neither the translation by itself nor the reflection by itself is a symmetry. This is why we say the glide reflection is not trivial.

Tips for Finding the Symmetries and Signature of a Band Ornament

Here are tips that can help you find the symmetries of a band ornament: • Determine the unit cell; it is the building block of the whole band and helps you understand the symmetry. • Determine the symmetries that the unit cell has on its own. This will help you find mirror lines and rotocenters of the band as a whole. • Is there a vertical mirror line? If there is one vertical mirror line, there will be a vertical mirror line of a second type, different from the first and exactly halfway between two adjacent vertical mirror lines of the first type. • Is there a horizontal mirror line? If there is, it must be the midline of the band. • Is there a non-trivial glide reflection? If it is a miracle, it will remind you of footprints. If there are gyrations and vertical mirror lines crossing the glide reflection, it is not a miracle.

38  Geometry for the Artist

4.5

MAKING A BAND ORNAMENT OF YOUR OWN

Here are steps to help you create a band ornament of your own. Using grid paper is very helpful in creating your own band ornament. • Choose one of the seven types. • Choose the width and height of the unit cell. The height of the unit cell will determine the height of the band. • Follow these instructions for the type of design you have chosen. ∞ ∞ Create or choose a design that is asymmetrical (with signature 1 •) to put into the unit cell. Repeat the unit cell along the band. ∞ ∗ Create or choose a design with bilateral symmetry (∗ 1 •) and place the design in the unit cell so that the mirror line is horizontal. Repeat the unit cell along the band. ∗ ∞ ∞ Create or choose a design with bilateral symmetry (∗ 1 •) and place the design in the unit cell so that the mirror line is vertical. Repeat the unit cell along the band. 2 2 ∞ Create or choose a design with point symmetry (2 •). Place the design with any orientation. Repeat the unit cell along the band. ∗ 2 2 ∞ Create or choose a design that has two mirror lines, with symmetry (∗ 2 •). Place the design so that one mirror line is vertical and one mirror line is horizontal. Repeat the unit cell along the band. 2∗∞ Create or choose a design that has bilateral symmetry (signature ∗ 1 •). Put it into one half of the unit cell. Reflect it across a horizontal mirror line and put it into the other half of the unit cell. Repeat this unit cell at equally spaced intervals. ∞ × Create or choose a design that is asymmetrical (with signature 1 •). Divide the unit cell into four quarters. Put the design into the upper left quarter of the unit cell. Put its reflection across a horizontal mirror line into the other lower right quarter of the unit cell. Repeat the unit cell at equally spaced intervals.

EXERCISES Understand

4.1 Classify. Classify each of the seven band ornaments in Figure 4.11 by finding its symmetries and giving its signature. 4.2 Bands in a Painting. Find as many band ornaments as you can in Figure 21.5, Greetings from a Manhattan Artist, by Ida Abelman. Copy and classify them. 4.3 Find and Classify. Find and copy some band ornaments in your environment. Classify each by identifying its symmetries and giving its signature. See if you can find one of each type.

Band Ornaments  39

Figure 4.11

Seven band ornaments to classify.

Apply

4.4 2by2. Are there any band ornaments in the two paintings you chose by two artists? If so, copy and classify them. 4.5 3by1. Are there any band ornaments in the three paintings you chose by one artist? If so, copy and classify them. Create

4.6 Create Your Own from One Motif. Choose or create a motif and then use it to make several band ornaments. Give the signature for each of your bands. 4.7 Create Your Own Band Ornaments. Create several of your own band ornaments and give the signature for each. 4.8 Create Your Own Band Ornaments of Type 2 2 ∞. Using the band ornaments in Figure 4.1 as models, create several band ornaments of signature 2 2 ∞. Make them look different from the ones shown.

40  Geometry for the Artist

4.9 Navajo Rug Bands. Copy one or two of the bands in the Navajo rug on page 9 and classify them. Create your own band ornament based on the bands in the rug. 4.10 Persian Rug Bands. Copy one or two of the bands in the Persian rug on page 9 and classify them. Create your own band ornament based on the bands in the rug. Review

4.11 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 4.12 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

READINGS • Symmetries of Culture: Theory and Practice of Plane Pattern Analysis by Dorothy Washburn and Donald W. Crowe • Designs & Patterns from North African Carpets & Textiles by Jacques Revault

ART • Indian sari borders • Rug borders

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Medieval mosaic tilings with symmetries based on triangles, squares, and hexagons. From L’Ornement Polychrome by Auguste Racinet. Figure 5.1

CHAPTER

5

The Regular Tilings

EXPLORATIONS • Study the different tiling patterns in Figure 5.1. Do you see how any of the tilings could be extended indefinitely to cover a plane? • What symmetries do the triangles, squares, quadrilaterals, and hexagons have? Do you think the symmetries of one of the shapes could be extended to the whole tiling containing it?

5.1

OVERVIEW

We have seen that the regular polygons—polygons whose sides have the same length and whose angles have the same measure—are important for studying finite designs. They are also important for studying tilings—symmetric patterns that cover the whole plane, also called tessellations or wallpaper patterns. In this chapter, we will study the three tilings shown in Figure 5.2. Each one uses one type of regular polygon—equilateral triangle, square, or regular hexagon— arranged so that edges of polygons match up exactly and each vertex has the same number of regular polygons coming together. Thus, there are four squares coming together at each vertex of the left tiling, six equilateral triangles coming together at each vertex of the center tiling, and three regular hexagons coming together in the right tiling. These are the three regular tilings.

Figure 5.2

Three regular tilings.

There are seven different symmetry classifications for band ornaments, but there are seventeen classifications for tilings—even though they have the same four symmetries as band ornaments: rotations, reflections, translations, and glide reflections. DOI: 10.1201/9781003110972-5

43

44  Geometry for the Artist

In this chapter, you will learn the symmetries and the signatures of the three regular tilings and how to identify symmetries of other tilings with the same signatures. In the next chapter, we study the other tilings. We will see that the symmetries of the tiles in a tiling help us find symmetries of the tiling itself, just as knowing the symmetries of the motif used in a band ornament helped us understand the symmetries of the band ornament.

5.2

TRANSLATIONS OF A TILING

Like a band ornament, a tiling has translational symmetry; but for a tiling, the translations are in many different directions: up and down, left and right, and diagonally. The symmetries of a tile help us determine the symmetries of the whole tiling. Some of the symmetries of a tile will extend to the whole tiling; others may not extend to the whole tiling, just as the symmetries of a motif of a band ornament did not necessarily extend to the band as a whole. Other new symmetries may emerge from the arrangement of the tiles.

5.3

A TILING BY SQUARES

The left-hand tiling shown in Figure 5.2 consists of just squares. The tiling has translations—up, down, and diagonally. A square by itself has four mirror lines and four rotations and has signature ∗ 4 •. Figure 5.3 shows the mirror lines of a square and their extensions in the square tiling. It is easy to see that these extensions are all mirror lines of the square tiling; you can trace the tiling and mirror lines to conFigure 5.3 A square firm this. So, in this case, all of the mirror lines of a square tiling with mirror extend to the regular tiling by squares. If you rotate the tiling lines and rotocenters. around the center point of the square, you will see that these rotations are also symmetries of the square tiling. But there are more symmetries of the square tiling than just the extensions of symmetries of a tile. Look carefully, and you will see additional mirror lines between squares; these mirror lines are shown in red in Figure 5.3. The center of a square tile is a rotocenter of the square, and it is also a rotocenter of the square tiling, with period 4. Now, check to see if the intersections of mirror lines of the tiling are rotocenters, as they were for band ornaments. They do appear to be rotocenters, which you can confirm by tracing. This gives another rotocenter of period 4 at the corner where four mirror lines intersect and four squares come together. In addition, there is yet another rotocenter, this time with period 2, in the middle of an edge where two mirror lines intersect and two squares meet. This tiling, like every tiling, has translations in many directions—the easiest ones to see are vertical and horizontal, but there are also diagonal translations. In addition, there are glide reflections, but these are not so easy to see. A glide reflection on the square tiling is shown in Figure 5.4, where you can see the familiar “footprint”

The Regular Tilings  45

pattern. This glide reflection is not a miracle since it intersects mirror lines, so it won’t contribute a cross × to the signature of the tiling. The signature for this tiling is ∗ 4 4 2. The star (∗) indicates that there are mirror lines and the 4 2 2 after the star tells us that there are three different types of kaleidoscopic rotocenters: two with period 4 and one with period 2. These rotocenters are of different types because there is no symmetry that takes a rotocenter of one type Figure 5.4 A square to a rotocenter of the other type. We don’t put a dot (•) tiling with glide reflecat the end of the signature because there is no point that tion. is fixed by every symmetry of the tiling, and we don’t put an infinity symbol (∞) as we did for band ornaments since there are no special points like the north and south poles of the sphere. Look at the tilings in Figure 6.1. Eight of them have signature ∗ 4 2 2. Can you find them? Choose one and see if you can find the mirror lines and the rotocenters.

5.4

A TILING BY EQUILATERAL TRIANGLES

If we use just equilateral triangles, we get the middle tiling of Figure 5.2. Again, looking at the symmetries of the triangle helps us find the symmetries of the triangle tiling. The equilateral triangle has three mirror lines and a period 3 rotocenter, with signature ∗ 3 •, shown in Figure 5.5.

A triangle with mirror lines and rotocenter. Figure 5.5

A triangle tiling with mirror lines and rotocenters. Figure 5.6

A triangle tiling with glide reflection. Figure 5.7

Extending the mirror lines of the triangle over the whole tiling, we see that they become mirror lines of the tiling. Drawing in many mirror lines, we see that there are rotocenters of period 6 where six triangles come together and of period 3 in the center of each triangle. As with the square tiling, you can find new mirror lines along the edges where two triangles meet. The midpoint of such an edge is a rotocenter with period 2. This tiling also has glide reflections in three different directions, passing through the centers of the triangles. One glide reflection is shown in green in Figure 5.7. This glide reflection crosses mirror lines, so it is not a miracle and we won’t put a cross (×) in the signature. The signature is ∗ 6 3 2, indicating that there are mirror lines and kaleidoscopic rotocenters with periods 6, 3, and 2.

46  Geometry for the Artist

5.5

A TILING BY REGULAR HEXAGONS

The tiling at right in Figure 5.2 is made up of regular hexagons. A regular hexagon by itself has six mirror lines and signature ∗ 6 •, as shown in Figure 5.8.

A hexagon with mirror lines and rotocenter. Figure 5.8

A hexagon tiling with its mirror lines and rotocenters. Figure 5.9

A regular hexagon tiling with glide reflection. Figure 5.10

Again, we extend the mirror lines of the hexagon over the whole tiling, and we see that these become mirror lines of the whole tiling. Where they intersect, there are rotocenters of period 6 in the center of the hexagons and period 3 at corners where three hexagons come together. You can see that there are also mirror lines between hexagons, giving rotocenters of period 2 at the midpoint of each edge of the hexagon. There are also glide reflections, which are not miracles since they intersect mirror lines. The signature of this tiling is ∗ 6 3 2, the same as for the tiling by equilateral triangles.

5.6

WHAT WE HAVE LEARNED ABOUT REGULAR TILINGS

In the next chapter, we will look at fifteen more tilings, each with different signatures. Even though we have looked at only three tilings so far in this chapter and seen only two different symmetry types, we have learned quite a bit about how to analyze tilings. Let’s summarize what we know, as this will be very useful when looking at these other tilings. • Individual tiles in the tiling can give important information about the whole tiling. • Mirror lines of individual tiles may extend to the whole tiling pattern. • Mirror lines may go along edges where two tiles meet. • Rotocenters of individual tiles making up the tiling pattern may be rotocenters of the tiling. • There will be rotocenters where mirror lines intersect. • Rotocenters can be in the center of a tile, at the midpoint of an edge halfway between two corners, or at a corner where tiles meet. • A rotocenter on the middle of an edge will have period 2.

The Regular Tilings  47

• Glide reflections will always display some sort of “footprint” pattern. These are harder to find than mirror lines or rotocenters.

5.7

NEW TILINGS FROM OLD

We have seen three regular tilings: the equilateral tiling, the square tiling, and the regular hexagon tiling. We can use these tilings to create new tilings. One way is to attach tiles to each other, and another way is to break apart tiles. Figures 5.11, 5.12, 5.13, and 5.14 show examples of new tilings formed by attaching tiles of a tiling. The symmetries of the new tilings may be different from symmetries of the tilings you started with.

Figure 5.11

A tiling formed by adjoining adjacent squares.

Figure 5.12

A tiling formed by adjoining adjacent triangles.

Figure 5.14

Figure 5.13

Another tiling formed by adjoining adjacent squares.

A tiling formed by adjoining adjacent hexagons.

EXERCISES Understand

5.1 Triangles and Hexagons. Show how to break up a regular tiling by hexagons to get a tiling by triangles. Show how to erase edges on a tiling by triangles to get a tiling by hexagons. 5.2 Symmetries of New Tilings. See how many symmetries you can find for the new tilings in Figures 5.11, 5.12, 5.13, and 5.14. Which of them were tilings of the underlying regular tiling?

48  Geometry for the Artist

5.3 New Tilings from Triangles. Find a way to erase edges on a regular tiling by triangles to get a new tiling. Can you find some symmetries of the new tiling? 5.4 New Tilings from Squares. Erase edges on a regular tiling by squares to get a new tiling. For example, you could make new tiles that are three squares long or tiles that form a corner made of three squares. You could also make tiles of different sizes; perhaps some tiles are made of two squares, while some tiles are made of one square. Can you find some symmetries of your new tiling? 5.5 Triangle Tiling. Cut a triangle out of cardboard. The triangle you choose does not need to be equilateral or isosceles; in fact, this exercise works for any triangle and is more fun using a triangle with sides of different lengths. Trace it repeatedly on a large sheet of paper, with edges matching, so that you get a tiling. You must be careful, as not every way of arranging the triangles gives a new tiling. Can you find some symmetries of this tiling? Apply

5.6 Find Your Own. Find several tilings in your environment, for example, floor tilings or textile prints. Photograph or copy them. Are any of them based on the regular tilings that we saw in this chapter? Can you find any symmetries that these tilings have? Create

5.7 Create Your Own. Choose a tiling pattern that you like. Adapt it using your own ideas. Did you change the symmetries of the original pattern? Review

5.8 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 5.9 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

FURTHER READINGS • Introduction to Tessellations by Dale Seymour • The Grammar of Ornament by Owen Jones

ARTISTS • William Morris (1834–1896); wallpaper, tapestry, and fabrics • Lyubov Popova (1889–1924); textile designs

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Figure 6.1

Jones.

Tilings from ancient Egypt. From The Grammar of Ornament by Owen

CHAPTER

6

Tilings

EXPLORATIONS • Can you find any tilings in Figure 6.1 that have the same symmetry as the tiling by squares with signature ∗ 4 4 2? • Which tilings have symmetry different from the tiling by squares? Can you find some of their symmetries?

6.1

OVERVIEW

In the previous chapter, we looked at the three regular tilings, with signatures ∗ 4 4 2 and ∗ 6 3 2. Mathematicians have been able to show that the symmetry transformations we have already seen—translations; reflection; rotations of period 2, 3, 4, and 6; and glide reflections—are the only symmetries that a tiling can have. These symmetries interact with one another, as we saw with designs and band ornaments. This means, for example, that wherever we have intersecting mirror lines, we must have a rotocenter. These interactions limit the kinds of tilings we can have to only seventeen types of tilings. We have already seen two different types, so we need to find fifteen more tilings to get all seventeen symmetry types of tilings. This is challenging, but we start with the three regular tilings that we know already and build up new tilings from them. The regular tilings have many symmetries—mirrors, kaleidoscopes, and glide reflections. As we build up new tilings, we will lose some of these symmetries while creating new ones. The new tilings can be put in categories according to their symmetries in different ways. We first look at those with no rotations, then those with gyrations only, and finally those with kaleidoscopic rotations. Later, we will see categories based on the periods of the rotations. At the end of the chapter, we summarize with observations about properties of tilings and tips for finding the symmetries and signatures of new tilings that you may encounter. You will learn the following: • How to find the symmetries of a tiling • How to find the signature of a tiling • How to create a tiling of your own DOI: 10.1201/9781003110972-6

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52  Geometry for the Artist

6.2

TILINGS WITHOUT ROTATIONS—FOUR TILINGS

To get a tiling without rotations, we put a design that doesn’t have a rotation (either an asymmetric design with signature 1 • or a bilateral design with signature ∗ 1 •) in each tile. We get four different tilings without rotations in this way. A Tiling with Translations Only—A Wonder—Signature ◦

If we put an asymmetric design on each tile of the regular square tiling, as shown in Figure 6.2, we get a tiling with only translations. Getting a symmetric pattern from a design that doesn’t have any symmetry is a wonder, and its signature is just a circle (◦). This symmetry type is very common on printed textiles and wallpapers Figure 6.2 A tiling with only because it gives the most freedom for the artist. translations and signature ◦. However, wonders don’t necessarily appear “symmetric” to the eye, and the artist must be careful to make the edges of tiles connect smoothly with one another. A Tiling with Two Mirror Lines—Signature ∗ ∗

If we use a bilateral design instead of an asymmetric design on each tile of the regular square tiling, we get a tile with two different parallel mirror lines, shown in Figure 6.3. One type of mirror line is inherited from the bilateral design, and the other type goes between bilateral designs, just as we saw with band ornaments. The signature of this tiling is ∗ ∗ to indicate the two different types of mirror lines.

Figure 6.3

A tiling with two parallel mirror lines and signature ∗ ∗.

A Tiling with a Mirror Line and a Miracle—Signature ∗ ×

If we stagger the bilateral tiles of Figure 6.3, we get the tiling shown in Figure 6.4. We still have a vertical mirror line through the individual designs, but we now have a glide reflection between columns of designs. Since this glide reflection does not intersect any mirror lines, it is a miracle. The signature is ∗ ×.

Tilings  53

The tiling with a mirror line and a glide reflection that does not cross any mirror lines—a miracle. The signature is ∗ ×. Figure 6.4

A Tiling with Two Miracles—Signature × ×

If we put left and right footprints into the tiles of the regular square tiling, we get a miracle. And if we stagger two paths of footprints, we get a tiling with two miracles, as shown in Figure 6.5. The signature × × indicates that there are two different types of miracles, one between the inner edges of the feet and one between the outer edges of the feet.

Figure 6.5

6.3

A tiling with two miracles and signature × ×.

TILINGS WITH GYRATIONS ONLY—SIX TILINGS

In Chapter 3, we saw that a simple pinwheel composed of four triangles has one gyration point of period 4 and no mirror lines. To get a tiling with only gyrations and no kaleidoscopes, we can use pinwheels—some with four triangles as we saw before and others with two or three triangles. Gyrations of Period 2—Signature 2 2 2 2

To get a tiling with only period 2 gyrations, we use a pinwheel with two “blades” inside a rectangle, as shown in Figure 6.6. The pinwheel itself has just one gyration of period 2, but the tiling has three more gyration points, with period 2, one on the corners where tiles meet and two on edges between tiles. Since there are four gyration points of period 2, this tiling has signature 2 2 2 2.

54  Geometry for the Artist

Pinwheel tile with signature 2 •. Figure 6.6

Figure 6.7

Pinwheel tiling with signature

2 2 2 2.

Gyrations of Period 2 with a Mirror and Glide Reflection—Signature 2 2 ∗

Recall that the checkerboard band ornament had a gyration and a mirror line but no kaleidoscopes. Like that, we can make a tiling with gyrations and mirror lines but no kaleidoscopes. We use a block made up of four squares, each containing two triangles, shown in Figure 6.8. Copies of these four tiles make the chevron pattern in Figure 6.9. There are two vertical mirror lines as shown, so there is a star (*) in the signature. There are two different gyrations of period 2, but they are not on the mirror lines. Thus, there are two 2’s in the signature before the star (∗). This tiling pattern has a glide reflection, but since it crosses the mirror lines, it is not a miracle and doesn’t appear in the signature. Putting this all together, we see that the signature is 2 2 ∗.

Four square tiles, each with signature ∗ 2. Figure 6.8

Chevron tiling with signature 2 2 ∗. Mirror lines are in blue, and the glide reflection is in red. Figure 6.9

Gyrations of Period 2 with a Miracle—Signature 2 2 ×

To get a tiling with gyrations of period 2 and a miracle, we use the square pinwheel, shown in Figure 6.10. For the miracle, we reflect the triangles but then translate them just enough to create “footprints,” as shown in Figure 6.11. There are two different gyrations of period 2 along with the miracle glide reflection, so the signature is 2 2 ×.

Gyrations of Period 4 and 2—Signature 4 4 2

In order to get gyrations of period 4 without any mirror lines, we’ll use a different kind of “pinwheel” formed by adding four rectangles to the edges of a square. This shape, shown in Figure 6.12, has a gyration of period 4 in the center. Put one of these “pinwheels” into every square of a regular square tiling to get the tiling in

Tilings  55

6.10 Pinwheel with signature 4 •. Figure

Figure 6.11

Pinwheel tiling with signa-

ture 2 2 ×.

Figure 6.13. There is a gyration of period 4 in the center of the pinwheels and at the corners where four pinwheels meet. There is a gyration of period 2 at the midpoint of each edge where two pinwheels meet. Thus, the signature is 4 4 2.

A square “pinwheel” with signature 4 •. Figure 6.12

Figure 6.13

Brick tiling with signature

4 4 2.

Gyrations of Period 3—Signature 3 3 3

To get gyrations of period 3, we put a pinwheel of three triangles inside each hexagon of the regular hexagon tiling, as shown in Figure 6.14. There are two different types of corner rotocenters, each with period 3. The pinwheels break the mirror symmetries of the regular hexagon tiling and no new mirror symmetries are created, so the signature of this tiling is 3 3 3.

Figure 6.14

Pinwheel tiling with signature 3 3 3.

56  Geometry for the Artist

Gyrations of Period 6, 3, and 2—Signature 6 3 2

To get gyrations of period 6, we put a pinwheel of six triangles inside each hexagon of the regular hexagon tiling. This new tiling has no mirror lines, but we do have a period 3 rotation and a period 2 rotation as we did with the regular hexagon tiling. These rotations are now gyrations instead of kaleidoscopes. This means the signature is 6 3 2.

Figure 6.15

6.4

Hexagon and triangle tiling with signature 6 3 2.

TILINGS WITH KALEIDOSCOPES—FIVE NEW TILINGS PLUS TWO REGULAR TILINGS

We used designs and pinwheels to break the symmetry of mirror lines and get tilings without kaleidoscopes. To get new tilings with kaleidoscopes, we need to break the symmetry of a regular tiling in different ways. In the previous chapter, we saw a way that works very well—adjoin adjacent tiles to create a new tile that has less symmetry. So, for example, a square has a period 4 kaleidoscope in the center, but two squares joined give a rectangle, a kaleidoscope of period 2 in the center. We will use that technique to get more tilings. We also break the symmetry of the regular hexagon tiling by adding a triangle to the inside of each tile. Kaleidoscopes of Period 2—Signature 2 ∗ 2 2

If we offset rectangles formed by adjoining two squares as we stack them, we get the tiling shown in Figure 6.16. The two mirror lines of the rectangle extend to mirror lines of the whole pattern. There are vertical mirror lines between two adjacent rectangles, but there are no horizontal mirror lines between adjacent rectangles. Since there are mirror lines, we have a star (∗) in the signature. These mirror lines intersect in two different rotocenters, one in the middle of a brick and one on a vertical edge between two bricks. They contribute to two 2’s after the star (∗) in the signature. Are there any rotocenters that are not on mirror lines? We must look at the corners and the midpoints of the edges. It is easy to check that the corners aren’t rotocenters. But we can find a gyration of period 2, halfway between two vertices on the midpoint of a short horizontal edge. This contributes a 2 before the star in the signature. There are two non-trivial glide reflections, one horizontal and one vertical, shown in orange. Both of these cross mirror lines, so they are not miracles. The signature of this tiling is 2 ∗ 2 2.

Tilings  57

Figure 6.16

Offset brick tiling with signature 2 ∗ 2 2.

Kaleidoscopes of Period 2—Signature ∗ 2 2 2 2

For this tiling, we use the same rectangles, with two mirror lines, one horizontal and one vertical, and a rotocenter of period 2 in the center. We stack the rectangles differently as shown in Figure 6.17, getting a tiling with mirror lines as shown—two of them inherited from the rectangle and two (one horizontal and one vertical) where rectangles are adjacent. There are four different rotocenters, each at the intersection of two mirror lines. The signature is ∗ 2 2 2 2, to indicate that there are four rotocenters of period 2, all of which are on mirror lines.

Figure 6.17

Stacked brick tiling with signature ∗ 2 2 2 2.

Kaleidoscopes of Period 4 and 2—Signature 4 ∗ 2

If we take a block of two bricks and then change their orientation, alternating them vertically and horizontally, we get the tiling shown in Figure 6.18. There are mirror lines as shown, with a kaleidoscope where two mirror lines intersect, contributing 2 after the star (∗). Where four bricks meet at a corner, there is a rotocenter of period 4; this rotocenter is not on any mirror lines, so it is a gyration. We put the numeral 4 in the signature before the star (∗). There are vertical and horizontal glide reflections, as shown in orange, but they intersect mirror lines and so do not contribute to the signature, which is 4 ∗ 2.

Figure 6.18

Brick tiling with signature 4 ∗ 2.

58  Geometry for the Artist

Kaleidoscopes of Period 3—Signature ∗ 3 3 3

To get a tiling with highest period equal to 3, we create a compound tiling, putting a triangle inside each hexagon of the regular hexagon tiling, shown in Figure 6.19. This compound tiling has three kaleidoscopes of period 3. The glide reflection of the regular hexagon tiling is still a glide reflection of this tiling; it is not a miracle because it crosses mirror lines. Thus, the signature of this tiling is ∗ 3 3 3.

Figure 6.19

Hexagon and triangle tiling with signature ∗ 3 3 3.

Kaleidoscopes of Period 3—Signature 3 ∗ 3

There is another way to put triangles into the hexagons of the regular hexagon tiling, shown in Figure 6.20. Instead of having the vertices of the triangles at the vertices of the hexagon, now the vertices of the triangles are at the midpoints of the sides of the hexagons. This gives us a different tiling. The center of each triangle is still a kaleidoscope of period 3, but now each corner where three triangles meet is a gyration, not a kaleidoscope. Again, this tiling inherits glide reflections from the regular hexagon tiling, but they cross mirror lines, so don’t contribute to the signature, which is 3 ∗ 3.

Hexagon and triangle tiling with signature 3 ∗ 3. Mirror lines and rotocenters are shown in red, and a glide reflection is shown in blue.

Figure 6.20

6.5

TIPS FOR DETERMINING THE SIGNATURE OF A TILING PATTERN

You can use the following steps, in any order, to help you determine the signature of a tiling pattern. • Find rotocenters. A tiling can have rotocenters only with periods 2, 3, 4, or 6. However, if you find a rotocenter of period 4, you must find another rotocenter of period 2, and if you find a rotocenter of period 6, you must find another rotocenter with period 3 and one with period 2. • Find gyration points. Remember that the tiling around a gyration point has a definite clockwise or counterclockwise direction and gyration points never lie

Tilings  59

on mirror lines. Two gyration points are different if neither is the image of the other under a translation, rotation, or reflection. Determine the period of each gyration. List the gyration periods in order of size, with the largest first. • Find mirror lines and kaleidoscopic points. If there is a mirror line, you know that the signature will have a star (∗). Kaleidoscopic points are always located where mirror lines intersect; the number of intersecting mirror lines is the same as the period of the kaleidoscope. Two kaleidoscopic points are different if neither is the image of the other under a translation, rotation, or reflection. Determine the period of each kaleidoscope. Then list the kaleidoscopic periods after the star (∗) in order of size, with the largest first. • Look to see if the mirror image of the tiling is the same as the tiling. If so, there will be reflections or glide reflections and you will have to find them. • If you can find a path from one point to its mirror image that never crosses or touches a mirror line, the tiling has a miracle. Put a cross (×) for each different miracle. • If you find translations but don’t find any rotations, mirrors, or miracles, you have a wonder. The signature is just a circle (◦). • Trace the design on tracing paper or a transparent sheet. Move it around and turn the tracing over to see when it matches the original design. Use the tracing to test whether a suspected symmetry transformation works. If you turn the tracing over and it looks different from the original tiling, there are no mirror lines. • Determine whether regions of the tiling appear to have a direction, either clockwise or counterclockwise. Look to see if these regions also appear in the tiling in the reverse direction. If there are tiles, regions, or parts that have a direction and do not appear with the opposite direction, there are no mirror lines. If such patterns appear with the opposite direction, see if you can find a mirror line between the pattern and its reflection. • Look at an individual tile of the tiling. What symmetry does that tile have? Do the symmetries of the tile extend to symmetries of the whole pattern? • You can find a rotocenter only in the middle of a tile, at a vertex where tiles meet, or on an edge midway between two vertices, or where mirror lines intersect. • You can find a mirror line only through the center of a tile, through a vertex where tiles meet, or along the edge between two tiles. • Glide reflections are harder to find. Look for a pattern that looks like “footprints,” a checkerboard pattern, a chevron, or a zigzag. The reflection line will be centered between the two lines of footprints, and the glide or translation will move the design halfway from one “footprint” to the next matching one.

60  Geometry for the Artist

6.6

SUMMARY OF TILINGS

There are exactly seventeen different symmetry classifications for symmetric tiling patterns. The symmetry transformations that a tiling can have are translation; reflection; rotations of period 2, 3, 4, or 6; and glide reflection. Translations leave the whole pattern unchanged; rotations and reflections are determined by the point or line that they leave unchanged. All seventeen tilings and their signatures. Rotocenters are in red, mirrors are in blue, and glide reflections are in orange. TABLE 6.1

◦

∗∗

∗×

××

2222

2 ∗ 22

22×

22∗

∗2222

442

4∗2

∗442

333

3∗3

∗333

632

∗632

The two complementary tables in this chapter will help you when finding symmetries and signatures or when creating a tiling. Table 6.1 gives an example of each

Tilings  61

Signatures for all seventeen tiling patterns, organized according to the rotation of largest period. TABLE 6.2

Signature

Gyrations

Mirrors

Kaleidoscopes Glide Reflections

None None None None

None Yes Yes None

None None None None

None Trivial Miracle Two miracles

2, 2, 2, 2 2, 2 2, 2 2 None

None Yes None Yes Yes

None None None 2, 2 2, 2, 2, 2

None Non-trivial Miracle Non-trivial Trivial

442 4∗2 ∗ 4 42

4, 4, 2 4 None

None Yes Yes

None 2 4, 4, 2

None Non-trivial Non-trivial

333 3∗3 ∗333

3, 3, 3 3 None

None Yes Yes

None 3 3, 3, 3

None Non-trivial Non-trivial

632 ∗632

6, 3, 2 None

None Yes

None 6, 3, 2

None Non-trivial

◦ ∗∗ ∗× ×× 2222 22∗ 22× 2 ∗ 22 ∗2222

tiling along with the rotocenters, mirror lines, and glide reflections marked; it is easy to determine translations. Table 6.2 gives the signatures for each of the seventeen tilings along with the symmetries of each tiling—mirror lines, periods of gyrations, periods of kaleidoscopes, and types of glide reflections—that determine the signature. This table can help you determine the signature of a tiling if you have found all the symmetries or help you if you have found some symmetries and want to know if there are any more that you have not found.

EXERCISES Understand

6.1 Signatures. For each of the following properties that a signature might have, say whether it could belong to a finite design, a band ornament, a tiling, or to more than one type of pattern or that it does not belong to any possible pattern. The whole signature is not given here—there may be other parts to the signature besides what is given here.

62  Geometry for the Artist

a. There is a dot (•). b. There are two crosses (× ×). c. There are a 6 and a 2. d. There are a 6 and a 4. e. There are two stars (∗ ∗).

f. There are neither a dot (•) nor an infinity symbol (∞). g. There are a dot (•) and an infinity symbol (∞). h. There are two infinity symbols (∞ ∞).

6.2 Tiling Signatures and Symmetries. Find the symmetries and signatures of the tilings in Figure 6.21.

Figure 6.21

Eight tilings.

6.3 Symmetries of New Tilings. Find the symmetries of the new tilings in Figures 5.11, 5.12, 5.13, and 5.14 in the previous chapter. Classify them by giving their signatures. Are the signatures the same as or different from the underlying regular tiling? Apply

6.4 The Tiling with Four Gyrations of Period 2. Look at the tiling with signature 2 2 2 2 shown in Figure 6.6. Make a similar tiling with pinwheels having two

Tilings  63

triangles inside a square instead of a rectangle. This tiling does not have signature 2 2 2 2. Why does it not have signature 2 2 2 2? What is its signature? 6.5 Find Your Own. Find one or more tilings in your environment, for example, a floor or textile pattern. Copy and classify. Create

6.6 Create Your Own. Choose a tiling pattern that you like. Adapt it however you choose. Classify your new tiling—it may or may not have the same signature as the tiling you started with. 6.7 Archimedean Tilings. A regular polygon has all sides equal in length and all angles equal in measure. Create a tiling using two or more types of regular polygon, making sure that each vertex has the same arrangement of polygons. These are the semiregular or Archimedean tilings. How many do you get? Classify each by identifying its symmetries and give the signature. 6.8 Tilings with Regular Polygons. See if you can create a tiling with two or more types of regular polygons, but now all vertices do not have to be the same. Classify each by identifying its symmetries and give the signature. Review

6.9 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 6.10 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

READINGS AND RESOURCES • Pattern Design by Lewis F. Day; artistic techniques for creating tilings • TesselManiac, www.TesselManiac.com; software for creating new tiling patterns

ART • Chinese lattices • Islamic wall tilings • Kente cloth from Ghana

Figure 7.1

Holland).

c M. C. Escher Heirs/Cordon Art—Baarn— MC Escher. Cycle, 1938 (

The tumbling blocks tiling pattern. Construction lines show how this tiling was made from the regular tiling by equilateral triangles. Figure 7.2

CHAPTER

7

Symmetry in the Work of MC Escher

EXPLORATIONS • What does Escher show in Figure 7.1? • The tumbling blocks tiling in Figure 7.2 looks like stacked cubes—either with the white faces on top or on bottom. Can you see how Escher used the tumbling blocks tiling as a basis for Cycle? Do you think he made effective use of the tiling?

7.1

MC ESCHER’S TILINGS

Maurits Cornelis Escher (1898–1972) was a Dutch graphic artist who incorporated many mathematical ideas into his work. His father was an engineer, but Maurits’s talents and interests were in art. He studied art and graphic design for two years after high school and then made several trips to Italy and Spain. In Spain, Escher was captivated by the mosaic tilings that decorated walls, ceilings, and floors with elaborate mosaic patterns like those shown in Figure 7.3 from the Alhambra in Granada. Escher and his wife meticulously copied many of these patterns into notebooks. After he returned to the Netherlands, Escher began to incorporate the symmetric de- Figure 7.3 Tiled wall in the Alhamsigns from the Alhambra into his work and bra in Granada, Spain. to develop original and innovative tilings. In 1937, he showed some of this work to his brother Berend Escher, a geologist. The mathematical classification of tilings is of great interest to chemists and geologists DOI: 10.1201/9781003110972-7

65

66  Geometry for the Artist

for the study of the regular patterns of crystals, and Berend was aware of the mathematical classifications of tiling patterns. He referred Maurits to the 1922 paper by Hungarian mathematician George Pólya that described the seventeen different classifications of tiling patterns that we studied in Chapter 6. Escher carefully copied the diagrams in Pólya’s paper into a notebook and began using these new patterns in his work. In this chapter, we will learn Escher’s method of creating new tilings from simpler tilings and then we will analyze two examples, Cycle and Day and Night.

7.2

HOW TO CREATE ESCHER-TYPE TILINGS

Escher began with a tiling with only one kind of tile, like the regular tilings and many of the other tilings that we saw in the previous chapter. He developed a method for altering the tile carefully so that the new tile still gives a tiling of the plane. Not every shape will cover the plane without gaps or overlaps—for example, the regular pentagon won’t tile the plane without gaps or overlaps. But, using Escher’s method, you will always get a new tile that will work. Our first example of Escher’s method for changing a tile using the regular square tiling is shown in Figure 7.4. Take the square, cut out a triangle square on left side of the square, and translate it to the right side of the square, as shown. Then cut out a triangle on the bottom of the square and translate it to the top, as shown.

Figure 7.4

Making an Escher-style tile step-by-step.

This systematic method guarantees that the resulting tile at far right of Figure 7.4 will tile the plan without any gaps or overlapping, as you can see in Figure 7.5. This tiling is a wonder with only translation symmetries and signature ◦. Perhaps you can imagine something that has this shape—maybe a person with a sharp nose looking to the right or a person running and carrying an umbrella. If you put in appropriate details and copy the tile over and over, you will have your own Escher-stye tiling.

Figure 7.5

A tiling with alternating colors using tiles from Figure 7.4.

Symmetry in the Work of MC Escher  67

The principle is this: start with the tile of a tiling and then alter it, step-by-step, so that it will still tile the plane. You can do this by tracing on cardboard, cutting, and taping, or with the software app TesselManiac. Here are the steps: 1. Start with a tiling that uses a simple shape. Determine the symmetries of the whole tiling. 2. Choose one edge of a tile and its image by a symmetry transformation of the tiling. (It could be the edge itself if there is a rotocenter at the midpoint of the edge.) 3. Cut out a piece of the tile along that edge. 4. Add that piece to the tile along the image, being careful to match it exactly according to the symmetry that you chose. We’ll look at some more examples. In Figures 7.6 and 7.7, you see what happens when you cut out a triangle from one side of a square and translate it to another side of the square. This also gives a wonder, a tiling with only translations. Again, we alternate the color of the tiles so that they can be more easily distinguished.

Figure 7.6

An altered square.

The tiling by the altered square with alternating colors. Figure 7.7

Figures 7.8 and 7.9 show what happens when you alter edges of the square by translating the triangle as before, but in addition flipping it across a vertical mirror line through the center of the square. If we ignore colors, this tiling has mirror lines and glide reflections as well as translation, with signature ∗ ×.

Figure 7.8

Altered square.

The tiling by the altered square with alternating colors. Figure 7.9

In Figures 7.10 and 7.11, we alter all four edges of a rectangle. The triangle cut from the vertical edge on the right is translated to the vertical edge on the left. The

68  Geometry for the Artist

triangle cut from the top edge is translated to the bottom edge and reflected across a vertical mirror line through the center of the rectangle. A tiling by stacked rectangles has signature 2 2 2 2. Ignoring colors, the new tiling has a different signature, × ×.

Figure 7.10

Altered rectangle.

Figure 7.11

Tiling by altered rectangles.

In the next example, we see how to alter a square by a rotation that changes only one edge, shown in Figure 7.12. Cut out a piece on onside and rotate it 180◦ around the center point of the edge. This altered square will tile the plane, with translations and four period 2 gyration points. Ignoring colors, the tiling has signature 2 2 2 2.

Altered square and tiling with alternating colors. Rotocenters that the uncolored tiling would have are marked. Figure 7.12

Our final example is rotation around a corner of a square tile. A triangle is cut out from one edge of the tile and rotated clockwise by 90◦ , as shown in Figure 7.13. The resulting tiling, with alternating colors, is shown in Figure 7.14. In Figure 7.15, the rotocenters, ignoring colors, are shown. Since there are two rotations of period 4 and one of period 2, the signature is 4 4 2.

. Figure 7.13

square.

Altered

Tiling by altered squares. Figure 7.14

Gyrations that the uncolored tiling would have are marked. Figure 7.15

Symmetry in the Work of MC Escher  69

7.3

DAY AND NIGHT

In the woodcut Day and Night, Figure 7.16, we see a checkerboard tiling of black and white fields becoming a pattern of dark and light birds. The dark birds are flying to the left, contrasting with the light colors of the daytime landscape, while the light birds fly to the right, contrasting with the dark colors of the nighttime landscape. Another feature of the woodcut is the approximate mirror image of the towns and rivers, colored with contrasting light and dark colors. As with Cycle, Escher surprises us with the unexpected connection between very different objects, shown here by the connection of the square fields with the organic form of the birds.

c M. C. Escher Heirs/Cordon Art— MC Escher. Day and Night, 1938 ( Baarn—Holland). Figure 7.16

We can see how Escher might have created the bird tiling by looking at a square tile, Figure 7.17. We first alter two adjacent edges using rotation of a small triangle about the midpoint of the edge, shown second from left. Then, for the two remaining edges, use rotation of a larger triangle about the midpoint of those edges. The construction technique guarantees that the tile will cover the plane with no gaps or overlapping. Color the tiles with alternating colors to make them stand out and add details to create birds. Escher uses topological transformations (see Chapter 20) of the tiles to create the final picture.

Figure 7.17

plane.

Creating a bird that will tile the

Figure 7.18

The tiling by birds.

70  Geometry for the Artist

7.4

CYCLE

In the lithograph Cycle, shown in Figure 7.1, Escher shows a little man merging into a tiling pattern that becomes a stack of cubes. Escher uses the tiling that is known as tumbling blocks to quilters, Figure 7.2, made by combining adjacent triangles of the regular equilateral triangle tiling in Chapter 5. To create a little girl tile like the little man in Escher’s picture, start with the quadrilateral of the tumbling blocks tiling, at far left in Figure 7.19. On the upper right side of the tile, cut out the cap to define the face and the region under the girl’s left arm. To ensure we get a tiling, we must add those on somewhere else, so we rotate them by 120◦ . The cutouts land on the lower right side and give the left foot of the girl. This is shown in the second part of the figure. To define the girl’s waist, take out a piece and rotate it about the adjacent vertex; this piece will become her left shoe. This strange tile in the middle of the figure will still tile the plane, since we have subtracted according to our rules. We now only need the left arm, so we take out another larger piece and rotate it around the same vertex, but in the opposite direction, shown in the fourth part of the figure. After erasing construction lines, we get a shape that looks like a little girl, at the right of Figure 7.19. Our careful step-by-step construction ensures that this shape will tile the plane, as shown in Figure 7.20. We use three colors so that adjacent shapes are colored differently and can be distinguished one from another. The next step is to add details to the tiles, getting the tiling pattern shown in Figure 7.21

Figure 7.19

Creating a tile shaped like a little girl that will tile the plane.

EXERCISES Understand

7.1 Symmetry of Bird Tiling. Classify the bird tiling pattern in Figure 7.18 ignoring the coloring. (If you don’t ignore the coloring, the classification is ◦, a wonder.) 7.2 Symmetry of Little Girl Tiling. Classify the little girl tiling pattern in Figure 7.20 ignoring the coloring. (If you don’t ignore the coloring, the classification is ◦, a wonder.)

Symmetry in the Work of MC Escher  71

Figure 7.20

A tiling of the shapes.

The tiling with features and details to make each tile into a little girl (illustration by Emilio Wherry, @y2cakey). Figure 7.21

7.3 Identifying a Symmetry by Escher. Find a picture by Escher that is based on a tiling pattern. Draw the underlying tiling pattern and determine its symmetry classification. Create

7.4 Create Your Own Escher Tiling. Using the instructions given in this chapter or the software TesselManiac (www.TesselManiac.com.), create your own Escher-type tiling. Add details to the tiles. 7.5 Create Your Own Escher-Style Picture. Incorporate the tiling that you created in Exercise 7.4 into an imaginative picture, as Escher does in Figures 7.1 and 7.16.

72  Geometry for the Artist

Review

7.6 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 7.7 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

READINGS AND RESOURCES • Escher on Escher: Exploring the Infinite by MC Escher • Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher by Doris Schattschneider • The official MC Escher website, http://www.mcescher.com/ • Introduction to Tessellations by Dale Seymour and Jill Britton • TesselManiac, www.TesselManiac.com; software for creating new tiling patterns

III Perspective

René Magritte. La condition humaine, 1933. National Gallery of Art. Gift of the c 2022 C. Herscovici / Artists Rights Society (ARS), New Collectors Committee. York

73

Figure 8.1

John Singer Sargent. In a Medici Garden, 1905.

Figure 8.2

Canaletto. Venice: The Doge’s Palace and the Riva degli Schiavoni,

late 1730s.

CHAPTER

8

Introduction to Linear Perspective

EXPLORATIONS • Where are you with respect to the fountain shown in the picture In a Medici Garden? How do you know? Is the fountain taller than the trees in real life? Why do you think the artist put the viewer where he did? • Where do you feel that you are in the picture Venice: The Doge’s Palace and the Riva degli Schiavoni? What clues help you determine your location? Why do you think the artist located the viewer where he did?

8.1

OVERVIEW

The goal of perspective is to create a picture of a scene that looks—to the viewer— exactly like the real scene. In fact, this goal is impossible, but using the mathematical techniques of perspective, we can come quite close. The geometry of perspective uses the properties of straight lines, planes, and their intersections. Using perspective, an artist shows the viewer a scene as the artist sees it—the artist is taking the viewer into the artist’s own personal, subjective world. The kind of symmetric designs that we saw in the previous chapters seem objective and impersonal when compared with a perspective drawing like that in Figure 8.1, which shows us what the artist was looking at and wanted us to see. In the chapters of this part, we focus on linear perspective. In this chapter, we outline the general principles of linear perspective and you will learn how to find the location of the artist or viewer with respect to the painting. You will also learn how to draw simple pictures using linear perspective. The next chapter covers grids or checkerboards in perspective, an important tool that is used for much more than just checkerboards. You will learn how to use a grid or checkerboard floor in perspective to evenly space fenceposts or trees along a road that recedes from the viewer. In Chapter 10, we learn about the conic sections and how they can be used to draw circles in perspective. We go beyond linear perspective with one vanishing point DOI: 10.1201/9781003110972-8

75

76  Geometry for the Artist

to using two-point and three-point perspective, which gives much greater freedom to the artist in choosing location and viewpoint, in Chapter 11. In the final chapter of this part, we will see how MC Escher adapted perspective in innovative ways to create his “impossible worlds.”

8.2

INTRODUCTION TO LINEAR PERSPECTIVE

In this section, we establish general principles of perspective that will guide us as we develop the geometric techniques of perspective. Take a few moments to look around you and pay attention to how things appear to you, what you see, and where you see it in your visual field. You can also look at the two pictures on page 74 to see how the artists have portrayed what they were seeing. Verify the following observations for yourself. • When you see two objects overlapping, the closer object overlaps the one that is farther from you. In Figure 8.1, the fountain is closer than the trees. • When you look down at objects on a table (or on the ground), closer objects are lower in your field of vision than ones farther away. In Figure 8.2, we are looking down on the plaza and the people on the lower left side are closer than the people in middle. • When you look up at objects on the ceiling or in the sky, the closer objects are higher in your field of vision than the ones farther away. In Figure 8.1, we are lower than the fountain basins, and the top of the fountain is closer than the tops of the trees. • When you are above an object, you see its top. In Figure 8.2, you are above the boats in the lower right and you see their tops. • When you are below an object, you see its bottom. In Figure 8.1, you are below the two large basins and you see the bottoms of them. • If two objects are the same size but at different distances, the one closer appears to be larger. In Figure 8.2, there are many people, of approximately the same size in real life, but the ones closer appear larger than those farther away. • If there are two lines that are parallel and go away from you, like receding edges of a table or the sides of a road, they appear to get closer to each other as they recede. In Figure 8.2, the lines on the receding face of the buildings are parallel in real life but appear to get closer together in the picture. • If there are two lines that are parallel but do not go away from you, like the front and back edges of a table that you are looking at, they appear parallel. In Figure 1.1, the horizontal edges of the skating ponds do not recede and they appear parallel in the picture.

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8.3

THE GEOMETRY OF LINEAR PERSPECTIVE

The woodcut in Figure 8.3, from a textbook on perspective by Albrecht D¨ urer (1479– 1528), illustrates the geometry that makes perspective painting work. In the woodcut, an artist is standing in front of a pane of glass on which he is drawing a picture of a seated man. To keep the artist’s eye in the same location as he draws the picture, he is looking through a sight vane that is fixed to the table in front of him. The location of the artist’s eye is called the station point of the picture that he is painting; it is also where the viewer should look at the painting to see the scene from the artist’s point of view. The glass pane that the artist is drawing on is called the picture plane (PP) of the painting. The station point and picture plane (PP) for the picture that the artist is painting are labeled in the sketch shown in Figure 8.4. Note that the station point is outside the picture plane. The artist paints on the glass exactly what he sees through the glass. If there is a blue object in the scene, a ray of blue light will leave the object and travel through the picture plane to the eye of the artist; the artist paints a spot of blue on the picture plane where he sees the blue light from the scene. The line connecting the eye of the artist through the picture plane to the object in the scene is called a visual ray. Once the painting is done, a viewer who puts their eye up to the sight vane will see the scene just as the artist did when painting the picture. This is the goal of perspective: the viewer will see exactly what the artist had seen.

Albrecht D¨ urer. An Artist Drawing a Seated Man, 1525. Figure 8.3

Sketch of D¨ urer’s picture. Drawing by Emilio Wherry, @y2cakey. Figure 8.4

You can try this yourself by standing in front of a window (or clear pane of glass or plastic) at a fixed location and drawing the scene that you see through the window by marking on the window, with erasable markers, whatever you see at each point on the window. Once you are finished, the scene will appear on the windowpane exactly as you saw it.

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Figure 8.5 is a diagram of a situation like that in D¨ urer’s picture of an artist. In this diagram, the station point is shown on the right with visual rays (dashed lines in green, orange, purple, and red) extending to four objects, two on the ground and two on the ceiling. We see a side view of the picture plane between the station point and the objects—this appears as a straight line.

Figure 8.5

Diagram showing station point, picture plane, and visual rays.

There is a fifth visual ray—a solid blue line—extending out straight, parallel to the ground and the ceiling. Since it is parallel to the ground and the ceiling, it never meets either of them. The point on the picture plane corresponding to this visual ray is between the ground and the ceiling, on the horizon, and called the central vanishing point (CVP). The line going across the picture plane at the level of the central vanishing point is called the horizon line (HL) and is at the height of the eye of the artist or the viewer. As we can see from the diagram, visual rays from objects on the ground meet the picture plane below the horizon line, while visual rays from objects on the ceiling meet the picture plane above the horizon line. The object on the ground that is closer to the picture plane is lower on the picture plane than the object farther away. Like that, the visual ray from the object on the ceiling that is closer to the picture plane is higher on the picture plane than the visual ray from the object farther away. We can combine these results by saying that visual rays from objects farther away from the station point are closer to the horizon line on the picture plane. The picture in Figure 8.6 shows us the painting that the artist in Figure 8.5 would paint. The ground is brown, and the ceiling is blue. They meet at the horizon line. The red object is lower in the picture than the purple object, and the green object is higher than the orange object. The red object and the green object are larger since they are closer. The purple object is closer to the horizon line than the red object, and the orange object is closer to the horizon line than the green object.

Summary of Principles of Linear Perspective

We can summarize these observations in the following principles of linear perspective:

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Figure 8.6

Picture created by the boy in Figure 8.5.

Picture Plane The picture plane is the flat surface on which the artist creates the painting of a scene. Station Point The station point is the location of the eye of the artist or viewer. It is outside the picture plane. Central Vanishing Point The central vanishing point is the point on the picture plane directly in front of the station point. Horizon Line The horizon line is the horizontal line on the picture plane that goes through the central vanishing point. It is the eye level of the artist or viewer. Objects on the Ground in the Scene Objects on the ground in the scene that are closer to the artist appear lower on the picture plane. Objects on the ground in the scene that are farther from the artist appear higher on the picture plane. Objects on the Ceiling in the Scene Objects on the ceiling in the scene that are closer to the artist appear higher on the picture plane. Objects on the ceiling in the scene that are farther from the artist appear lower on the picture plane. Objects of the Same Size If two objects are the same size in real life and are at the same distance from the picture plane, they appear to be the same size on the picture plane. If one is farther away from the picture plane, it will appear smaller in the picture than the one closer.

8.4

THE CENTRAL VANISHING POINT

The central vanishing point is the most important point in a perspective picture. In this section, we investigate its properties. We start by examining how straight lines and parallel lines look in a perspective picture. Recall that parallel lines are two straight lines that never intersect—the distance between them is constant. Straight

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lines and parallel lines are very important because most perspective pictures show scenes with straight lines such as edges of furniture, edges of buildings, and sides of roads. The important ideas that we will use over and over in this section are the following: • If two objects in the scene have the same size, the one farther from the picture plane will appear smaller in the picture. • Two objects of the same size in real life appear to be the same size on the picture plane when they are at the same distance from the picture plane. Transversals

A line that does not recede from the picture plane is parallel to it; we call such a line a transversal. The distance between two parallel lines is constant, so if they are both transversal—they don’t recede—then the distance between them on the picture plane will also be constant and they will look parallel on the picture plane. For example, the front and rear edges of the large building on the left in Canaletto’s painting are parallel to the picture plane and parallel to one another—so they are parallel in the painting. Transversals are often found as the vertical and horizontal lines of furniture and buildings, but transversals can be diagonal as well. In D¨ urer’s picture, the short diagonals bracing the pane of glass to the table are transversals. Receding Parallels

If two lines are parallel in the scene—so that the distance between them is constant— but they recede from the picture plane, the distance between them will appear to get less and less as we see them receding in the picture. If they are above the horizon line, they will get lower and lower in the picture plane, closer and closer to the horizon line. If they are below the horizon line, they will get higher and higher in the picture plane, closer and closer to the horizon line. The point on the horizon line where they appear to meet is called a vanishing point. Three sets of receding parallels are shown in Figure 8.7. Orthogonals

Of all the lines that recede from the picture plane, there are some that are special— the ones that go directly back, at a right angle to the picture plane. A line that is perpendicular to the picture plane is an orthogonal; we also call its image in the picture plane an orthogonal. This should not be confusing since we always know whether a line is in the scene or on the picture plane when we are talking about it. Any two orthogonals are parallel to one another in the scene, so they meet at the same vanishing point on the horizon line. The vanishing point on the horizon line where the orthogonals appear to meet is called the central vanishing point. This

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A perspective picture of three sets of receding parallel lines. Their vanishing points are all on the horizon line. Figure 8.7

is the point directly in front of the viewer’s eye—the point that you are looking at directly when you are looking at the picture. When there are two main directions of transversals—up and down, and right and left – then there is one main direction of receding lines, orthogonal to the transversals and orthogonal to the picture plane, and we have one-point perspective. We summarize the observations of this section with the following additional principles of linear perspective: Transversals Lines in the scene that are parallel to the picture plane will appear parallel in the perspective picture. Receding Parallel Lines Lines in the scene that are parallel and recede from the picture plane will appear to meet at a vanishing point on the horizon line (HL). Orthogonals and the Central Vanishing Point Lines that recede at right angles—perpendicularly—to the picture plane appear to meet at the central vanishing point (CVP), which is located on the horizon line.

If one should sit down in a room in front of a glass window, he should if he looked out of one eye only and kept his head still, trace upon the window glass a correct perspective drawing of the opposite houses or other objects outside, by making all the lines of his drawing exactly coincide with the real lines outside: in other words, by making each point of the drawing on the glass part of a straight line drawn from the corresponding part of the real object to the pupil of the eye. Thomas Eakins A Drawing Manual

8.5 ANALYSIS OF A PERSPECTIVE PICTURE Analysis of a perspective picture will tell you where the artist and the viewer are with respect to the scene. In addition, you will see in the next chapter that you can sometimes determine how far away the artist is from the picture plane.

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Where are you?

The first step is to estimate where you are just by looking. Sometimes, your feeling about where you are is accurate. Visual clues, such as whether you see the top or bottom or left side or right side of an object, are a big help. For example, in D¨ urer’s picture, Figure 8.3, you feel like you are in the room with the artist and seated man. You can see the top of the table and the chest, so you are above them. You can see the underside of the molding of the seated man’s chair, so you are below it. You can see the back of the seated man’s chair, so you are to the right of the chair. You can just barely see the right side of the pane of glass, so you are a little to the right of it. In the painting by Canaletto, Figure 8.2, you can see into the boats in the front, so you are above them. You cannot see the roofs of the buildings, so you must be below the roofs. You are to the right of all the buildings, and, because you can see the right side of the plaza, you are to the right of it as well. In the watercolor by Sargent, Figure 8.1, you can see the bottom of the two water basins, so you are below them. The picture has no straight lines, so you cannot continue with geometrical analysis. Geometrical Analysis

To begin geometrical analysis, look for transversals, the lines that are parallel to the picture plane. In D¨ urer’s picture, the vertical and horizontal edges of the chest and the back of the seated man’s chair are transversal, as are the front and back edges of the pane of glass. The front and back edges of the table are also transversal. We can’t say anything for sure about the legs of the table, as they seem to be angled. In the picture by Canaletto, Figure 8.2, vertical corners and vertical architectural details on the buildings are transversal. We can’t say for sure that the edges of the column are transversal, because it appears that the column tapers toward the top. These transversals help you identify the orthogonals because in the scene the transversals and orthogonals meet at right angles. In D¨ urer’s picture, the receding edges of the table and chest are orthogonals. In the painting by Canaletto, the receding edges of the buildings are orthogonal as are the receding edges of the plaza. Once you have identified several orthogonals, extend them until they meet—they will meet at the central vanishing point. The horizontal line through the central vanishing point will be the horizon line. Orthogonals, the central vanishing point and horizon line for the pictures by D¨ urer and Canaletto are shown in Figures 8.8 and 8.9.

Finding the Central Vanishing Point

We can summarize the steps of finding the central vanishing point—the location of the artist of viewer—and the horizon line as follows: 1. Identify two or more directions of transversals.

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8.8 Central Vanishing Point and Horizon Line of An Artist Drawing a Seated Man. Figure

Central Vanishing Point and Horizon Line of Canaletto’s picture of Venice. Figure 8.9

2. Identify two or more orthogonals, lines perpendicular to the transversals in the scene. 3. Extend the orthogonals until they meet; the meeting point is the central vanishing point. 4. Draw a horizontal line through the central vanishing point—this line is the horizon line. Tips When Finding the Central Vanishing Point and Horizon Line

• Many diagonal lines are not orthogonals. Some diagonals, like the diagonal supports of the glass pane in D¨ urer’s picture, are actually transversals. Other diagonals, like the oars of the gondoliers in Canaletto’s picture, are lines that recede at an angle, but not obviously at a right angle. • Use several orthogonals to find the central vanishing point. Even though two lines are sufficient to determine an intersection point, using several orthogonals gives a safety factor in case one of the lines was mistakenly taken for an orthogonal. Sometimes, perspective may only be approximate, and the lines will not all meet at one point. • The horizon line may not be within the picture plane—it can be above or below the picture frame; in such cases, the central vanishing point will also not be within the picture frame. Even though the horizon line itself is within the picture plane, the central vanishing point may be off to the left or right, outside the picture. Such placements will create a dramatic feel, as we will see when we look at the work of MC Escher in Chapter 12. • Artists often adapt perspective for their own purposes. The possibilities are endless, but if you are clear about the rules for perspective, you will be able to see when those rules have been broken for artistic purposes. Cubists like

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Braque and Picasso sometimes used multiple perspectives of the same object in one painting. • Early artists, before the twelfth century, did not know the rules of linear perspective. Their pictures may look three dimensional but usually do not use perspective correctly. • Even when an artist is painting an interior scene or cityscape, they may not be using linear perspective but rather their own judgment. Where Do Artists Locate Central Vanishing Points?

Artists are very deliberate about where they put the horizon line and the central vanishing point of a perspective picture. Knowing the location of the central vanishing point and the horizon line will help you better understand the intentions of the artist. In the picture of the fountain by Sargent, you are below the main part of the fountain. This vantage point gives the feeling that the fountain is large and impressive. In D¨ urer’s picture, you are between the artist and his picture plane. This is appropriate since the picture is illustrating how to draw a picture in perspective. The viewer is at eye level with the artist, so you feel that you are standing in the room with him. In Canaletto’s picture, you are to the right of center, giving you a panoramic view of the buildings along the water. The horizon line is in the lower half of the picture, showing more of the sky, which creates a balancing and calming influence. This location is very effective for creating an interesting landscape painting. Studying different artists who use linear perspective will give you insight into how different locations for horizon lines and central vanishing points can create very different experiences for the viewer.

8.6

HOW TO DRAW A PICTURE IN ONE-POINT PERSPECTIVE

What we have learned about perspective gives the information needed to draw a picture in perspective. The steps to drawing a simple perspective picture are shown in Figure 8.10. 1. The first step is to draw the horizon line, HL. This line is at the eye level of the artist or viewer. This can be anywhere, even outside the picture plane. If you are looking up, the horizon line is low. If you are looking down at the scene, the horizon line is high. Mark the central vanishing point, CVP, on the horizon line. This is the point directly in front of the viewer and can be anywhere on the horizon line; it also can be outside the picture. Putting the horizon line or central vanishing point outside the picture creates a dramatic effect. 2. Draw the front edge or face of each object in the picture. In this picture, the front faces of two solids are shown along with the front edge of a road at the very bottom of the picture plane.

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Figure 8.10

Steps in drawing a simple perspective picture.

3. Connect the endpoints or corners to the central vanishing point. These lines are construction lines that will be erased later. 4. Draw the rear edges of objects in the picture. We will see in the next chapter how far back these edges should be, but for now, make an estimate of where to put them so they look good to you. 5. Erase construction lines and points—lines and points that will not appear in the final picture. 6. Add details. In this picture, the road and the faces of the solids are colored in.

8.7

AERIAL OR ATMOSPHERIC PERSPECTIVE

Aerial or atmospheric perspective uses color to show depth. The principles of aerial perspective are the result of the air or atmosphere between the artist and the objects being depicted. Take a few moments to look around you at objects that are close and objects that are farther away; do this outdoors as well. Try to verify for yourself the following features of aerial perspective: 1. Objects closer are clearer and more distinct, while objects farther away appear less distinct. 2. There is more contrast between light and dark areas in objects that are closer than objects that are farther away. 3. Dark objects (like trees) that are farther away will appear bluer than they are, while bright objects (like bright lights) that are farther away will appear redder than they are.

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Figure 8.11

Albrecht D¨ urer. Saint Jerome in His Study, 1514.

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EXERCISES Understand

8.1 Features of Linear Perspective. For each of the features of linear perspective on page 79, describe how The School of Athens on page 90 displays that feature. 8.2 Features of Aerial Perspective. For each of the features of aerial perspective on page 85, describe how Hunters in the Snow on page 2 displays that feature. 8.3 Finding Orthogonals, Transversals, Vanishing Point, and Horizon Line. For the engraving Saint Jerome in His Study by Albrecht D¨ urer shown in Figure 8.11, do the following: a. Draw at least four transversals and label them. b. Draw at least four orthogonals and label them. c. Use the orthogonals to find the central vanishing point and label it CVP. d. Draw the horizon line and label it HL. e. Where is the viewer in relation to the scene? f. Why do you think D¨ urer put the viewer in that location? Apply

8.4 Analyzing Perspective. For the painting Hunters in the Snow on page 2, answer the following questions. a. Where are you, the viewer, in relation to everything else? b. What clues help you to determine your location? c. How does the artist make use of aerial or atmospheric perspective? d. Why do you think the artist located the viewer where he did? 8.5 La condition humaine. How does the painting La condition humaine by René Magritte on page 73 comment on the theme of perspective and the picture plane? 8.6 2by2. Look at the two works of art by different artists that you have chosen and answer the following questions for each of them. The purpose of these questions is to help you determine whether the artist used perspective or not. Note that many artists do not use perspective and intentionally give a flat feeling to a picture. a. Is there a feeling of depth or are the pictures flat? b. Does the artist use linear perspective in any of the pictures?

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c. If the artist uses linear perspective, see if you can find orthogonals, transversals, the central vanishing point, and the horizon line. If you find them in any of the pictures, label them. d. Whether or not the artist used perspective, determine where the viewer is in relation to the scene or subject matter. Why do you think the artist put the viewer in that location? What role does the location of the viewer have for the picture? 8.7 3by1. Look at the three works of art you have chosen and answer the following questions for each of them. The purpose of these questions is to help you determine whether the artist used perspective or not. Note that many artists do not use perspective and intentionally give a flat feeling to a picture. a. Is there a feeling of depth or are the pictures flat? b. Does the artist use perspective in any of the pictures? c. If the artist uses perspective, see if you can find orthogonals, transversals, the central vanishing point, and the horizon line. If you find them in any of the pictures, label them. d. Whether or not the artist used perspective, determine where the viewer is in relation to the scene or subject matter. Why do you think the artist put the viewer in that location? What role does the location of the viewer have for the picture? Create

8.8 Create Your Own. Draw a picture using linear perspective that includes some furniture, buildings, or other rectilinear shapes. Pay careful attention to where you put the horizon line and central vanishing point. 8.9 Perspective around You. Take a photograph that shows many of the features of linear perspective. Be aware of where the horizon line and central vanishing point are in your picture and what impression that will give to the viewer.

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Review

8.10 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 8.11 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

READINGS • How to Draw Comics the Marvel Way by Stan Lee • Perspective Drawing Handbook by Joseph D’Amelio • Viewpoints by Marc Frantz and Annalisa Crannell • On Painting by Leon Battista Alberti

ARTISTS • Renaissance painters • Rene Magritte (1898–1967) • Computer-generated imagery

Figure 9.1

Raphael. The School of Athens, 1509–1511.

Figure from the 1804 edition of Della pittura by Leon Battista Alberti showing the construction of a grid in perspective.

Figure 9.2

CHAPTER

9

Drawing Grids in Perspective

EXPLORATIONS • Find the central vanishing point of The School of Athens by Raphael. Approximately where are you? Does the floor appear accurate to you? • Look at Alberti’s construction of the floor. What do you notice about the diagonals of the quadrilaterals, which are squares drawn in perspective?

9.1

OVERVIEW

In this chapter, you will learn how to draw a checkerboard or grid in perspective. We see a floor pattern based on a checkerboard grid in Raphael’s School of Athens and many other Renaissance pictures. However, the construction of grids is used for more than drawing checkerboard floors, because a grid can be used to determine correct sizes and distances in perspective pictures. The squares in both pictures show an important feature of perspective pictures— the side and rear edges of a square are shorter than the front edge of the square. This feature is called foreshortening. The problem of how short to make the receding edges—foreshortened edges—is solved using the techniques of this chapter.

The vanishing points for angles of 45◦ , that is for lines sloping like the diagonals of a square on the floor, two sides parallel with the picture, would be as far to the right or left of the central plane as the picture is forward of the eye. Thomas Eakins A Drawing Manual

Renaissance artist Leon Battista Alberti (1404–1472) developed the technique shown in Figure 9.2 to draw a square grid in perspective. This technique uses principles of perspective that we already know and is very simple to implement. In the DOI: 10.1201/9781003110972-9

91

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next section, you will learn how to use this technique to draw a grid in perspective. Then, we will use square grids for placing objects at regular intervals in perspective pictures and for drawing irregular shapes in perspective. In the next chapter, you will learn how to use grids to draw a circle in perspective.

9.2

DRAWING A SQUARE GRID IN PERSPECTIVE

We first saw the square grid in Chapter 5, where we studied its symmetries. The key to drawing a square grid in perspective is to determine the foreshortening of the squares—that is, to determine the distance between the horizontal lines of the grid. It is easy to determine the placement of the receding lines of the grid since they are orthogonals and will meet at the central vanishing point. To help with determining the foreshortening of the squares in perspective, we add diagonals, shown in Figure 9.3, to help us. There are two important features of these diagonals. First, the diagonals are parallel to one another on the grid—and since they recede from the picture plane, they will meet at a vanishing point. The vanishing point where the diagonals of a square grid meet is called the diagonal vanishing point (DVP) or the distance point Second, the diagonals intersect the vertical and horizontal grid lines. Lines that intersect in the scene will also intersect on the picture plane. To understand this principle, draw two intersecting lines on a piece of paper and then look at it from different perspectives. You will see that the angle between the two lines can change and the lengths of the two lines can change, but they will always intersect. Thus, the diagonals in the picture plane will intersect the corners of the grid in perspective and they will meet at the diagonal vanishing point. At each corner of the grid in the scene, three lines intersect—one vertical, one horizontal, and one diagonal. This means that where the orthogonals and the receding diagonals intersect is also where the transversals intersect the orthogonals. And this completely determines the foreshortening. To draw a checkerboard in perspective, we start with the grid and its diagonals that we wish to draw, Figure 9.3. Then, mark off the corners of the squares along the bottom edge of the grid that will be on the picture plane and connect them to the central vanishing point on the horizon line. Mark a diagonal vanishing point on the horizon line and connect a lower corner of the grid to the diagonal vanishing point; draw transversals where this line intersects the orthogonals. Now you have the necessary lines for the grid in perspective, Figure 9.4. Erase unneeded construction lines and complete the details of the picture, Figure 9.5.

9.3

WHERE IS THE DIAGONAL VANISHING POINT?

The diagonal vanishing point (DVP)— also called the distance point—is on the horizon line where the diagonals of a square grid drawn in perspective meet. What does the distance from the diagonal vanishing point to the central vanishing point in the

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A square grid with diagonals. Figure

9.3

A line to the DVP determines foreshortening. Figure 9.4

Figure 9.5

Grid in per-

spective.

picture correspond to in the scene? The bird’s eye view of a grid and picture plane shown in Figure 9.6 helps us answer this question. The station point is directly in front of the central vanishing point on the picture plane. The artist sees a grid on the floor beyond the picture plane. Visual rays are drawn from the station point to a diagonal of the grid. As the artist looks farther and farther out along the diagonal of the grid, the point on the picture plane appears farther to the right and closer to the horizon line. Finally, when the artist looks in a direction parallel to the diagonal of the grid, the visual ray no longer intersects the diagonal of the grid, and there is no longer a point on the picture plane. This is where the diagonal vanishing point is located on the horizon line. As you can see in Figure 9.6, there is a right triangle, with the Figure 9.6 Bird’s eye view of a picture plane and central vanishing point, the di- grid. agonal vanishing point, and the station point as vertices. The sides of this triangle are parallel to the sides of the smaller triangles of the grid; these smaller triangles have an angle of 90◦ , and they are isosceles—they have two equal sides. This means that the larger right triangle will also have two equal sides, namely the side from the station point to the central vanishing point and the side from the central vanishing point to the diagonal vanishing point.

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These observations taken together tell us that the distance from the central vanishing point to the diagonal vanishing point is the distance from the station point to the canvas. To see the picture exactly as the artist intended, stand directly in front of the station point at a distance equal to the distance between the central vanishing point and a diagonal vanishing point.

Figure 9.7 Masquerade Ball by Albrecht D¨ urer with orthogonals, grid diagonals, central vanishing point, diagonal vanishing point, and horizon line shown.

9.4

FINDING THE DIAGONAL VANISHING POINT IN A PAINTING

If a painting has a grid or checkerboard floor drawn accurately, it is straightforward to find the diagonal vanishing point, as shown in Figure 9.7. Extend several orthogonals, shown in green, to find the central vanishing point, shown in red, and the horizon line, shown in light blue. Then, for the squares of the grid that is shown in perspective, draw and extend the diagonals, shown in purple. If the grid has been drawn accurately in perspective, the diagonals will meet at the diagonal vanishing points—one on the left and one on the right. The two diagonal vanishing points will always be equidistant—at equal distances—from the central vanishing point. In this picture, the central vanishing point is not centered horizontally, and one of the diagonal vanishing points is inside the picture, while the other diagonal vanishing point is outside the picture. This is a small picture, about 9 inches on a side, and the distance from the central vanishing point to a diagonal vanishing point is about 6 inches. This means that the ideal viewing position is 6 inches in front of the central vanishing point, which is fine for a small picture.

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Figure 9.8

9.5

A perspective picture with fence and evenly spaced trees.

FORESHORTENING DISTANCES IN A PERSPECTIVE PAINTING

You can use grids to determine the location of features that appear at regular intervals, such as fence posts, columns, or trees lining a road. The first step is to draw a grid in perspective. Then determine the placement of the features. In Figure 9.8, the foreshortened transversals of a grid in perspective are shown. Fence posts are placed at every other grid line, and trees are placed at every third grid line. The tops of the fence posts and trees are at the same height, determined by orthogonals drawn from the top of the front fence post and front tree. In Figure 9.9, you see the grid used by Thomas Eakins to ensure that waves are properly foreshortened as they recede from the picture plane. Figure 9.10 shows the resulting painting.

Thomas Eakins. Perspective drawing for The Pair-Oared Shell, 1872. Figure 9.9

Thomas Eakins. The PairOared Shell, 1872.

Figure 9.10

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9.6

USING GRIDS TO DRAW SHAPES IN PERSPECTIVE

You can use a grid in perspective to easily draw any polygon in perspective. To draw a shape like that shown at left in Figure 9.11, put it inside a grid, shown at lower center. Draw the grid in perspective and then put corresponding vertices into the grid, estimating the location of points that do not lie exactly on corners of the grid. Connect the vertices and color the interior of the polygon. Finally, as shown at right, erase construction lines and complete the details of the picture. With this technique, you can draw any shape in perspective. Your grid can have as many squares as you like; the finer the grid, the more accurate is the foreshortening. In the next chapter, we will see how to use this technique to draw a circle in perspective.

Figure 9.11

Drawing a simple shape in perspective using a grid in perspective.

EXERCISES Understand

9.1 Drawing a Grid in Perspective. Draw a grid or checkerboard floor. Show all of your construction lines. 9.2 Steps of Drawing a Grid in Perspective. List the steps used to draw a grid in perspective. 9.3 Drawing a Shape in Perspective. Draw a simple shape inside a grid. Then draw the shape in perspective, using the steps given in Section 9.6. Apply

9.4 False Perspective. Study the picture The importance of knowing perspective by William Hogarth shown in Figure 9.12 and list as many different mistakes in the use of perspective as you can. You may find some things that look wrong even if

Drawing Grids in Perspective  97

Figure 9.12

The importance of knowing perspective. William Hogarth, 1754.

you do not know why; include these as well. The legend underneath the picture says, “Whoever makes a Design without the Knowledge of Perspective will be liable to such Absurdities as are shown in this Frontispiece.” Create

9.5 Create Your Own Checkerboard. Draw a picture that includes a checkerboard floor in perspective. Do not erase your construction lines. 9.6 Create Your Own Perspective Drawing. Draw a picture that uses a grid in perspective as a basis for arranging objects. 9.7 Create Your Own Perspective Absurdities. Using ideas from Hogarth’s picture shown in Figure 9.12, draw a picture that includes many “absurdities” of incorrect perspective. Review

9.8 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 9.9 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

c 2022 Janet Fish / Licensed Janet Fish. Painted Water Glasses, 1974. by VAGA at Artists Rights Society (ARS), NY.

Figure 10.1

Figure 10.2

John Sloan. Travelling Carnival, Santa Fe, 1924.

CHAPTER

10

The Conic Sections

EXPLORATIONS • Painted Water Glasses in Figure 10.1 shows glasses, which are circular, but are seen at an angle. Describe the shape of their rims. • Travelling Carnival, Santa Fe in Figure 10.2 shows a merry-go-round and a Ferris wheel—both of which are circular—and shown at an angle. Describe their shapes.

10.1

OVERVIEW

We see circles everywhere: the rim of a water glass, a merry-go-round, a Ferris wheel, the sun, and ripples in water. Most of the time, we don’t see circles straight on, but rather from the side where they take on the shape of an ellipse, which we see in Figures 10.1 and 10.2. There are other shapes around us that are also related to circles: water cascading from a fountain takes the shape of a parabola, Figure 10.3, and the shadow on a wall cast by a lamp takes the shape of a hyperbola, Figure 10.4. These curves are the conic sections: circle, ellipse, parabola, and hyperbola. In this chapter, we will see how they are related to one another and where they appear around us.

10.2

THE CONIC SECTIONS

Imagine that you are looking at a circle and imagine all the visual rays from your eye to the circumference of the circle. The visual rays form a cone, as shown in Figure 10.6. There are many examples of cones around us—an ice cream cone, a party hat, a traffic cone, a funnel, and a newly sharpened pencil tip. If you want to draw a circle, imagine the visual rays between your eye—the station point—and the circle. The intersection of the visual rays with the picture plane will determine how the circle looks on the picture plane. We call the intersection of a plane with a cone a conic section. Look at the circles around you—the rim of a cup or jar and the top or bottom of a lamp shade—and you see the conic shape known as the ellipse. The glass rims are ellipses in Painted Water Glasses, Figure 10.1, and DOI: 10.1201/9781003110972-10

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Figure 10.3

Water fountain.

Figure 10.4

Lamp with hyperbolic

shadow. the Ferris wheel and merry-go-round in Travelling Carnival, Santa Fe, Figure 10.2, are ellipses. Figure 10.6 shows different planes at other angles with different cross sections, giving the four types of conic sections. If the plane is parallel to the circle at the base of the cone, the conic section is a circle. If the plane is tilted a bit, somewhere between being parallel to the circle at the base of the cone and being parallel to the edge of the cone, the conic section is an ellipse. If the plane is parallel to a straight line from the vertex of the cone along the surface of the cone, the conic section is a parabola. At a steeper angle than for the parabola, the plane gives a hyperbola. A hyperbola will meet both parts of a double cone, at right in Figure 10.7, and has two pieces that are separate from one another.

10.3

THE CIRCLE

A circle is defined by its center and radius, shown in Figure 10.8. All points on the circumference of a circle are at the same distance from the center. We see circles everywhere around us: wheels, clocks, coins, plates, ripples on water, tree trunks, sliced fruit, and so on. You can use a compass to draw a circle on paper; a stake and

Looking at a circle. The visual rays form a cone with the eye at the vertex of the cone. Figure 10.5

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The four conic sections given as the intersection of planes with a cone. The angle of intersection determines which conic section is formed. Figure 10.6

The four conic sections given as the intersection of planes with a cone. The angle of intersection determines which conic section is formed. Figure 10.7

string create a circle on the ground; a lathe or potter’s wheel creates a circular bowl or pot.

10.4

THE ELLIPSE: A CIRCLE IN PERSPECTIVE

An ellipse is formed when a plane intersects a cone as shown in Figures 10.6 and 10.7. The most common example of an ellipse is a circle in perspective. The shadow of a sphere is an ellipse, as shown in Figure 10.9. In the same way, a sphere in perspective is also an ellipse, although most artists draw a sphere in perspective as a circle. Besides a circle in perspective, other examples of ellipses are a carrot sliced at an angle, a whispering gallery, and the orbit of a planet around the sun. An ellipse has two important interior points, the foci of the ellipse. Find the distances from any point on the ellipse to each focus and add these two numbers

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A circle with center, radius, and circumference labeled. Figure 10.8

Elliptical shadow of spherical light bulb. Figure 10.9

up—you will get the same number for each point on the ellipse. We use this property to construct an ellipse, Figure 10.10. Put a pushpin at each point that will be a focus. Tie each end of a string, somewhat longer than the distance between the two pushpins, to one of the pushpins. Take a pencil and draw the ellipse as shown, keeping the string taut. The total length of the string is the sum of the distances of a point on the ellipse to the two foci.

Figure 10.10

How to draw an ellipse.

The planets travel around the sun with the sun located at one of the foci. In a whispering gallery, someone whispering at one of the foci can be heard very clearly by someone at the other focus, even if it is quite far away.

10.5

THE PARABOLA: A FOUNTAIN OF WATER

An object that is thrown up in the air and let fall to the ground takes a path that is a parabola, Figure 10.11. This is commonly seen in fountains, Figure 10.3. The geometric properties of a parabola make it useful for suspension bridges like the Golden Gate Bridge in San Francisco, Figure 10.12; solar cookers, Figure 10.13; and satellite dishes. A parabola has one focus, shown in red in Figure 10.14, and a special line, called a directrix, the dashed line in blue. A point on the parabola has the property that the distance from the focus to the point on the parabola is the same as the distance

The Conic Sections  103

from the directrix to the point on the parabola. You can see these distances as dashed lines in the figure. A silvered parabola has the property that rays from a light at the focus will bounce off the parabola and then go out straight. This is why parabolic mirrors are used in headlights and flashlights. This property works the other way as well—light or radio waves coming in straight bounce off the sides of the parabola and meet at the focus. This properly makes solar cookers and satellite dishes work—whatever light rays or signals coming in are reflected and meet at the focus, where their strength gets amplified many times. There is a solar furnace in France that can reach over 6,000◦ Fahrenheit.

10.6

THE HYPERBOLA: A SHADOW ON A WALL

A common example of a hyperbola is a shadow cast on a wall by a circular lampshade, Figures 10.4 and 10.15. Rattan stools (Figure 10.16) and nuclear power plants (Figure 10.17) have the shape of a hyperbola rotated around a central axis. In the rattan stool, straight rattan canes are slanted in different directions as they go across each other, creating the hyperbolic shape. This structure is much stronger than straight vertical canes with circular bands. For the same reason, the nuclear power plant has a hyperbolic structure.

10.7

DRAWING A CIRCLE IN PERSPECTIVE

The technique for drawing a polygon in perspective that we saw in the previous chapter in Section 9.6 can be used to draw a circle in perspective. This is shown in Figure 10.18. First, draw a square grid in perspective, left, directly above a square grid containing a circle. We show a grid of 16 squares, but more could be used. Next mark points in the picture plane corresponding to points on the circle in the grid below. Corresponding midpoints of the edges of the grids are shown in red. Four points are marked in the circle below where the circle intersects horizontal grid lines; extend vertical segments connecting them until they meet the perspective grid. Draw orthogonals from these meeting points up to the central vanishing point. Where these orthogonals intersect the corresponding horizontal grid lines, there will be points on the ellipse of the picture plane.

Basketball lit by a strobe light. Figure 10.11

Figure 10.12

Gate Bridge.

Golden

Figure 10.13

cooker

Solar

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Figure 10.14

Parabola with focus and directrix.

Figure 10.15

Hyper-

bolic shadow.

Rattan stool. Illustration by Emilio Wherry, @y2cakey. Figure 10.16

Figure 10.17

Nuclear

power plant.

You now have eight points on the ellipse in the picture plane. This is enough for you to draw a good approximation to the circle in perspective.

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Figure 10.18

How to draw a circle in perspective.

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EXERCISES Understand

10.1 Drawing a Circle in Perspective. Draw a circle in perspective. Show all construction lines. Apply

10.2 In a Medici Garden. Find circles in perspective in In a Medici Garden, Figure 8.1. 10.3 Saint Jerome in His Study. Find circles in perspective in Saint Jerome in His Study, Figure 8.11. 10.4 2by2. Look at the two works by two artists that you have chosen. Does either of them have circles in perspective? Is the perspective accurate? Does either of them have other conic sections? Explain. 10.5 3by1. Look at the three works by an artist that you have chosen. Do any of them have circles in perspective? Is the perspective accurate? Do any of them have other conic sections? Explain. Create

10.6 Create Your Own with a Circle. Draw a picture that incorporates a circle in perspective. 10.7 Create Your Own with a Shadow. Take a photograph of a circular object that has a shadow that is an ellipse or hyperbola. Which is it? 10.8 Create Your Own with a Parabola. Take a photograph that shows a parabola. 10.9 Create Your Own—Strobe Light. Use a strobe light app on your phone to photograph a moving object that takes the path of a conic section. Review

10.10 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 10.11 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

READINGS • Practical Conic Sections: The Geometric Properties of Ellipses, Parabolas and Hyperbolas by JW Downs

Taylor & Francis Taylor & Francis Group

http://taylorandfrancis.com

Figure 11.1

Vincent van Gogh. The Yellow House, 1888.

Figure 11.2

Skyscrapers in M¨ unster, Germany.

CHAPTER

11

Two-Point and Three-Point Perspective

EXPLORATIONS • What parts of the houses in Figure 11.1 do you see? Where are you with respect to the houses? How is your location different from pictures where the artist uses one-point perspective? Which directions recede from you? • What parts of the skyscrapers in Figure 11.2 do you see? Where are you located? How are you looking at the skyscrapers? How is your location different from pictures where the artist uses one-point perspective? Which directions recede from you?

11.1

OVERVIEW

We have already seen how to draw pictures in one-point perspective, when the artist is looking directly at the rectangular solids in the picture. There is one receding orthogonal dimension (depth) and two transversal dimensions (height and width) that do not recede from the picture plane. Drawing using one-point perspective requires us to be right in front of the rectangular solids in the scene. However, we constantly move around in the three-dimensional world and are not always directly in front of the objects—books, boxes, furniture, and buildings—around us. Two-point and three-point perspective give artists that same freedom to move around, so they can locate themselves anywhere in the scene and still draw an accurate representation. In this chapter, you learn how to draw using two-point and three-point perspective. To see the difference, look at the three artists shown drawing the same rectangular solid in Figure 11.3. The seated artist in the light blue shirt is directly in front of the pink face and is using one-point perspective; one direction of the solid (depth) is receding from the artist. The seated artist in the green shirt is directly in front of an edge of the solid and is using two-point perspective; two directions, breadth and depth, are receding from this artist. The artist on the step ladder is looking down at a corner of the solid and using three-point perspective; all three directions are receding. The pictures created by the three artists are shown in Figure 11.4. DOI: 10.1201/9781003110972-11

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Differences between one-point, two-point, and three-point perspectives. The artist in the light blue shirt is using one-point perspective; the artist in the green shirt is using two-point perspective; and the artist on the step ladder is using three-point perspective. Illustration by Emilio Wherry, @y2cakey. Figure 11.3

Figure 11.4

Pictures created by the three artists in Figure 11.3.

Two-point and three-point perspective techniques are based on one-point perspective, but the station point can be anywhere with respect to the objects in the scene being portrayed. In the next section, you will learn how to draw using two-point perspective, and in the following section, you will learn how to draw using three-point perspective.

Two-Point and Three-Point Perspective  111

11.2

TWO-POINT PERSPECTIVE

In two-point perspective, there are two vanishing points on the horizon line, corresponding to the two directions in space that recede from the artist, breadth and depth. The third direction, height, is transversal in two-point perspective. The following steps describe how to draw a rectangular solid using two-point perspective and are demonstrated in the frames of Figure 11.5. Vanishing Points. Draw the two vanishing points corresponding to the two receding directions; they determine the horizon line. The viewer is somewhere in front of the line segment determined by the two vanishing points. Front Edge. Draw the front edge of the rectangular solid that you are drawing in perspective. Connect Vertices. Connect each of the two vertices of the front edge to both vanishing points. Mark a point on one edge in each direction indicating how far back the receding edges go. Back Edges. Draw a vertical line, which will be a transversal, from each of these two points to the other receding line in the same direction. Finishing Details. Color the visible faces of the solid and the background. Erase construction lines and color the background.

How to use two-point perspective to draw the rectangular solid shown in Figure 11.3. Additional solids can be added to the picture using the same steps, but be careful to ensure that closer solids obscure solids that are farther away. Figure 11.5

11.3

ESHER’S CYCLE —AN EXAMPLE OF TWO-POINT PERSPECTIVE

Recall Escher’s lithograph Cycle, which shows a little man turning into a tile, which then turns into the little man. This picture uses two-point perspective, shown in Figure 11.6, with the vanishing points far outside the picture plane, to make us feel

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that we are in a realistic location and that everything is normal. That is necessary for us to be surprised by the tiles turning into a little man and then back into tiles.

Figure 11.6

11.4

Transversals and receding parallels of Escher’s picture Cycle.

THREE-POINT PERSPECTIVE

In three-point perspective, there are three vanishing points, corresponding to the three directions in space, all of which recede from the viewer. There are three receding directions and no transversal directions. If one of the vanishing points is above the head of the artist or viewer, it is at the zenith, and if one is below the feet of the artist or viewer, it is at the nadir. On the picture plane, three vanishing points must satisfy the condition that they are the vertices of an acute triangle—a triangle each of whose angles is less than 90◦ . This acute triangle points up if you are looking toward the ceiling or sky and points down if you are above looking toward the floor or ground. In Figure 11.2, we are looking up, so the triangle formed by the three vanishing points will point upward and the top vanishing point is at the zenith. Vanishing Points. Draw the three vanishing points corresponding to the three receding directions. The viewer is directly in front of the center of the triangle formed by the three vanishing points. Front Corner. drawing.

Draw the front corner vertex of the rectangular solid that you are

Receding Lines. Connect the corner vertex to the three vanishing points. Mark on

Two-Point and Three-Point Perspective  113

How to use three-point perspective to draw the rectangular solid shown in Figure 11.3. Figure 11.7

each of these receding lines a point indicating the length of the corresponding receding edge. This gives three receding edges. Edges. Connect each of the endpoints of the receding edges to the other two vanishing points. The three intersection points of pairs of these lines are the three farther corners of the solid and determine six receding edges of the solid. Finishing Details. Color the visible faces of the solid and erase construction lines. Color the background. Additional solids can be added to the picture using the same steps, but be careful to ensure that closer solids obscure solids that are farther away.

11.5

USING TWO-POINT AND THREE-POINT PERSPECTIVE

One-point perspective is suitable when looking at a scene directly. Artists use onepoint perspective when they are more interested in the human activity going on than they are in the scene or when they want to present a more neutral and straightforward impression. Two-point and three-point perspective give great freedom in terms of the placement of the artist and viewer—they can be looking in any direction at all. Two-point perspective is used for rectangular objects that have only one transversal direction. There are two vanishing points corresponding to the two other directions. Two-point perspective is often used when a building itself is the subject of a painting, since this allows us to see two sides of the building. Van Gogh used two-point perspective in Figure 11.1, where the houses are the main features of the painting. This painting also shows another effective use of two-point perspective; we are looking at a corner where two streets or paths come together and can see down both streets at once.

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Receding lines going in three different directions for the skyscrapers shown in Figure 11.2. Note that the left and right vanishing points are far outside the picture plane. We are looking up, so the vertical vanishing point is above the other two vanishing points and makes a triangle that points up. Figure 11.8

Three-point perspective gives a sense of drama or excitement. When creating a picture in three-point perspective, the artist is above or below the objects being depicted. When above the objects, the viewer feels free and unbounded. When below the objects, the objects can seem dominating or threatening, overwhelming the viewer. In Figure 11.2, for example, three-point perspective emphasizes the enormity of the buildings. Action comics and graphic novels make extensive use of two-point and threepoint perspective in addition to using one-point perspective for the simpler images. Switching among the three types of perspective from one frame to the next gives variety and gives the feeling that the viewer is moving around in the scene.

In one-point you can only look straight ahead, dead-on. In two-point you can look around, but not up or down. In three-point, you can look anywhere, which is a good thing. Obviously, you need three-point for flying, swinging, and leaping superheroes. Jason Cheesman-Meyer Vanishing Point

Two-Point and Three-Point Perspective  115

EXERCISES Understand

11.1 Two-Point Perspective. Draw a picture of a simple rectangular solid using twopoint perspective. Show all construction lines. 11.2 Three-Point Perspective. Draw a picture of a simple rectangular solid using three-point perspective. Show all construction lines. 11.3 Using Two-Point and Three-Point Perspective. For each of the following, say what kind of perspective you would use. Briefly give your reasoning. For threepoint perspective, also tell whether you are above or below the scene. a. An exotic Martian city seen from your spacecraft b. The background of a formal group portrait c. An architectural rendering of a two-story duplex d. A superhero on the cover of an action-packed graphic novel e. A classical temple on the cover of a book of essays on Greek philosophy f. An amusement park from a child’s point of view Apply

11.4 2by2. Look at the two works of art by different artists that you have chosen. Do either of the artists use two-point or three-point perspective? If so, find the vanishing points, determine roughly where the viewer is, and explain why you think the artist used that type of perspective. 11.5 3by1. Look at the three works of art you have chosen. Do any of the paintings use two-point or three-point perspective? If so, find the vanishing points, determine roughly where the viewer is, and explain why you think the artist used that type of perspective. Create

11.6 Two-Point Perspective. Draw a picture or take a photograph using two-point perspective. Make sure that there are at least three main rectangular solids in your picture. Briefly explain why two-point perspective is good for the subject matter that you chose. 11.7 Three-Point Perspective. Draw a picture or take a photograph using three-point perspective. Make sure that there are at least three main rectangular solids in your picture. Briefly explain why three-point perspective is good for the subject matter that you chose.

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Review

11.8 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 11.9 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

READINGS • How to Draw Comics the Marvel Way by Stan Lee • Vanishing Point: Perspective for Comics from the Ground Up by Jason Cheeseman-Meyer

Taylor & Francis Taylor & Francis Group

http://taylorandfrancis.com

c M. C. Escher Heirs/Cordon Art— MC Escher. Relativity, 1953 ( Baarn—Holland). Figure 12.1

CHAPTER

12

Perspective in the Work of MC Escher

EXPLORATIONS • Where is the viewer of the picture Relativity by MC Escher shown in Figure 12.1? How can you tell? • What is impossible about this picture? • Does Escher use one-point, two-point, or three-point perspective in this picture? • Does Escher use aerial perspective in this picture? Which people or objects are closer? Which are farther away?

12.1

OVERVIEW

As an art student, MC Escher mastered the use of perspective. Many of his earlier pictures use one-point perspective. Escher’s more imaginative pictures, however, use two-point and three-point perspective to create his “impossible worlds.” We will look at three of Escher’s pictures to see how he used perspective in innovative ways. First, however, let’s review the important principles of perspective that we will use in this chapter. Picture Plane (PP) The picture plane is the flat surface on which the artist creates the painting of a scene. Station Point The station point is the location of the eye of the artist or viewer. It is outside the picture plane. Central Vanishing Point (CVP) The central vanishing point is the point on the picture plane directly in front of the station point. Horizon Line (HL) The horizon line is the horizontal line on the picture plane at the level of the central vanishing point.

DOI: 10.1201/9781003110972-12

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Objects on the Ground in the Scene Objects on the ground in the scene that are closer to the artist appear lower on the picture plane. Objects on the ground in the scene that are farther from the artist appear higher on the picture plane. Objects on the Ceiling in the Scene Objects on the ceiling in the scene that are closer to the artist appear higher on the picture plane. Objects on the ceiling in the scene that are farther from the artist appear lower on the picture plane. Objects of the Same Size Two objects of the same size in real life appear to be the same size on the picture plane when they are at the same distance from the picture plane. Receding Parallel Lines Parallel lines that recede from the picture plane appear to meet at a point on the picture plane. Objects at a Distance If two objects in the scene have the same size, the one farther from the picture plane will appear smaller in the picture. Aerial Perspective Objects farther away from the picture plane appear less distinct than objects closer to the picture plane.

12.2 RELATIVITY We begin our study of Escher’s use of perspective with Relativity, shown in Figure 12.1. At first the picture seems to be a confusing scene of people at odd angles, going up and down stairways that defy gravity. The picture has many architectural elements with lines that would be parallel in real life. These lines seem to be receding in many different directions. Extend these lines, as shown in Figure 12.2, and you see that Escher used threepoint perspective with all the three vanishing points located far outside the picture. No one of the three is clearly the zenith or nadir for the viewer. Look closely, and you will see that each group of people has one of the three vanishing points directly above their head, at their zenith. Escher achieved his “impossible world” in this picture by integrating three different worlds into one picture. As the eye of the viewer moves around the picture observing different people or groups of people, the zenith direction keeps changing, and the viewer is confused but also delighted—and this seems to have been Escher’s intention.

12.3

BELVEDERE

In this section, we look at the lithograph Belvedere, shown in Figure 12.3. Dominating the picture is a belvedere—a structure built to give visitors a good vantage point to view a lovely scene. There are a number of people, some observing the scene, some approaching the belvedere, and one unfortunate individual locked in a jail cell at the bottom of the belvedere. Look at it for a while, and you will notice that the bottom of the picture with the plaza and stairway looks fine, the mountain scenery in the back looks fine, the roof of the belvedere looks fine, and in fact, each floor of the belvedere

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Extensions of receding lines in Relativity. All three vanishing points are outside the picture plane.

Figure 12.2

looks fine—but there is something terribly wrong when we look at the belvedere as a whole. Escher based this picture on the Necker cube, an optical illusion created by Swiss crystallographer Louis Albert Necker in 1832. It is a cube drawn in isometric perspective, a kind of perspective used by engineers for technical drawing that does not show receding lines converging. Shown at the left in Figure 12.4, the Necker cube has no foreshortening, so all edges are of the same length. The Necker cube does not show which lines are in front or which are in the back. When we look at a Necker cube, we can’t tell which edges overlap other edges and we can’t determine its orientation. A cube drawn in two-point perspective is at right for comparison. Look at the Necker cube, and you see that any side can be the top or bottom, left or right, front or rear. Each way of looking at the Necker cube seems to give the same shaped cube. On the other hand, the perspective cube appears to be a cube only one way, with the smaller faces in the rear. Other ways of looking give an oddly shaped solid that does not appear to be a cube. Figure 12.5 shows two drawings of the Necker cube with shading so that they appear to open from the bottom or top. At far right in the picture is the impossible cube based on the Necker cube that was created by Escher. Escher’s construction

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c M. C. Escher Heirs/Cordon Art— MC Escher. Belvedere, 1960 ( Baarn—Holland).

Figure 12.3

confuses the overlapping edges in such a way that it does not represent a cube at all.

Figure 12.4

Necker cube on the left and cube in two-point perspective on the right.

Look closely at Escher’s picture again, Figure 12.3, and you will see that the man seated by the steps has a picture of the Necker cube on the ground in front of

Perspective in the Work of MC Escher  123

Necker cube at far left, Necker cube drawn to open from the bottom, Necker cube drawn to open from the top, and, at far right, Escher’s impossible cube that is examined by the seated man in Belvedere. Figure 12.5

him with the problematic crossings circled. In his hand, he holds a physical model of Escher’s impossible cube. This model seems to be the model for the belvedere itself, where you see the supporting columns of the upper floor and the roof crossing one another in a way that is as impossible as the model held by the seated man.

Transversals and extensions of receding lines in Belvedere. The picture uses two-point perspective with both vanishing points located outside the picture plane. Figure 12.6

The transversals and the extensions of receding parallels in Figure 12.6 show that Escher used two-point perspective with vanishing points outside the picture plane. Escher’s careful use of perspective makes everything look realistic except for the impossible columns. So again, Escher confuses and delights the viewer.

12.4

ASCENDING AND DESCENDING

Escher’s lithograph Ascending and Descending, Figure 12.7, is one of Escher’s most beloved works and depicts an intricate building that appears to be rectangular. There are monks traveling along a staircase on the roof, circling the inner courtyard. Some monks are always ascending as they travel the circuit, and some are always descending along the same route. This is, of course, impossible, but your impression is that you “see” this happening.

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c M. C. Escher Figure 12.7 MC Escher. Ascending and Descending, 1960 ( Heirs/Cordon Art—Baarn—Holland). To play this trick on us, Escher uses the Penrose staircase, Figure 12.8, developed by the British physicist Roger Penrose and his father, Lionel Penrose. The stairs are built on what appears to be a rectangular block that is drawn in what appears to be three-point perspective, with vanishing points far off the picture plane. When you look carefully at the staircase, however, you realize that these appearances are completely wrong. To understand how the Penrose stairs work, look at the left-hand and righthand sides separately. In our analysis, we will be assuming that the stairway is on a rectangular block. On the front left-hand side, stairs go up; the front center edge is shorter than the left-hand front edge, as we would expect for stairs that go up. We know that when two objects of the same height are drawn in perspective, the one farther away is higher on the picture plane. That, together with the fact that the stairs are going up, tells us that the corner in the far left should be higher on the picture plane, just

Perspective in the Work of MC Escher  125

Figure 12.8

Impossible staircase designed by Roger Penrose.

as we see it. So far, everything is fine. However, the front left face is narrower than the back right face—but the rules of perspective say that the side farther away on a rectangular block should be narrower, not wider. So we have already found something wrong with this picture. On the right-hand side, we see the stairs going down as they go back, and the front edge is longer than the back right-hand edge, just as it should be. If the stairs continue to go down once we turn the corner at right, the farthest corner would still appear to be higher, as objects farther away are higher. However, and this is the key to the Penrose staircase, if perspective were correct, the farther corner would be much higher than it is shown in the picture—and the rear right-hand stairs could not appear to be going down. So we see how this is achieved—Escher tricks us into thinking the top of the building is rectangular. The building as a whole uses three-point perspective, as you can see in Figure 12.9. The lower parts of the building are drawn accurately, and we believe that the whole building could be real. The upper circuit has a small tower on the right and a higher part of the building on the left, all realistically drawn and still convincing us that the building could be real. But the stairway itself follows the pattern of the Penrose staircase—and Escher has, in his own words, “caused the desired shock” and has fooled, confused, and delighted us all at once.

Whoever wants to portray something that does not exist has to obey certain rules. Those rules are more or less the same as for the teller of fairytales: he has to apply the function of contrasts; he has to cause a shock. The element of mystery to which he wants to call attention must be surrounded and veiled by perfectly ordinary every day self-evidences is that are recognizable to everyone. That environment, which is true to nature and acceptable to every superficial observer, is indispensable for causing the desired shock. MC Escher “Approaches to Infinity,” in Escher on Escher: Exploring the Infinite

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Figure 12.9

Receding parallels in Escher’s Ascending and Descending.

EXERCISES Understand

Note: You can find images for the exercises below on the official MC Escher website: http://www.mcescher.com/. 12.1 St. Peter’s Rome. In St. Peter’s Rome, Escher uses three-point perspective. Where are the vanishing points? What does Escher achieve with this engraving? 12.2 Tower of Babel. In Tower of Babel, Escher uses three-point perspective. Where are the vanishing points? What does Escher achieve with this woodcut? 12.3 Another World. In Another World, Escher uses one-point perspective. Where is the central vanishing point? Where is the horizon line? How does Escher create an imaginative world in this picture? 12.4 Relativity. On a copy of Escher’s lithograph Relativity, determine the zenith (overhead vanishing point) for each person. Choose three colors and color each person with the same zenith the same color. What do you notice about the

Perspective in the Work of MC Escher  127

people with the same zenith? Why do you think Escher located them the way he did? 12.5 Escher’s Waterfall. Describe what is depicted in Escher’s lithograph Waterfall. What is impossible? How did Escher use the principles of perspective to create this impossibility? 12.6 Escher Picture. Choose a picture by Escher that you like. What kind of perspective did he use? Why? 12.7 Creating a Shock. Choose a picture by Escher that you think gives a shock to the viewer. How does Escher give the “desired shock”? Does he follow his own advice that is given on page 125? Create

12.8 Creating an Escher-like Drawing. Use Escher’s techniques to create your own impossible world. You could start by using the Necker cube, the Penrose staircase, or some other optical illusion as a foundation for your drawing. Review

12.9 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 12.10 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

READINGS AND RESOURCES • The Magic Mirror of M.C. Escher by Bruno Ernst • Escher on Escher: Exploring the Infinite by MC Escher • The official MC Escher website: http://www.mcescher.com/

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IV Fractals

Fantasy fractal image by Stefan Keller from Pixabay, 2019.

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Figure 13.1

Vincent van Gogh. Starry Night over the Rhône, 1888.

Figure 13.2

Wassily Kandinsky. Circles in a Circle, 1923.

CHAPTER

13

Proportion and Similarity

EXPLORATIONS • Look at Starry Night over the Rhône, by Vincent van Gogh, Figure 13.1. What shapes are repeated almost exactly? What is the effect of the repetition of these shapes? • Look at Circles in a Circle by Wassily Kandinsky, Figure 13.2. How many circles do you see? Are any of the circles the same size? How many different sizes of circles do you see? What feelings does the repetition of circles give?

13.1

OVERVIEW

In the pictures by van Gogh and Kandinsky, the repetition of shapes gives a feeling of unity even in the presence of differences in size, color, or position. In this chapter, we look at figures that have the same shape. There are two key concepts: congruent figures have the same size and shape, and similar figures have the same shape but possibly different sizes. See Figures 13.3 and 13.4 for examples of congruent and similar figures. We have already seen congruence as a tool in creating symmetric designs—a symmetric design is congruent to a rotation or reflection of itself; band ornaments and tilings have infinitely many congruent copies of a basic motif. In this chapter, we see other ways that artists use congruence and similarity.

13.2

CONGRUENT FIGURES

Measurement, which can distinguish one figure from one another, is also used to determine how figures are alike. When two shapes have exactly the same measurements, we say they are congruent—they have the same shape and the same size—without regard to position or orientation. Four congruent shapes are shown in Figure 13.5; notice that they all have the same shape and size even though they have different positions or orientations. The two quadrilaterals ABCD and EF GH in Figure 13.6 are congruent. To confirm this, we measure the sides and the angles. The measurements have the same values, but that is not enough—we must compare the measurements for corresponding sides and angles—sides and angles that match or pair up. DOI: 10.1201/9781003110972-13

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Two congruent polygons with the same shape and size. Figure 13.3

Two similar polygons with the same shape but different sizes. Figure 13.4

Figure 13.5

Four congruent polygons with different orientations.

Figure 13.6

Two congruent quadrilaterals with side and angle measurements.

Each quadrilateral has a side of length 11, another side of length 9, and two sides of length 5. The two sides of length 11 should correspond, as should the two sides of length 5. But which side of length 5 in the left quadrilateral should correspond to which side of length 5 in the right quadrilateral? Looking at the vertices will make it easier, as all four angles have different measures. We see that vertex A corresponds

Proportion and Similarity  133

to vertex H, as the angle at A and the angle at H both have measure 131◦ . Also, vertex B corresponds to vertex G, vertex C corresponds to vertex G, and vertex D corresponds to vertex E. We write ABCD ∼ = HGF E to mean that polygon ABCD is congruent to polygon HGF E. The symbol ∼ = is read “corresponds to.” The order of vertices here is important, as we want to match up A with H, B with G, C with F , and D with E. We write the correspondences of the sides and angles this way: AB ∼ = GH ∼ BC = F G

CD ∼ = EF ∼ DA = HE

∠BAD ∼ = ∠GHE ∼ ∠ABC = ∠F GH

∠BCD ∼ = ∠EF G ∼ ∠ADC = ∠HEF

Artists use congruence in diverse ways. One way is to repeat the same shape over and over, as we saw with band ornaments and tiling patterns. This kind of repetition is shown in Poem Scroll with Deer, Figure 13.7, where we see congruent deer—bucks and does—repeated over and over. The repetition conveys the feeling of a herd of deer, each like the other and all of them closely connected to one another. Another way to use repetition is shown in Starry Night over the Rhône by Vincent van Gogh, Figure 13.1. In that picture, van Gogh repeats stars over and over and the reflections of light over and over, giving light to the scene that would otherwise be dark. Even though the shapes are only approximately congruent, the eye sees them as having the same shape and size. Their repetition gives a peaceful and orderly quality to the picture, contributing to the overall settled feeling of nighttime.

Tawaraya Sotatsu. Poem Scroll with Deer, 1576–1643. Detail (Seattle Art Museum. Gift of Mrs. Donald E. Frederick).

Figure 13.7

13.3

SIMILAR FIGURES

The two quadrilaterals in Figure 13.8 are similar but not congruent—they have the same shape but different sizes. Because they have the same shape, corresponding

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angles have the same measure. The correspondence of angles is written as for the congruent quadrilaterals: ∠LIJ ∼ = ∠M N P ∠IJK ∼ = ∠N P Q

∠JKL ∼ = ∠P QM ∼ ∠KLI = ∠QM N Because they are of different sizes, pairs of corresponding sides are not equal, but rather proportional. Proportions of similar figures allow us to compare them in a precise way using measurement. This is shown in Figure 13.8. You see that corresponding angles have the same measurements, while length of each side of quadrilateral M N P Q is exactly 1/2 the length of the corresponding side of quadrilateral IJKL. We will see ratios of corresponding sides of similar polygons again when we study fractals in the next chapter, Chapter 14.

Figure 13.8

13.4

Two congruent quadrilaterals with side and angle measurements.

CONGRUENCE AND SIMILARITY IN ART

Artists often use shapes that are only approximately congruent or similar; the viewer still sees the connection even when congruence or similarity is not exact. This means that when analyzing a work of art, you can use your judgment about whether two shapes appear congruent or similar. If you choose, you can also make measurements, but that is not necessary for your analysis—your impression of how shapes appear is what matters most. Any two circles are similar. For any circle, the circumference and diameter have the same ratio, the number 3.14159. . . , symbolized by the Greek letter pi, π. Kandinsky uses the similarity of circles in Circles in a Circle, Figure 13.2, to tie the elements of the picture together. Some of the circles appear to be congruent, but for sure they are all similar. The Dutch painter Johannes Vermeer (1632–1675) made effective use of similarity in the two paintings shown in Figures 13.9 and 13.10. He used similarity to tie the elements of each painting together, but he also used similarity to convey the differing impressions he wanted to give of the two women he portrayed.

Proportion and Similarity  135

The Milkmaid gives an overall impression of warmth, nourishment, and comfort, while Young Woman with a Water Pitcher gives an impression of coolness, elegance, and higher social status. In the first painting, the repetition of circles and rounded curves enhance the feeling given by the warm colors and the pouring of milk, which provides nourishment. In the second picture, the repeated sharp angles in the lip of the pitcher and the basin, the corner of the head covering, and the negative space between the edges of the collar contribute to the more sophisticated feeling associated with the elegant clothing and furnishings. The coolness of the water in the pitcher reinforces this impression. These two pictures show that even approximate similarity is enough to create a significant effect on the viewer.

Figure 13.9 Johannes Vermeer. The Milkmaid, circa 1660.

13.5

Johannes Vermeer. Young Woman with a Water Pitcher, circa 1662–1665.

Figure 13.10

IMPORTANT EXAMPLES OF CONGRUENCE AND SIMILARITY

In the next sections, you will see several important examples of congruence and similarity used by artists. Body-to-Head Ratio or Number of Heads Tall

To ensure that drawings of an individual at different scales will look like the same individual, artists use ratios of different parts of the body to one another, ensuring that the drawings at different scales will be similar. Many ratios have been given by Palladio, D¨ urer, and others, including ratios comparing the parts of the face (eyes, eyebrows, nostrils, and lips) and ratios comparing

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the hands and feet to other parts of the body for men, women, and children. Examples are given in Exercises 13.8 and 13.9. A commonly used ratio is the body-to-head ratio; this is the total height of the body (from soles of feet to top of head) divided by the height of the head (from tip of chin to top of head). In Figure 13.11, the total height of the individual is 14 units and the head height is 2 units, so the body-to-head ratio is total height 14 = =7 head height 2 and we say that the person is 7 heads tall.

The total height of this individual is 14 units, and the head height is 2 units, so they are 7 heads tall. Illustration by Emilio Wherry, @y2cakey. Figure 13.11

The ratio of body to head varies as an individual grows, with infants and young children having a larger head in proportion to their body and thus a lower body-tohead ratio and fewer heads tall. Body-to-head ratios can also vary between individuals of the same age. Typically, adults are 7 1/2 heads tall. Artists exaggerate the bodyto-head ratio to create different effects. To give heroic appearance, classical Greek statues have larger body-to-head ratios. Action heroes have exaggerated body-tohead ratios so that their powerful bodies are emphasized. Fashion models, who are sometimes drawn to be 9 or even 10 heads tall, look dramatic and elegant. Cute or child-like cartoon characters have larger heads relative to their bodies, portrayed as

Proportion and Similarity  137

anywhere from 2 heads tall to 4 or 5 heads tall. Various body-to-head ratios are given in Table 13.1. TABLE 13.1

Examples of body-to-head ratios used by artists in different contexts. Figure

Chibi Cartoon Character Childish Cartoon Character Baby Young Child Adult Classical Greek Sculpture Vitruvian Man Action Heroes (female) Action Heroes (male) High Fashion Model

Body-to-Head Ratio (Number of Heads Tall) 2 or 3 2, 3, or 4 4 6 7.5 7 8 8 9 9 or 10

Vitruvian Man

The Roman architect and engineer Vitruvius Pollio (c. 70–c. 25 BCE), author of a Roman manual of architecture, emphasized the importance of using harmonious proportions in the design of temples, to mimic how nature designed the human body with precise natural proportions. He gave an elaborate system of proportions of the human body, involving everything from the location of the navel to the size of the hands and feet, that influenced Renaissance portrayals of the human figure. He located the navel at the center of a circle passing through the hands and feet extended and also at the center of a square reaching vertically from the soles of the feet to the top of the head and horizontally from the fingers of the outstretched hands, as shown in Figure 13.12.

. . . since nature has designed the human body so that its members are duly proportioned to the frame as a whole, it appears that the ancients had good reason for their rule, that in perfect buildings the different members must be in exact symmetrical relations to the whole general scheme. Vitruvius Pollio The Ten Books of Architecture

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A person drawn using proportions given by Vitruvius. Illustration by Emilio Wherry, @y2cakey. Figure 13.12

The Golden Rectangle

The golden rectangle, one of the most recognized shapes in geometry, has the property that cutting a square away leaves a rectangle similar to the original golden rectangle. This property of a part being similar to the whole is the origin of the term “golden” that is used to describe this rectangle. We see this property in Figure 13.13, where a golden rectangle is shown on the left. Cutting off square AF ED leaves the golden rectangle √ F BCE. For both rectangles AF ED and F BCE, their base-to-height ratio is (1 + 5)/2 ≈ 1.618, which is called the golden ratio. Construction of the Golden Rectangle.

To construct a golden rectangle, start with a square and find the midpoint of the base, as shown in Figure 13.15. Draw a circle having its center at this midpoint and passing through a vertex on the opposite side of the square. Extend the base of the square until it intersects this circle; this extended base is the base of the golden rectangle. Complete the construction, giving the golden rectangle. Construction of a Golden Spiral.

Using the golden rectangle, we can construct a close approximation to the golden spiral, as shown in Figure 13.15. Start with the construction of a golden rectangle,

Proportion and Similarity  139

A golden rectangle on the left. It is shown on the right with the square AF ED cut off, leaving the golden rectangle F BCE. The base-to-height ratio for √ both rectangles is the golden ratio (1 + 5)/2 ≈ 1.618. Figure 13.13

Construction of the golden rectangle. Construct a square AF ED and find the midpoint M of side DE; extend the base of the square DE as shown at left. Construct a circle with center M passing through corner F . Construct a perpendicular to the extended line DE at intersection point C of the square, as shown in the center. The intersection of this perpendicular with the extension of edge AE gives the upper right corner of the golden rectangle. Erase construction lines, as shown at far right, to get the golden rectangle.

Figure 13.14

and erase all construction lines except edge EF of the original square. Construct a quarter circle with center at corner E and radius equal to edge EF , as shown at left. Rectangle EBCF is also a golden rectangle; construct a square in the upper part of this rectangle and draw a quarter circle as shown. Continue in this way, constructing a quarter circle in each smaller golden rectangle, for as many steps as you like. The spiral constructed in this way will be a close approximation of a golden spiral.

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Figure 13.15

How to construct a golden spiral.

EXERCISES Understand

13.1 Evaluating Cartoon Characters. Find the ratio of the total height of the figure to the height of the head—the number of heads tall—for each character in Figure 13.16. What impression does each character give?

Figure 13.16

Six characters. Illustration by Emilio Wherry, @y2cakey.

13.2 Escher’s Little Man. Find the approximate ratio of the total height of the figure to the height of the head—the number of heads tall—for the little man in Escher’s lithograph Cycle, shown in Figure 7.1. What impression does this person give?

Proportion and Similarity  141

13.3 Finding Cartoon Characters. Find three pictures of different interesting cartoon characters. The best pictures for this exercise show the full length of the character standing up straight. Find the ratio of the total height of the figure to the height of the head—so you will know how many heads tall each figure is. What impression does each character give? Explain. 13.4 Similarity and Congruence in Art. Find examples of approximate similarity and approximate congruence in one of the following pictures. Does the similarity or congruence connect different objects? Why do you think the artist made the connections? • Pictorial Quilt by Harriet Powers, Figure 21.3 • Painted Water Glasses by Janet Fish, Figure 10.1 • Street to Mbari by Jacob Lawrence, Figure 21.2 • Wheat by Thomas Hart Benton, Figure 21.4 Apply

13.5 2by2. Copy or trace each of the two pictures you have chosen by two artists and do the following: a. Label some objects in each picture (if any) that are congruent. b. Label some objects in each picture (if any) that are similar. c. For each work, discuss the contribution to the work as a whole that is given by the congruence and similarity of the objects that you have found. If no objects are similar or congruent, explain why you think the artist did that and what effect the lack of congruence and similarity has. 13.6 3by1. Copy or trace each of the three paintings by one artist that you have chosen and do the following: a. Label some objects in each picture (if any) that are congruent. b. Label some objects in each picture (if any) that are similar. c. Discuss the contribution to the work as a whole that is given by the congruence and similarity of the objects that you have found. If no objects are similar or congruent, explain why you think the artist did that and what effect the lack of congruence and similarity has. Create

13.7 Creating Cartoon Characters. Create at least three cartoon characters with exaggerated head-to-body ratios. Using grid paper makes it easy to get the head-to-body ratio that you want. For each character, give the head-to-body ratio and tell what effect you wanted to achieve.

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13.8 Drawing a Vitruvian Face. Create a Vitruvian face, with proportions given as follows: If we take the height of the face itself, the distance from the bottom of the chin to the underside of the nostrils is one third of it; the nose from the underside of the nostrils to a line between the eyebrows is the same; from there to the lowest roots of the hair is also a third, comprising the forehead. For this exercise, grid paper with three, six, or nine rows will help you easily achieve the correct proportions. 13.9 Drawing a Vitruvian Man. Create your own version of Vitruvian Man using the proportions given by Vitruvius: For the human body is so designed by nature that the face from the chin to the top of the forehead and the lowest roots of the hair, is the tenth part of the whole height; . . . the head from the chin to the crown is an eighth . . . The length of the foot is one sixth of the height of the body; of the forearm one fourth; and the breadth of the breast is also one fourth. For this exercise, grid paper with 40 rows will help you most easily achieve the correct proportions. Review

13.10 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 13.11 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

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Figure 14.1

Safari. Trees in Hawai’i, 2019.

Figure 14.2

Romanesco broccoli.

CHAPTER

14

Fractals

EXPLORATIONS • Can you apply the concept of similarity from the previous chapter to the branches of the trees in Figure 14.1? How are the smaller branches related to the larger branches? • Can you apply the concept of similarity from the previous chapter to the Romanesco broccoli shown in Figure 14.2? How are the larger “mountains” related to the smaller “mountains”? How are the larger spirals related to the smaller spirals?

14.1

OVERVIEW

In the previous chapter, we saw that two different figures are similar to one another if they have the same shape but possibly different sizes. In this chapter, we will consider shapes that are similar, or approximately similar, to a part of themselves. This property is called self-similarity, and such figures are called fractals. Many structures in nature, such as plants, clouds, coastlines, and mountain ranges, shown in Figure 14.3, have fractal structure. Benoit Mandelbrot, who coined the term fractal, called fractal geometry “the geometry of nature.”

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. Benoit Mandelbrot The Fractal Geometry of Nature

In the final section of this chapter, we consider a characteristic property of fractals: their dimension is equal to a fraction rather than an integer. We are familiar with the dimensions 0, 1, 2, and 3 for a point, line, plane, and space. A shape with fractional dimension has properties that place it somewhere between a point and a line, between a line and a plane, or between a plane and space. DOI: 10.1201/9781003110972-14

145

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Understanding the structure of fractals helps artists draw natural shapes. Natural and geometric fractal patterns provide inspiration for artists to create abstract designs, like the fractal landscape on page 129.

Natural fractals: clouds, a coastline in Thailand, sea ice near Japan (NASA), lightning, and a mountain range. Figure 14.3

14.2

CONSTRUCTING GEOMETRIC FRACTALS

Complex geometric fractals are built up from a simple geometric shape by repeating— or iterating—a geometric construction over and over. The fractal itself is the result of infinitely many iterations and can only exist in your consciousness, not on paper. While each stage of the fractal will have parts that are similar to other parts, the fractal obtained after infinitely many steps will have perfect self-similarity—a part of the fractal is similar to the complete fractal. We see this in the next section with the Cantor set, one of the earliest fractals discovered.

Fractals  147

Cantor Set

The Cantor set was developed by the German mathematician Georg Cantor (1845– 1918) to study relationships between a line segment and individual points. A geometric fractal begins with a simple geometric shape, which is called stage 0 of the fractal; at this stage, it is not clear at all what fractal will be created. For the Cantor set, stage 0 is a line segment. To get stage 1 from stage 0, a construction, procedure, or step is given. For the Cantor set, this step is to remove the middle third, except for its endpoints, from the segment. This results in stage 1 of the Cantor set. Notice that stage 1 of the Cantor set consists of two copies of stage 0 of the Cantor set. See Figure 14.4. Continue in this way as shown, applying the procedure of removing the middle third without endpoints from each new segment. Notice that at each stage the Cantor set consists of twice as many intervals as before and the new stage has two halves, each similar to the previous stage.

Figure 14.4

The first four stages of the Cantor set.

The Cantor set itself is the result of removing middle thirds infinitely many times. Of course, this cannot be done on paper but only in your consciousness. The result is that the Cantor set consists only of the endpoints of segments and does not contain any complete segment at all. Because of this property, the Cantor set is sometimes called Cantor dust. Koch Snowflake

The Koch snowflake shown in Figure 14.5 was developed by the Swedish mathematician Helge von Koch (1870–1924) in his search for a curve that had corners everywhere. We start by constructing the Koch snowflake curve. Begin with a line segment, which is stage 0. Replace the middle third with two segments that form an equilateral triangle with the missing Figure 14.5 The Koch snowflake. third, getting stage 1, which consists of the four line segments shown in Figure 14.6. Continue by replacing the middle third of each of these four segments as before, getting stage 2. Continue in this way for as many stages as you like. Notice that at each stage, there

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are four segments for every one segment in the previous stage. Three copies of this Koch snowflake curve make the Koch snowflake.

Figure 14.6

The first three stages of the Koch snowflake curve.

´ Sierpinski Triangle

The Sierpiński triangle or Sierpiński gasket was developed by the Polish mathematician Waclaw Sierpiński (1882–1969). Start with a triangle, stage 0. Mark the midpoint of each of the three edges of the triangle and connect to form four new triangles. Color in the middle triangle, getting stage 1. Continue in this way for each empty triangle that gets created. Notice that at each stage, there are three smaller empty triangles and one smaller colored triangle for every one empty triangle in the previous stage. See Figure 14.7. The construction shown here starts with an equilateral triangle, but you could use any triangle.

Figure 14.7

14.3

The first three stages of the Sierpiński triangle.

NATURAL FRACTALS

Benoit Mandelbrot (1924–2010), one of the first mathematicians to study fractals, saw fractals everywhere in the natural world while finding the traditional shapes of geometry—circles, triangles, and squares—hard to find. A natural fractal will not

Fractals  149

have infinitely many stages in the way that a mathematical fractal does, so selfsimilarity will only be approximate. A tree starts as a shoot; then branches appear. As the branches grow larger, new branches grow from them. All branches have similar shapes that depend on the species, but they have different sizes that depend on their age—this gives the appearance of self-similarity, as shown in the trees in Figure 14.1. Many other natural structures, as we saw in Figure 14.3, have fractal-like shape including lightning, coastlines, clouds, mountains, riverbeds, ferns, cauliflower, broccoli, the surface of the brain, and the surface of the lungs.

14.4

FRACTAL DIMENSION

Mathematicians use fractal dimension to help them understand the complexity and structure of a fractal. To compute fractal dimension, we first look at the familiar dimensions 0, 1, 2, and 3. We apply our observations about dimensions 0, 1, 2, and 3 to the structure of fractals to get the dimension of a fractal. Dimensions 0, 1, 2, and 3

A point has dimension 0, a line segment has dimension 1, a plane has dimension 2, and the space we live in has dimension 3. One way to understand dimensions is in terms of degrees of freedom. So at a point, there is no possible motion and we can say 0 degrees of freedom. Along a line, there is one degree of freedom, back and forth for a horizontal line. On a plane, there are two degrees of freedom: up and down as well as left and right. In space, there are three degrees of freedom, up and down, forward and backward, left and right. Figure 14.8 A point, interval, Another way to think of dimension is square, and cube cut into pieces usto take a shape—point, segment, square, or ing scaling factors 2 and 3. cube—and cut it into similar smaller shapes, as shown in Figure 14.8, where the scaling factors—the ratios of lengths of edges— are 2 and 3. Notice that when cut using scaling factor 2, the point still has one piece, the segment consists of two pieces, the square four, and the cube eight. When cut using scaling factor 3, the point still has one piece, the segment now consists of three pieces, the square nine, and the cube twenty-seven. These numbers are shown in Table 14.1. The numbers in the table satisfy the following equation: (scaling factor)dimension = number of pieces This pattern belongs to the familiar shapes—point, line, surface, and space—and corresponds to our usual idea of dimension. For this reason, mathematicians have

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TABLE 14.1

Scaling Factor and Number of Resulting Pieces.

Scaling Factor

Point

Segment

Square

Cube

1 2 3

1 1 1

1 2 3

1 4 9

1 8 27

decided to use this pattern to determine the dimensions for fractals as well. We see how this works in the next section. Dimensions of Geometric Fractals—The Koch Snowflake

We define the dimension of a geometric fractal using the pattern we found for the segment, square, and circle: (scaling factor)dimension = number of pieces Going from one stage to the next for each edge of the Koch snowflake, the scaling factor is 3 and the number of pieces that result is 4. Therefore, the equation giving the dimension d of the Koch snowflake is 3d = 4, To solve for the exponent d in this equation, we must use logarithms; taking a logarithm is the inverse operation of exponentiation. The logarithm of 105 , for example, is 5. You can calculate with logarithms using the scientific calculator found on most phones, getting, for example, log 4 ≈ 0.602. Logarithms have many mathematical properties that help in their calculation. The property that we will use to find the dimension of a fractal is that log ad = d log a for the number a. To solve the equation 3d = 4, take the logarithm of both sides, obtaining log 3d = log 4. Simplifying this using properties of logarithms gives d log 3 = log 4 so the dimension d of the Koch snowflake is d=

log 4 log 3

which is approximately equal to 1.262. This dimension tells us that the Koch snowflake is slightly more than a one-dimensional line but not yet a two-dimensional plane.

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Dimension of the Cantor Set

To go from one stage to the next stage of the Cantor set, the scaling factor is 3 and the number of pieces that we get is 2. This means that the dimension d of the Cantor set is 3d = 2. Again, we take the logarithm of both sides to obtain log 3d = log 2. Simplifying this gives d log 3 = log 2 so the dimension d of the Cantor set is d=

log 2 log 3

which is approximately equal to 0.631. This tells us that the Cantor set is somewhere between a point, which has dimension 0, and a line, which has dimension 1. Dimensions of Natural Fractals

Mathematicians have developed several empirical techniques to determine the dimension of a natural fractal like broccoli or the human lungs. Table 14.2 gives the fractal dimensions of several natural fractals along with the dimensions of geometrical fractals for comparison. TABLE 14.2

Dimensions of Natural and Geometric Fractals. Fractal Cantor Set Coastline of South Africa Coastline of Australia Coastline of Ireland Coastline of Great Britain Koch Snowflake Curve Coastline of Norway Sierpiński Triangle Broccoli Surface Human Brain Surface Cauliflower Cross Section Human Lung Surface

14.5

Dimension 0.631 1.05 1.13 1.22 1.25 1.262 1.52 1.585 2.7 2.79 2.8 2.97

APPLICATIONS OF FRACTALS

Fractals have many applications. Computer graphics software uses geometrical fractals to create artificial landscapes that mimic real landscapes. Fractal image compression software takes advantage of the fractal nature of many natural structures to

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reduce the size of image files to as much as one-fiftieth their original size. Fractal analysis of medical images can detect the presence of abnormalities in a patient’s organs. Antennas shaped like fractals are used in cell phones because they work the same as larger antennas but take up far less space. In engineering applications, space-filling fractal devices are used for high-precision fluid mixing.

EXERCISES Understand

14.1 Sierpiński Carpet. Construct the Sierpiński carpet fractal starting with a square as stage 0. Divide the square into nine parts, similar to a tic-tac-toe grid; color the middle square to get stage 1. Iterate (repeat) this procedure in the eight empty squares to get stage 2. Continue. If you use grid paper and draw the stage 0 square with 27 squares on a side, the grid squares will help you to construct stages 1, 2, and 3. 14.2 Dimensions. Cut a point, interval, square, and cube by scaling factor 4. How many pieces do you get in each case? What about if the scaling factor is 5? 14.3 Dimension of the Sierpiński Triangle. What is the scaling factor at each stage of the Sierpiński triangle? How many new pieces do you get from each previous piece? Use this information to find the dimension of the Sierpiński triangle. 14.4 Dimension of the Sierpiński Carpet. What is the scaling factor at each stage of the Sierpiński carpet, described in Exercise 14.1? How many new pieces do you get from each previous piece? Use this information to find the dimension of the Sierpiński carpet. 14.5 Coastlines. Compare the fractal dimensions of the coastlines in Table 14.1. Why do you think the coastline of Norway has the highest fractal dimension? Looking at pictures of the different coastlines can help you answer this question. Apply

14.6 2by2. Do you see any fractal structures in the two paintings by two artists that you have chosen? What do they contribute to the works? 14.7 3by1. Do you see any fractal structures in the three paintings by one artist that you have chosen? What do they contribute to the works? Create

14.8 Fractals around You. Go to an interesting natural place and sketch or photograph some natural fractal-like structures that you see around you. Create a painting or collage featuring these fractals.

Fractals  153

14.9 Creating with Fractals. Create an imaginary scene using fractals that you have seen already or new ones that you invent yourself. Build up the fractals in stages, using an iterative process. Review

14.10 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 14.11 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

READINGS • Viewpoints by Marc Frantz and Annalisa Crannell • African Fractals: Modern Computing and Indigenous Design by Ron Eglash

ARTISTS • Jackson Pollock (1912–1956); fractals • Salvador Dalí (1904–1989); fractals

Figure 15.1

Katsushika Hokusai. The Great Wave off Kanagawa, c. 1829–1833.

Mary Jo Toles. from: (notations): grinding wheels, 2020. Suite of four c Mary Jo Toles, 2020, all rights 14 × 11 silver gelatin high-voltage shadowgraphs. reserved.

Figure 15.2

CHAPTER

15

Dynamical Systems and Chaos

EXPLORATIONS • We regard the whole ocean as a dynamical system because it changes and moves according to natural laws. The woodblock print The Great Wave off Kanagawa by the Japanese printmaker Katsushika Hokusai (1760–1849) shows the surface of the ocean rising up in waves. How are the waves like a fractal? • The images in Figure 15.2 were created naturally, by exposing selected objects placed on a light-sensitive surface to high-voltage discharge from a horizontal resonating Tesla coil with added ambient light. What characteristics of fractals do you see in the images?

15.1

OVERVIEW

All around us, there is change—dynamism—that is governed by natural laws. We describe and understand this kind of change by using a dynamical system. “Dynamical” means changing, and “system” means orderly and following precise rules. However, as we can see in these images of water waves and electrical discharge, the states of a dynamical system may seem unruly or chaotic. In this chapter, we look at examples of dynamical systems, both mathematical and natural. We will see how the orderliness of a dynamical system can lead to what mathematicians call chaos. You’ll also see that dynamical systems can give rise to fractals.

15.2

DYNAMICAL SYSTEMS

A dynamical system is a natural system or a mathematical process that changes or evolves according to fixed laws. A dynamical system has two parts: first, the conditions—called states or phases—of the system, and second, the laws, rules, or procedures that transform one or phase of the system into the next state or phase. The weather is an important dynamical system—it is constantly changing according to natural laws. The state of the weather is described by the information contained DOI: 10.1201/9781003110972-15

155

156  Geometry for the Artist

in a weather report: temperature, humidity, wind direction and speed, cloud cover, probability of precipitation, and so on. The rules governing the weather are the laws of meteorology: hot air rises, a high pressure area moves into a low pressure area, and so on. We are interested in weather predictions—what the weather will look like in the future. Weather in the short term—a minute or an hour from now—is easy to predict; it is harder and more important to predict the weather for a week or a month from now. Similarly, the solar system is a dynamical system. It is constantly changing because each component is moving along its own orbit. Every aspect of the solar system is governed by natural laws like Newton’s laws of motion and the law of gravitational attraction. Astronomers are very good at predicting the motion of the components of the solar system for days, months, and years, but the very long-term state of the solar system is unknown. When studying a dynamical system, we start with an initial condition—the current or beginning state of the dynamical system—and we want to know the long-term behavior—what state or states the system is likely to be in the future. Weather prediction in the long-term is difficult not because of lack of knowledge of the rules governing the system—they are very well known to meteorologists—but because it is very difficult to measure accurately the current state of the weather. We will see that sensitivity to initial conditions is very important in natural dynamical systems and is the origin of chaos, an important concept that we consider in this chapter. Examples of dynamical systems around us include the following: Pendulum. Each state is a position of the pendulum along its trajectory. The rules governing the motion of the pendulum are gravity and friction. A Population. Each state is given by the number of individuals in the population. The rules governing the change in population are all factors that impact birth rate and death rate. An Ecosystem. Each state consists of all living and non-living aspects of the ecosystem. The rules of the system describe the movement of energy and matter within the ecosystem. Planetary Motion. Each state corresponds to the position of the sun, the planets, and their moons. Newton’s laws of motion and the laws of gravity, both classical and relativistic, govern their motion. Water Flowing in a Pipe. Each state corresponds to the direction and speed of the water measured at every point in the pipe. Water flow is governed by well-known laws of fluid dynamics. An Economy. The state of an economy can be described by a wide range of economic indicators such as gross national product, consumer price index, unemployment rate, interest rates, and rate of inflation. The rules governing an economy include government regulations and the law of supply and demand.

Dynamical Systems and Chaos  157

As you see by these examples, dynamical systems can be quite complicated. Even with sophisticated computer modeling, systems like the weather and the economy are very hard to predict. In this chapter, we will first look at very simple mathematical dynamical systems, but even those very simple dynamical systems will help us understand the sorts of behavior that can happen in natural systems.

15.3

MATHEMATICAL DYNAMICAL SYSTEMS

The simplest mathematical dynamical systems are based on the arithmetic operations of the real numbers. The Real Numbers

The real numbers correspond to the numbers along an infinite straight line, called the real number line, shown in Figure 15.3 with the numbers −3.5, −3, −2, −1, 0, 3, and 5.5 marked on the number line. Larger numbers are on the right, smaller on the left. We will use the arithmetic operations of addition, subtraction, and multiplication to create dynamical systems on the real numbers.

The real number line with points corresponding to the numbers −3.5, −3, −2, −1, 0, 3, and 5.5 indicated in red.

Figure 15.3

The Dynamical System Defined by Squaring

Our first dynamical system is the squaring operation: 22 = 4, 32 = 9, 12 = 1, 02 = 0, and so on. Remember that when you multiply two fractions, you multiply their numerators and their denominators, so  2

1 2

=

1 1 1 × = 2 2 4

 2

=

1 1 1 × = 3 3 9

1 3

And when you multiply two negative numbers together, you get a positive number, so −12 = −1 × −1 = 1 −22 = −2 × −2 = 4 

1 − 2

2

1 1 1 =− ×− = 2 2 4

The states or phases of the squaring dynamical system are all the real numbers.

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The rule that transforms one state to another is squaring. This is simple enough, but what is more interesting—as when we look at the weather or the economy—is what happens in the long term, when we repeat, or iterate, squaring. Let’s start with 2 as input. Squaring 2 gives 4 as output; we write this as 2 7→ 4 and say “2 maps to 4.” Squaring 4 gives output 16; squaring 16 gives 256, and so on. We write it in this way 2 7→ 4 7→ 16 7→ 256 . . . It is easy to see that iterating the squaring operation here will lead to ever larger numbers. To understand the behavior of this dynamical system as a whole, we must look at what happens to many other numbers. The number we start with is called the initial condition or seed, and the collection of all numbers that result from iterating the squaring operation on one particular seed is called the orbit of the seed. We look at these seeds to see what their orbits look like. 1 7→ 1 7→ 1 7→ 1 . . .

0 7→ 0 7→ 0 7→ 0 . . .

0.5 7→ 0.25 7→ 0.0625 7→ 0.00390625 . . .

−1 7→ 1 7→ 1 7→ 1 . . .

−2 7→ 4 7→ 16 7→ 256 . . . The number 0 keeps going to 0 at each step, and the number 1 keeps on going to 1 at each step; each of these numbers is a fixed point of the dynamical system. Like 2, the number −2 maps to ever larger and larger numbers; we say that 2 and −2 each have an unbounded orbit. Numbers like −1 and 0.5 have bounded orbit. Continuing to look at more seed values for this dynamical system, we get the behavior shown in Figure 15.4. The numbers with unbounded orbit are shown in blue, with arrows indicating the direction they move under iterations of the squaring operation. The two fixed points are in green; and numbers (except −1) that are not fixed points but still have a bounded orbit are in red with arrows indicating that they move toward 0 under iterations of the squaring operation. The special number −1 is in orange, and its orbit is just the number 1. The numbers −1, 0, and 1 have special

The phase line of the squaring dynamical system. The numbers with unbounded orbit are shown in blue, with arrows indicating the direction they move under iterations of the squaring operation;. The two fixed points are in green. All numbers (except −1) that are not fixed points but still have a bounded orbit are in red with arrows indicating that they move toward 0 under iterations of the squaring operation. The special number −1 is in orange, and its orbit is just the number 1. Figure 15.4

Dynamical Systems and Chaos  159

behavior that is important to look at. Numbers very close to 0 are mapped closer and closer to 0; we call 0 an attractor for this dynamical system. Since it is also a fixed point, we can also call it an attracting fixed point. Numbers very close to −1 or 1 are mapped farther and farther away, so each is unstable or a repellor. Since 1 is also a fixed point, it is a repelling fixed point. The behavior of numbers close to −1 and 1 is very interesting. Just a slight change in a seed near −1 or 1 can result in a large change in the outcome. The seeds −1.1, −1, and −0.9 have very different orbits: −1.1 has unbounded orbit, −1 maps to 1 and stays there, and −0.9 has bounded orbit, getting closer and closer to 0. This phenomenon is called chaos: a very small change in the initial condition or seed of a dynamical system can result in a large change in the long-term outcome. The Dynamical System x 7→ x2 + 1

The dynamical system x 7→ x2 + 1 has very simple behavior. Every number gets mapped to a larger number, as you can see from these examples. There are no fixed points. 0 7→ 1 7→ 2 7→ 5 7→ 26 . . .

1 7→ 2 7→ 5 7→ 26 . . .

−1 7→ 2 7→ 5 7→ 26 . . .

0.5 7→ 1.25 7→ 2.5625 7→ 7.5664 7→ 58.2505 . . .

−0.5 7→ 1.25 7→ 2.5625 7→ 7.5664 7→ 58.2505 . . . 2 7→ 5 7→ 26 . . .

−2 7→ 5 7→ 26 . . .

From these examples, you can also see that a number x and its negative −x have the same orbit. Every number has unbounded orbit, as shown in Figure 15.5.

The phase line of the dynamical system defined by x 7→ x2 + 1. Every number has unbounded orbit. Figure 15.5

The Dynamical System x 7→ x2 − 1

The dynamical system x 7→ x2 − 1 has some new behavior that we haven’t seen before. The number 0 maps to −1, which then maps to 0 and so on, back and forth. We say that the number 0 has a periodic orbit. It has period 2, since there are two different numbers in the orbit. The number −1 maps to 0, then back to −1, and so

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on; it also has a periodic orbit of period 2. 1 7→ 0 7→ −1 7→ 0 7→ −1 . . .

0 7→ −1 7→ 0 7→ −1 7→ 0 7→ −1 . . .

−1 7→ 0 7→ −1 7→ 0 7→ −1 7→ 0 . . . 2 7→ 3 7→ 8 7→ 63 7→ 3968 . . .

−2 7→ 3 7→ 8 7→ 63 7→ 3968 . . .

0.5 7→ −0.75 7→ −0.4375 7→ −0.8086 7→ −0.3462 7→ −0.880162075 . . .

The phase line of the dynamical system x 7→ x2 − 1. Numbers from −1 to 1 are shown in red and have bounded orbits. The numbers 0 and 1 have periodic orbits, alternating between −1 and 1. Numbers greater than 1 and less than −1 are shown in blue and have unbounded orbits.

Figure 15.6

We find the feature of chaos in this dynamical system near the points −1 and 1. Just a slight change in the seed value when we are at either of these points can result in a dramatic change in the long-term behavior, either bounded or unbounded.

15.4

CHAOS

To understand the mathematics idea of chaos better, let’s look more closely at the repellors −1 and 1 for the squaring dynamical system. If a number is near either one of them, it could have an unbounded orbit or move in toward 0, depending on which side of the repellor the number is on. This is an example of mathematical chaos— it doesn’t mean the dynamical system is disorderly because, in fact, the dynamical system is governed by completely orderly rules. Mathematical chaos has the specific meaning of extreme sensitivity to initial conditions. If there is a slight change in a number near −1 or 1, the long-term outcome of iterations under the rule of the dynamical system can be completely different—bounded or unbounded. Chaos results if there is an initial condition that has the property that a slight change leads to a completely different outcome. A dynamical system is completely determined by the rules of the system, and the short-term and long-term outcomes are completely determined by the initial condition. So we may be surprised by discovering chaos, but we see it in many dynamical systems that are completely determined by precise rules.

“Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set off a Tornado in Texas?” Edward Lorenz Title of address to the America Association for the Advancement of Science, 29 December 1979.

Dynamical Systems and Chaos  161

For example, meteorologists use precise mathematical equations to predict the weather. The initial condition—the current state of the weather—used to make predictions consists of many different measurements, such as temperature, humidity, and wind speed and direction. Physical measurements inevitably have some uncertainty due to the nature of the measuring devices. The weather model is very complex and has the property that it is chaotic at every point in the phase space; unlike the squaring dynamical system which is chaotic only for initial values −1 and 1, the weather is chaotic for every initial condition. This means that a slight change in the weather conditions at any one instant will change dramatically the long-term prediction of the weather—and leads to the now-classic description of chaos for the weather: the flap of a butterfly’s wings in Brazil can set off a tornado in Texas. The mathematical idea of chaos in a dynamical system is just this: chaos is extreme sensitivity to an initial condition. A dynamical system is chaotic if it is extremely sensitive to initial conditions; this means that a very small change early in the evolution of a system can produce a very great effect later on. The first example of chaos in a dynamical system was described by Edward Lorenz (1917–2008), a meteorologist at MIT, who used a dynamical system to describe weather patterns. In 1961, he was using a computer to describe the weather when he discovered the phenomenon of chaos. He found that if he changed the initial condition put into the dynamical system, even as little as a ten-thousandth, the longterm outcome of the dynamical system was changed dramatically. This implied that the weather could be changed by something as small as the flapping of a butterfly’s wings. Lorenz termed this outcome the “butterfly effect.”

15.5

DYNAMICAL SYSTEMS AND CHAOS IN ART

Understanding dynamical systems and chaos can help artists appreciate the fractals around us in nature—clouds, lightning, ferns, waves, and so on—in terms of the behavior of the natural processes that create them. Many artists have found ways to use dynamical systems to create art. For example, Jackson Pollack used paint dripped on canvas to create fractal-like images, and MJ Toles (Figure 15.2) used electric discharges in her work. Some artists find ways to show extreme sensitivity to initial conditions, whether in human life or in the natural world. Sandro Botticelli shows an event from the Greek classical epic the Iliad in The Judgement of Paris, Figure 15.7. The Trojan warrior Paris can choose which of the three goddesses, Hera, Athena, or Aphrodite, is the most beautiful. His choice of Aphrodite, the goddess of love, led to the Trojan war. His two other possible choices would likely have resulted in dramatically different outcomes.

15.6

CHAOS AND CHANCE

It is natural to ask about the difference between chaos and chance. A dynamical system is chaotic if we know the outcome of each step of the system but can’t predict the long-term outcome with accuracy.

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Figure 15.7

Sandro Botticelli. The Judgement of Paris, 1445–1510.

Instances where chance is involved, like flipping a coin, are different. We have no idea at all what the next outcome will be, but we can predict the long-term behavior with quite good accuracy using the laws of probability.

EXERCISES Understand

15.1 Dynamical System with Negatives. Compute orbits like the ones shown in Section 15.3 for the dynamical system defined by x 7→ −x. What do you notice about this dynamical system? What numbers have bounded orbits? Draw a phase line for this dynamical system. 15.2 Dynamical System with Reciprocals. Compute orbits like the ones shown in Section 15.3 for the dynamical system defined by x 7→ 1/x. You can use any number for x except 0. What do you notice about this dynamical system? What numbers have bounded orbits? Draw a phase line for this dynamical system. 15.3 The Dynamical System x 7→ x2 + x. Compute orbits like the ones shown in Section 15.3 for the dynamical system defined by x 7→ x2 + x. What numbers have bounded orbits? What numbers have unbounded orbits? Are there any fixed points? Does this dynamical system exhibit chaos at any points? Draw a phase line for this dynamical system. 15.4 The Dynamical System x 7→ x2 − x. Compute orbits like the ones shown in Section 15.3 for the dynamical system defined by x 7→ x2 − x. What numbers have bounded orbits? What numbers have unbounded orbits? Are there any fixed points? Does this dynamical system exhibit chaos at any points? Draw a phase line for this dynamical system.

Dynamical Systems and Chaos  163

Apply

15.5 In Daily Life. Can you think of situations in daily life that exhibits chaos—in other words, a situation where a very small change early on results in a very different outcome? Explain. Create

15.6 Natural Fractal. Draw or photograph a natural fractal that you think is the result of a natural dynamical system. You can use Figures 15.1 and 15.2 for ideas. Say what dynamical system you think created the fractal. 15.7 Create Your Own. Create an abstract fractal. For ideas, use any of the fractals or dynamical systems that you have seen so far. Review

15.8 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 15.9 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

READINGS • Arcadia by Tom Stoppard • Chaos by James Gleick

Robert W. Brooks and Peter Matelski. The first published image of the Mandelbrot set, 1978. Figure 16.1

Wolfgang Beyer. The Mandelbrot set in black with its surrounding region in color. GNU Free Documentation License. Figure 16.2

CHAPTER

16

The Mandelbrot Set

EXPLORATIONS • How are the two images in Figure 16.1 and Figure 16.2 alike? How are they different? • Watch an online video of Mandelbrot zooms or create your own zoom at https: //math.hws.edu/eck/js/mandelbrot/MB-info.html. What properties does the Mandelbrot set have that make it appear to be a fractal?

16.1

OVERVIEW

The Mandelbrot set, the most iconic mathematical image of recent times, is based on a very simple formula that we will look at in this chapter. Unlike the simple dynamical systems of the previous chapter that use the properties of the real numbers, the Mandelbrot set uses the properties of a two-dimensional extension of the real numbers, the complex numbers. Iteration of a very simple formula using complex numbers results in the Mandelbrot set. In this chapter, we first study complex numbers and then use them to develop a dynamical system that leads to the Mandelbrot set.

Think not of what you see, but what it took to produce what you see. Benoit Mandelbrot

16.2

THE COMPLEX NUMBERS

The real numbers are constructed in stages from the whole numbers, 0, 1, 2, 3, . . . . First, negative numbers were added to make subtraction possible. Then, √ √ fractions were added to make division, except by 0, possible. Then numbers like 2 and 3 DOI: 10.1201/9781003110972-16

165

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were added to find square roots of positive numbers. The number π was added to be able to find the area of a circle. Even with all the real numbers, you cannot find the square root of a negative number. But mathematicians often want the square root of a negative number, so they once again extended the number system, getting the complex numbers. In the complex number system, every number has a square root. To get answers to all possible subtraction problems, we only need to add only one number, −1, to the positive numbers and then use multiplication to get numbers like −10 and −1/2. To get square√ roots of all possible negative numbers, we also just need to adjoin only one number, −1, and then use addition, subtraction, multiplication, and division to get all of the complex numbers. The square root of −1 was at first considered to be imaginary when compared with the real numbers and was represented by i. The special property that the number i has is this: √ √ −1 × −1 = i × i = i2 = −1 We take all possible sums, products, and differences of real numbers with i, getting complex numbers like 5, 3i, and 5 − 2i. The convention is that for the sum of a real number with a multiple of i, the real number is written before the multiple of i. The result is the complex numbers, where every number—positive or negative—has a square root. Computing with complex numbers follows the same rules as ordinary arithmetic with the additional rule that i × i = −1. Here are a few examples. 3i + 4i (4 − 3i) + (−6 + 7i) 2i × 3 2i × 3i i(3 + 4i) (4 − 3i)(−6 + 7i)

= = = = = =

7i −2 + 4i 6i −6 −4 + 3i −3 + 46i

With complex numbers, there is a square root for every number. A few examples are shown here. √ 2i = 4i2√= 4 × −1 = −4 √ since 2i ×√ √−4 = 2i, −8 = 2 2i, since 2 2i × 2 2i = 4 × 2 × i2 = 8 × −1 = −8 √ 16 + 30i = 5 + 3i, since (5 + 3i) × (5 + 3i) = 25 + 15i + 15i + 9i2 = 16 + 30i

16.3

GRAPHING COMPLEX NUMBERS

Similar to the way real numbers can be located on a straight line, we can locate complex numbers in a plane, called the complex plane, as shown in Figure 16.3. The horizontal line is the real axis, with units of 1. The points B = 0, C = 1, and E = 4 are real numbers and are plotted on the real axis. The vertical line is the imaginary axis with units of i, and the point A = 3i is on the imaginary axis.

The Mandelbrot Set  167

Complex numbers plotted in the complex plane. A is the point corresponding to the complex number 3i, B to 0, C to 1, D to 3 + 2i, E to 4, F to 3 − 2i, G to −3 − 2i, and H to −5 + 4i. Figure 16.3

16.4

CONSTRUCTING THE MANDELBROT SET

When studying dynamical systems that use complex numbers, Benoit Mandelbrot (1924–2010) discovered a new set, now called the Mandelbrot set, that was intricate and beautiful and has become the icon of the study of fractals and chaos. Recall that in the previous chapter we looked at dynamical systems like x 7→ x2 , x 7→ x2 + 1, x 7→ x2 − 1, and so on. Mandelbrot was interested in looking at properties of dynamical systems like z 7→ z 2 , z 7→ z 2 + 1, and z 7→ z 2 − 1, where the variable z represents a complex number instead of a real number. In fact, he was interested in studying all possible dynamical systems of this form, for every possible complex number. Just as with the dynamical systems we saw before, it is the longterm behavior of these complex systems that is interesting. So Mandelbrot asked for which initial conditions are orbits bounded, for which is the orbit unbounded, and whether the dynamical system exhibits chaos. However, as you might imagine, complex dynamical systems are more difficult to study than real dynamical systems. Luckily, Mandelbrot found that he could get the information that he wanted about the particular dynamical system z 7→ z 2 + c just by studying what happens to the initial condition 0, determining only whether 0 has a bounded orbit or unbounded orbit. If the orbit of 0 for the dynamical system z 7→ z 2 + c is bounded, then the complex number c is in the Mandelbrot set. If the orbit of 0 for the dynamical system z 7→ z 2 + c is not bounded, then the complex number c is not in the Mandelbrot set. Let’s look at a few examples, shown in Table 16.1. We give the first few points in the orbits of 0 for the dynamical system z 7→ z + c for nine different values of c. It is easy to see whether the orbit is bounded or unbounded for these numbers, even though we only look at a few points in the orbits.

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TABLE 16.1 Values of complex numbers c and the first three points in the orbits of the complex number 0 under the mapping z 7→ z 2 + c. The last column gives long-term behavior.

Value of c

First Iteration

Second Iteration

Third Iteration

Long-term Behavior

0 1 i −1 −i 1+i −1 + i 1−i −1 − i

0 2 −1 + i 0 −1 − i 1 + 3i −1 − i 1 − 3i −1 + i

0 5 −i −1 i −7 + 7i −1 + 3i −7 − 7i −1 − 3i

0 26 −1 + i 0 −1 − i 1 − 97i −9 − 5i 1 + 97i −9 + 5i

Stable Unstable Stable Stable Stable Unstable Unstable Unstable Unstable

Figure 16.4 The Mandelbrot set and points 1, −1, i, −i, 1 + i, 1 − i, −1 + i, and c 2022 Geek3 / GNU-FDL. −1 − i from Table 16.1. Image of Mandelbrot

The Mandelbrot set is usually shown in black, while the points not in the Mandelbrot set are shown in color. Points that get larger at approximately the same rate are shown in the same color, Figure 16.5.

The Mandelbrot Set  169

The Mandelbrot set. Points belonging to the Mandelbrot set are in black, while points with unbounded orbits are in color. Points escaping at approximately the same rate are in the same color. Figure 16.5

16.5

PROPERTIES OF THE MANDELBROT SET

The properties of the Mandelbrot set are truly amazing. Mathematicians have discovered many things about the Mandelbrot set, but there are still many questions they have still not resolved. Artists have been inspired by the Mandelbrot set since its discovery. Here are some well-known properties of the Mandelbrot set. • The Mandelbrot set has bilateral symmetry with the real axis as horizontal mirror line. • All points outside the circle with radius equal to 2 centered at the point 0 have unbounded orbits and are not in the Mandelbrot set. • The interval from −2 to 1/4 is in the Mandelbrot set. • The Mandelbrot set is approximately self-similar, though not exactly selfsimilar. Nevertheless, it is still considered to be a fractal. • The Mandelbrot set is connected. This means there are fine threads or tendrils between black regions of the Mandelbrot set, even if they are too fine to be seen on a computer. • The fractal dimension of the boundary of the Mandelbrot set is 2. This means that the boundary looks like a plane rather than like a curve. • For every fraction m n in lowest terms, there is an associated limb attached to the main part of the Mandelbrot set. • Parts of the Mandelbrot set have names based on their appearance, including seahorse valley, seahorse tail, antenna, double hook, and satellite.

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• The number π and the Fibonacci sequence can be found inside the structure of the Mandelbrot set. • The Mandelbrot set exhibits chaos near its boundary, where a very small change in position can determine whether a point c is in the Mandelbrot set or not.

EXERCISES Understand

16.1 Computations with Complex Numbers. Compute the following. Remember that √ i = −1 and i2 = −1. a. 8i + 3i

d. 2i × (4 − 2i)

b. 12i − 7i

e. (2 − i)(5 + 2i)

c. 3i × 5i

f. (5 + 2i)2

16.2 Square Roots. Show that 3 − i is the square root of 8 − 6i by squaring 3 − i. 16.3 Mandelbrot Set. Show that 0.5 is not in the Mandelbrot set by iterating z 2 + 0.5 for z = 0. Three iterations are enough. 16.4 Mandelbrot Set. Show that −0.5 is in the Mandelbrot set by iterating z 2 − 0.5 for z = 0. Three iterations are enough. Create

16.5 Inspired by the Mandelbrot Set. Create an image inspired by the Mandelbrot set. 16.6 Create Your Own Zoom. Locate a web application such as the Mandelbrot Viewer at https://math.hws.edu/eck/js/mandelbrot/MB-info.html that allows you to create zooms of the Mandelbrot set using your choice of colors. Use the web app to find a zoom of the Mandelbrot set that you find interesting or appealing. Save it as a picture and give it a title that captures your feelings or impressions about the image. Review

16.7 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 16.8 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

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READINGS • Chaos by James Gleick • The Fractal Geometry of Nature by Benoit Mandelbrot

Taylor & Francis Taylor & Francis Group

http://taylorandfrancis.com

V Curves, Spaces, and Geometries

Georgia O’Keefe. Hibiscus with Plumeria, 1939. (Smithsonian American Art Museum. c 2022 Georgia O’Keefe Museum / Artists Gift of Sam Rose and Julie Walters. Rights Society (ARS), New York)

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Figure 17.1

Franz Marc. Tiger, 1912.

Figure 17.2

Franz Marc. Red Deer II, 1912.

CHAPTER

17

Lines and Curves

EXPLORATIONS • The painting Tiger by Franz Marc, shown in Figure 17.1, shows a tiger in an abstract landscape. Where are there straight lines in the picture? The straight lines are vertical, horizontal, or diagonal—which predominate? Are there any curves? What impression do the straight lines give you? • The painting Red Deer II, shown in Figure 17.2, also by Franz Marc, shows two deer in an abstract landscape. Can you find any straight lines in the picture? Some of the curves in the picture are gentle, while others are strongly curved. Which predominate, the gently curved or the sharply curved? What impression do the curves give you? • Compare and contrast the two paintings by Franz Marc. Why do you think he used predominantly straight lines for the tiger and predominantly curved lines for the deer?

17.1

OVERVIEW

Lines and curves in a painting give details to individual shapes and boundaries between shapes. They move the viewer’s eye in, around, and out of a work of art. Understanding their properties gives insight into how a work of art creates an impression on the viewer. For example, in Tiger, Figure 17.1, the repetition of straight lines conveys the sharpness of the tiger. The many diagonal lines give the feeling of the potential for rapid movement by the tiger. In Red Deer II by Franz Marc, Figure 17.2, there are curves everywhere, even in the mountains, conveying the gentle feeling of two deer playing in a field. Curves of different sizes and shapes along with strongly contrasting colors keep the eye of the viewer engaged. We have already seen lines and curves in many contexts—mirror lines, orthogonals and transversals, circles, and conic sections. In this chapter, you will learn how to describe lines and curves, and we will look at examples in art and nature. You will learn how to DOI: 10.1201/9781003110972-17

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• Describe a line using its slope • Describe the curvature of a circle or curve • Recognize many kinds of curves that we see in nature • Evaluate the effect of lines and curves in a painting

17.2

STRAIGHT LINES AND THEIR SLOPES

A straight line looks like a stretched string and can be finite or infinite in length. We use the term “segment” or “line segment” specifically for a straight line with finite length. Straight lines and line segments are clean, simple, severe, and architectural. While there are many curves in nature, straight lines and line segments occur only rarely in nature: a ray of light, the edge of a crystal, and a stretched string. In art, lines convey a feeling of simplicity. Piet Mondrian uses line segments as the rigid boundaries of blocks of solid colors in Composition (No. 1) Gray-Red, Figure 17.3, which is an abstract arrangement of squares and rectangles. The picture has a balanced and restful feeling. The differences in the sizes of the rectangles and the single red rectangle create enough liveliness to sustain the viewer’s interest.

Figure 17.3

Piet Mondrian. Composition (No. 1) Gray-Red, 1935. The Art Institute

of Chicago. There are many ways to draw a straight line. Artists painting frescoes during the Renaissance used a stretched string. You can fold a piece of paper or just use a ruler. To completely describe a straight line, we can • Give two distinct points • Give the two endpoints • Give one point and the direction of the line

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The slope of a line is a convenient way to give its direction. For any two points on a line, the rise is the vertical distance between the two points and the run is the horizontal distance between the two points. For the third line in Figure 17.4, the rise between the two points is 3 and the run is 4. The slope is defined to be the rise divided by the run for any two points on the line, so the slope of this line is 3/4. A steep line has a large slope since there is a large rise for a given run; a horizontal or flat line has slope 0 since the rise is 0 for any run. The slope of a vertical line is infinite since the run is 0 for any rise. A line that goes down to the right instead of going up to the left has negative rise for a positive run, so its slope is negative. Different lines and their slopes are shown in Figure 17.4.

Figure 17.4

Straight lines with different slopes.

The direction of a line segment in a picture creates an influence in the viewer; many line segments with similar directions reinforce that influence. By looking at straight lines in different pictures, you will see how they give different impressions. Artists can use lines in many ways: • A horizontal line such as a person lying down can be calm and restful. • A horizontal line such as the horizon line of a perspective picture can give a feeling of expansion. • A horizontal line such as a floor can give equilibrium or stability. • A vertical line such as a tree or column with its basis in the ground can convey height and upward expansion or support, balance, and firmness. • A diagonal line conveys movement, dynamism, or even a sense of agitation or of being off-balance. Look carefully at the vertical, horizontal, and diagonal lines in Conversation Piece, by Charles Sheeler, shown in Figure 17.5. Notice how important the diagonal lines of the roofs and shadows are in the picture. They enliven the stability of the horizontal and vertical lines and tie the different elements of the picture together. In Greetings from a Manhattan Artist, most of the lines are diagonal, creating the impression of the constant movement and dynamism of Manhattan.

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Charles Sheeler. Conversation Piece, 1952 (Reynolda House Museum of American Art, Museum Purchase).

Figure 17.5

17.3

CURVES AND CURVATURE

Looking around us in nature or the man-made environment, we see a great variety of curves—some as unique as the outline of a friend’s face and some as commonplace as the edge of a coin. To understand the many curves that we see around us, we compare them to what is simplest and most familiar: the straight line and the circle. We use a tangent to a curve to compare the curve to a straight line and a circle of curvature to compare a curve to a circle. Tangent Line

The straight line that “looks more like” a curve than any other straight line at a point is the tangent line at that point, shown in Figure 17.6. The tangent line gives us the direction of the curve. If we were walking along the curve, looking straight ahead, we would be looking along the tangent line. The slope of the tangent line at a point on the curve tells us whether the curve is increasing, decreasing, or level; if the tangent line is steep, it tells us that the increase or decrease is rapid.

Circle of Curvature

To get an intuitive idea of curvature, think in terms of driving along a curved road, where large curvature makes us turn the wheel more sharply and small curvature requires less turning. If we want to describe how “curved” a line is, we compare it to a circle. The circle that “looks more like” a curve than any other circle at a point on

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Figure 17.6

A curve with two tangent lines.

the curve is the circle of curvature at that point. The circle of curvature is the circle that fits the curve at that point better than any other circle. Larger circles have less curvature, and smaller circles have greater curvature. Since a large circle has a large radius and small curvature, while a small circle has large curvature and small radius, a good measure of the curvature of a circle is the reciprocal of the radius of the circle. Recall that the reciprocal of a number is just 1 divided by the number. Thus, curvature of a circle =

1 . radius

Figure 17.7 shows several circles and their curvature computed using this formula. Notice that the smaller the circle is, the larger the curvature is, and when the circle is larger, the curvature is smaller.

Figure 17.7

Four circles, each with a different radius and curvature.

If you have a curve that is not a circle, you can determine the curvature at any specific point by finding the circle that is “closest to the curve” at that point and then determining the curvature of that closest fit circle. For each of the points A, B, and C on the curve in Figure 17.8, we show the circle of curvature. To determine the curvature, measure the radius and then find its reciprocal. You can also just compare the curvature at different points by comparing the circles. This means that the curvature at point A is largest while the curvature at point C is smallest.

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Figure 17.8

The circle of curvature at each of the points A, B, and C on a curve.

Line of Beauty or S-Curve

A curve in the shape of the letter “S” has circles of curvature that change size as we move along the curve, but more significantly, the circle of curvature changes from one side of the curve to the other. In fact, between the two parts of the curve where the circles of curvature lie on opposite sides, there is a point where the circle of curvature would have infinite radius, in other words where the curve is best approximated by a straight line, not a circle. Such a point is called a point of inflection.

An S-curve. The circles of curvature at A and C are on opposite sides of the curve, while the circle of curvature at B is so large that its curvature is close to 0 and it looks almost like a straight line; the point B is a point of inflection of the curve. William Hogarth (1697–1764) called any S-shaped curve, implied or explicit, a line of beauty. He felt that an S-shaped curve was beautiful and harmonious. We see S-curves in the foreground of Wheat, Figure 21.4, where dried leaves of the wheat have graceful S-curves. They add a feeling of grace and delicacy to the otherwise formal painting. In Red Deer II, Figure 17.2, there are many S-curves, drawing attention to the beauty and grace of the deer. In contrast, in Greetings from a Manhattan Artist, Figure 21.5, the letter “S” on the word “EATS” is the only S-shape, but it is not used as a line of beauty. The picture is not meant to convey delicacy and grace, but rather the commotion of Manhattan. Figure 17.9

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17.4

SPECIAL CURVES

We now look at some of the special curves we see around us. Parabola

The arc of a fountain or the path of a bouncing ball is a parabola, one of the conic sections we studied in Chapter 10. Catenary

A chain that is allowed to hang freely from two points forms a curve called a catenary, Figure 17.10. This is the shape that a suspension bridge will fall into by the force of gravity. While the catenary may at first look like a parabola, they are two very different curves, as shown in Figure 17.11.

A hanging chain forming a catenary curve (image by Jim Fairchild). Figure 17.10

Figure 17.11

A catenary and a

parabola.

Caustic Curves

The path a ray of light takes depends on what it encounters as it travels. Light that hits a mirror or other reflective surface is bounced back or reflected. Light that travels from one medium to another is refracted or bent. Reflection and refraction create a variety of curves, called caustic curves. The term “caustic” means burning and refers to the burning quality of light. Caustic curves are all around us in our daily life even if we may not have paid attention to them. Light reflected from the smooth sides of a cup containing water, coffee, or tea creates the familiar looped curves shown in Figure 17.13. Light reflected and refracted by the irregular surface of water in a pond, lake, or the ocean creates a web-like pattern of intersecting curves, like the pattern shown in Figure 17.12. Artists find many ways to represent these curves in their work. In still-life paintings, the caustic curves may be represented accurately, but more often caustic curves are shown in a stylized or abstract way, as in Painted Water Glasses, by Janet Fish, Figure 10.1, or David Hockney’s patterns of light on water in swimming pools.

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Spiral

You are familiar with many kinds of spirals—the arrangement of seeds in a sunflower, the curl of a plant tendril, the arms of a galaxy, and the stripes on a candy cane. A spiral is the result of circular motion combined with outward or upward motion; for example, to paint a stripe on a candy cane, you must move your brush up while moving it around the cane. Spirals combine the closed feeling of a circle along with outward or upward expansion. In patterns and designs, spirals contribute a feeling of controlled expansion.

Figure

17.12

Caustic

curves in water (image by Jim Fairchild).

17.5

Figure

17.13

Caustic

Figure 17.14

Spiral.

curves (image by Jim Fairchild).

HOW ARTISTS USE LINES AND CURVES

Artists use lines in pictures to move our eye around the picture—they take us from here to there. How the artist moves our eye around the picture contributes to our feelings when seeing the picture. Does our eye flow around the picture gracefully? Are we pulled around in different directions, up and down, back and forth? Do we move slowly or quickly? Are we forced to stop somewhere along the way? Understanding the properties of lines and curves helps us answer these questions, and answers to these questions help us understand a picture.

EXERCISES Understand

17.1 Tangent Lines for Three Points on a Curve. Figure 17.15 shows a curve with points A, B, and C marked. Sketch the line tangent to the curve at each of the three points. 17.2 Comparing Curvature for Three Points on a Curve. Figure 17.15 shows a curve with points A, B, and C marked. For each of these points, the circle of curvature

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is shown. Arrange the points A, B, and C in order of the size of the curvature, from largest to smallest. 17.3 Comparing Curvature for Six Points on a Curve. For the curve in Figure 17.16, arrange the points A, B, C, D, E, and F in order of the size of the curvature, from largest to smallest.

Figure 17.15

A curve with points A, B, and C and their circles of curvature.

Figure 17.16

A curve with the points A, B, C, D, E, F , and G and their circles of

curvature. 17.4 Find S-curves in the pictures Poem Scroll with Deer by Tawaraya Sotatsu, Figure 13.7, and Jeanne Hébuterne in yellow sweater by Amedeo Modigliani, Figure 20.8. For each picture, explain why S-curves are appropriate. 17.5 Mark and label five points on the curve shown in Figure 17.17 and list them in order of the size of the curvature, from largest to smallest. Also, mark and label an inflection point. 17.6 Book Cover. Look at the lines and curves on the cover of this book. What do you notice about the curvature of nearby curves? Why is this important for the design as a whole? 17.7 Catenary Curves. Make a catenary by hanging a chain (such as a chain necklace or keychain) between two points and draw or photograph it. Now change the

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Figure 17.17

A curve.

two points several times, making them closer or farther apart, and draw or photograph them each time. Where is the curvature on the catenary the greatest? How does the curvature of the catenary change when the two points are closer or farther apart? Apply

17.8 Yellow-Red-Blue. Trace some important lines and curves in the picture RedYellow-Blue, shown in Figure 1.2; the lines can be explicit or implied. Discuss the use of lines and curves in the picture by answering some of the following questions: are the lines vertical, horizontal, or diagonal? Do the curves have large curvature or small curvature? Are there any S-shaped curves? Are there any other special curves? How do the lines and curves move your eye around the picture? How do these lines and curves create the impression that you get from the picture? 17.9 2by2. Trace some important lines and curves in the two pictures by two artists that you have chosen. Trace some important lines and curves in the picture; the lines can be explicit or implied. Discuss the use of lines and curves in the picture by answering some of the following questions: are the lines vertical, horizontal, or diagonal? Do the curves have large curvature or small curvature? Are there any S-shaped curves? Are there any other special curves? How do the lines and curves move your eye around the picture? How do these lines and curves contribute to what you already know about the pictures? 17.10 3by1. Trace some important lines and curves in the three pictures by one artist that you have chosen. Trace some important lines and curves in the picture; the lines can be explicit or implied. Discuss the use of lines and curves in the picture by answering some of the following questions: are the lines vertical, horizontal, or diagonal? Do the curves have large curvature or small curvature? Are there any S-shaped curves? Are there any other special curves? How do the lines and curves move your eye around the picture? How do the lines and curves contribute to what you already know about the pictures?

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Create

17.11 Lines and Curves around You. Sketch some of the lines and curves you see around you. Label vertical, horizontal, and diagonal lines. Label curves with large curvature and ones with small curvature. Do you see any of the special curves studied in this lesson? 17.12 Lines and Curves in the Environment. Go to an interesting place on campus or near your home. Sketch pictures (or take photographs) of some of the straight lines and curves that you see around you. Label vertical, horizontal, and diagonal lines. Label curves with large curvature and ones with small curvature. Do you see any of the special curves studied in this lesson? 17.13 Create Your Own. Draw an abstract picture based on the straight lines and curves that you found in Exercise 17.11 or Exercise 17.12. Review

17.14 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 17.15 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

READINGS • Point and Line to Plane by Wassily Kandinsky • Pedagogical Sketchbook by Paul Klee • A Seduction of Curves by Allan McRobie • Exploring Curvature by James Casey

ARTISTS • Wassily Kandinsky (1866–1944), Russian painter; lines, curves, and circles • Piet Mondrian (1872–1944), Dutch painter; straight lines • Minnie Evans (1892–1987), American artist; curves and spirals • David Hockney (b. 1937), English painter; caustic curves

Grant Wood. Spring Turning, 1936. Reynolda Museum. Gift of Barbara c 2021 Figge Art Museum, successors to the Estate of Nan B. Millhouse Copyright Wood Graham/licensed by VAGA at Artist Rights Society (ARS), NY.

Figure 18.1

Figure 18.2

Robert S. Duncanson. Ellen’s Isle, Loch Katrine, 1871.

CHAPTER

18

Surfaces

EXPLORATIONS • We see hills and valleys in Spring Turning by Grant Wood, Figure 18.1. What would be your experience traveling over a hill and through a valley? • The rolling hills in the background of Ellen’s Isle, Loch Katrine by Robert S. Duncanson, Figure 18.2, are contrasted with the flatness of the lake. What is the overall feeling you get from the picture? What is the difference between the feeling of the lake with the feeling of the hills?

18.1

OVERVIEW

So far, we have been looking at shapes extrinsically—we are looking at them from the outside. In this chapter we look at surfaces intrinsically, with reference to themselves rather than to the space that contains them. In Chapter 17, we studied lines and curves that are drawn on a flat plane. This is an extrinsic way to look at them—they are on the plane, and we are looking at them from the outside to see how they fit into the space around them. Looking at surfaces intrinsically means that we look at them from within the surface, without reference to how they fit into the surrounding space. One way we can compare these two viewpoints is to think about how a horse and an ant look at an apple. The horse sees it in your hand, and with one bite, it is gone—this is the extrinsic way of looking at the apple. An ant crawls around the surface of the apple, paying no attention to the surroundings outside the apple—this is the intrinsic way of looking at the apple. When we look at surfaces intrinsically, we find two new geometries, quite different from Euclidean geometry—elliptic geometry and hyperbolic geometry; we will find these geometries everywhere around us. First, we consider what a surface is; then we review Euclidean geometry, the geometry that is most familiar to us; then we’ll consider these two new geometries.

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18.2

SURFACES

As we turn our attention to surfaces, we will be comparing them to the simplest and most familiar surfaces, the plane and the sphere, just as we compared curves to the simplest and most familiar curve—the straight line. For us, the sphere will be the boundary of the ball—a beach ball rather than a bowling ball. A plane, as we saw in Chapter 14, is two dimensional since it has two independent degrees of freedom— back and forth or up and down. A sphere is also two dimensional, even though we cannot construct or even imagine a sphere except inside three-dimensional space. A sphere is a surface because you have two independent degrees of freedom—back and forth (east and west) or up and down (north and south).

18.3

EUCLIDEAN GEOMETRY

Euclidean geometry takes its name from the great geometer Euclid (circa 330–circa 270 BCE) of Alexandria, Egypt. Euclid compiled all of the geometry known at the time, along with some new results of his own, into a book called The Elements. The sources of his geometry have been lost over time, but most likely included Greece, Egypt, Babylon (preset-day Iran), and even China and India. The way that he systematized geometry was so effective that The Elements completely supplanted previous writings in geometry. Euclidean geometry studies the geometry of the flat plane with its lines and curves. We have studied many aspects of Euclidean geometry so far: symmetry transformations, the conic sections, similarity and proportion, and fractals are all part of Euclidean geometry. Some of the most important properties of Euclidean geometry are as follows: 1. Two points determine a line. 2. A line can be extended infinitely. 3. If you are given a point for a center and a length for a radius, you can draw a circle with that center and that radius. 4. The sum of the measures of the three angles of a triangle is 180◦ . 5. If you are given a line and a point not on the line, there is a unique line parallel to the given line through the given point. This knowledge of Euclidean geometry will help us as we look at two new geometries: elliptic geometry and hyperbolic geometry. Both of these geometries share many properties with Euclidean geometry but are quite different.

18.4

ELLIPTIC GEOMETRY

Elliptic geometry is geometry on the surface of a sphere—like the ant traveling over the surface of an apple. We don’t have straight lines, but we still can take the shortest path between two points. We call such a path a geodesic, and it is part of a great

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circle, which is a circle whose center is at the center of the sphere, thus dividing the sphere into two equal hemispheres. The equator of the earth or a line of longitude is a great circle on the earth. Figure 18.3 shows a sphere with two great circles as well as curves that are not great circles. Surprisingly, if we replace “straight line” in Euclid’s postulates with “great circle,” many of the properties of Euclidean geometry have close analogs on the sphere. This geometry is called elliptic geometry and has been used by pilots for navigation, both by sea and by air. Albert Einstein used it to develop general relativity. Any two points determine a straight line on the plane, and any two points on a sphere determine a great circle. On the plane, a straight line can be extended infinitely. A great circle can be “extended” indefinitely—we just keep going round and round.

A sphere with two great circles in black. The red curves are circles in elliptic geometry, but they are not great circles since they are smaller than the equator of the sphere. Figure 18.3

A sphere with triangle whose sides are parts of great circles. Angles A and B have measure 90◦ and angle C has measure 100◦ , so the angle sum for the triangle is 280◦ . Figure 18.4

On a plane, either two lines are parallel or they intersect in one point. On the sphere, two different great circles always intersect at two points on opposite sides of the sphere. Such points, like the north and south poles, are called antipodal points; you can see two antipodal points in Figure 18.3, where the two black great circles intersect. So for this property, we do not have analogs between elliptic geometry and Euclidean geometry. On the plane, we can draw two parallel lines, lines which never meet. But—and this makes all the difference—any two great circles on a sphere must intersect. This means that two great circles can never be parallel. And so, the geometry of the surface of a sphere—elliptic geometry—is very different from Euclidean geometry, and we have a new geometry. Another feature of elliptic geometry, which mathematicians understand to be a consequence of the fact that every two great circles have two points of intersection, is that triangles have angle sum greater than 180◦ . An example of this is shown in

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Figure 18.4, where the triangle has angle sum 280◦ . In fact, the larger a triangle on a sphere is, the greater is its angle sum. This is quite different from Euclidean geometry!

18.5

HYPERBOLIC GEOMETRY

Hyperbolic geometry is the geometry of a saddle-shaped surface, as shown in Figure 18.5, or a region of hills and valleys, as shown in Figure 18.1. If you are on a saddle and move back and forth in one direction, you go higher; if you move back and forth in a different direction, you go lower. Instead of the straight lines of Euclidean geometry, hyperbolic geometry has geodesics that flow along the surface. Figure 18.5 shows hyperbolic geometry with a triangle with three geodesic sides and several geodesic lines. Just as in elliptic geometry, many properties of Euclidean geometry are still valid in hyperbolic geometry. Any two points determine a geodesic, and any geodesic can be extended indefinitely. We can draw circles on the saddleshaped surface, although they will look warped from the outside. However, when we consider parallel lines, we find very interesting behavior. Figure 18.5 Saddle-shaped surface with As before, two lines are parallel if they triangle made of geodesics and geodesics do not intersect. However, on the saddle, with three parallel lines through the same for any given geodesic and any point not point. on the geodesic, there are many, in fact, infinitely many, lines that are parallel to the given geodesic—that means, they do not intersect the given geodesic. In Figure 18.5, you can see a point with three geodesics passing through it, each one of which does not intersect the line below. And now we have a third geometry, hyperbolic geometry.

18.6

THE THREE GEOMETRIES AND CURVATURE

Besides different properties of parallels and angle sum for triangles, there are other ways to distinguish the three geometries. One way is the “squash” test. Euclidean geometry is flat. Think about what happens if we try to flatten, or squash, a round orange peel; it will break apart. If we try to flatten an elliptic surface, it breaks apart. On the other hand, if we try to flatten or squash a saddle, it will double back on itself. If we try to flatten a leaf of kale, we can only succeed if the kale folds back on itself. If we try to flatten a hyperbolic surface, it folds over on itself. Another is the “wrap” test. Again, we compare the two new geometries to Euclidean geometry. If we try to wrap a piece of paper around an orange, it folds over on itself. If we try to wrap a piece of paper over a saddle, it tears.

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Both of these tests are related to the fact that these three types of geometry have different curvatures. Recall that for curves, we give the curvature at a point, based on how sharply the curve changes direction at the point. Surfaces also have curvature. The curvature of a plane is 0, analogous to the curvature of a straight line being 0. A point on a small sphere will have large curvature and a point on a large sphere will have small curvature, analogous to the curvature of a curve. And the curvature of a point on surface will be the curvature of the sphere that best fits the surface at that point. An important point to note here is that the curvature of a sphere does not depend on whether we are looking at it from the inside, as if we are looking at the inside of a bowl, or whether we are looking at it from the outside, as we would look at a beach ball. So both the inside of a bowl and the outside of a bowl have positive curvature. How can we make sense of curvature of a hyperbolic surface? If we try to fit the surface of a saddle, say, with a sphere, which side do we put the sphere? In one direction, a sphere on top of the saddle would fit better, but in the other direction, a sphere on the bottom of the saddle would fit better. To deal with this situation, we say that the curvature is negative. And the curvature of a saddle is negative whether we look from above or below. We can also look at curvature of a surface in terms of a tangent plane, the plane that looks more like the surface than any other plane. At any point on a plane, the tangent plane is exactly the plane itself. At a point in elliptic geometry, the tangent plane is on one side of the surface. At a point in hyperbolic geometry, the tangent plane cuts through the surface. You can see this in Figure 18.6.

At left, a sphere with a tangent plane. At right, hyperbolic space with a tangent plane. Another way to think of the three geometries is that elliptical geometry “doesn’t have enough space,” so geodesics curve toward one another and we don’t have any parallel lines. Hyperbolic geometry “has too much space,” so geodesics curve away from one another, and for each geodesic and each point not on the geodesic, there are infinitely many other geodesics through the point parallel to the geodesic. Euclidean geometry is “just right,” and for each line and each point not on the line, there is exactly one line through the point parallel to the line. If we have an irregular surface, we can find the curvature at each point on the surface, just as we did for curves. The surface of the horse Whistlejacket, Figure 19.2, has many different curvatures, depending on the part of the body of the horse you are Figure 18.6

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looking at. The horse’s back where a saddle is positioned is hyperbolic. The horse’s rump is elliptical, and the bottoms of the feet are flat.

18.7

THE THREE GEOMETRIES AROUND YOU

When you look at the surfaces around you, you will see many different examples of the different surfaces: flat, round like a sphere, or saddle-shaped. Artists use the variety of geometries in their work. In Ellen’s Isle, Loch Katrine, Figure 18.2, Duncanson contrasts the flatness of lake with the roundness of the hills and the contours of the shoreline. Grant Wood emphasizes the saddle-like geometry of the rolling hills of the Iowa farms in Spring Turning, Figure 18.1. Stubbs shows change in curvature of the body of the horse in Figure 20.2 using light and shadow. Powers flattens everything in the process of quilt making, giving a feeling of simplicity and innocence in Pictorial Quilt, Figure 22.3.

EXERCISES Understand

18.1 Familiar Shapes. Classify these common shapes as having elliptical geometry, hyperbolic geometry, or flat geometry: a. Pringles potato chips

f. A kale leaf

b. A basketball

g. The inner side of a donut

c. A windowpane

h. The outer side of a donut

d. A book cover

i. The inside of a bowl

e. The top of your head

j. The outside of a bowl

18.2 Georgia O’Keefe. What is your overall impression of Hibiscus with Plumeria by Georgia O’Keefe? What kinds of lines and curves do you see? How do they contribute to your impression? Analyze

18.3 Surfaces of Your Body. Sketch or photograph some of the surfaces you see on your arms, hands, face, legs, or feet. Classify them as flat, elliptic, or hyperbolic. Pay attention to how the light reflects from different surface. 18.4 Surfaces around You. Sketch or photograph some of the curves you see around you. Classify them as flat, elliptic, or hyperbolic. Pay attention to how the light reflects from different surface. Here are some places you could find examples. a. Fruit, vegetables, plants, and animals b. Vases, lamps, dishes, and furniture

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c. Electronic devices 18.5 2by2. In the two pictures by the two artists that you have chosen, see if you can find some surfaces that are flat, some that are positively curved (elliptic), and some that are negatively curved (hyperbolic). Which type predominates? What effect does that have? 18.6 3by1. In the three pictures by one artist that you have chosen, see if you can find some surfaces that are flat, some that are positively curved (elliptic), and some that are negatively curved (hyperbolic). Which type predominates? What effect does that have? Create

18.7 Hyperbolic Art. Make a hyperbolic tiling from your own artwork using the app at http://www.malinc.se/m/ImageTiling.php. 18.8 Using Planes around You. Draw a picture that includes some of the flat planes that you see around you. 18.9 Using Surfaces around You. Draw a picture that includes different kinds of surfaces that you see around you. Label each surface as flat, hyperbolic, or elliptical. Review

18.10 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 18.11 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

READINGS • Poetry of the Universe: A Mathematical Exploration of the Cosmos by Robert Osserman • The Shape of Space by Jeffrey Weeks • Crocheting Adventures with Hyperbolic Planes by Daina Taimin, a

ARTISTS • Fernando Botero Angulo (b. 1932), Colombian artist; elliptic geometry

John A. Woodside. Still Life: Peaches and Grapes, circa 1825 (Metropolitan Museum of Art. Rogers Fund, 1941).

Figure 19.1

Figure 19.2

George Stubbs. Whistlejacket, circa 1762 (National Gallery).

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Euclidean and Non-Euclidean Geometries

EXPLORATIONS • Where do you see surfaces of positive curvature in Figure 19.1? Notice how light reflects from the grapes. • Look at the different parts of the horse in Figure 19.2. Can you find parts of the surface that have positive curvature and parts that have negative curvature? Are any parts flat? Notice how light reflects from different parts of the horse.

19.1

OVERVIEW

In the previous chapter, we saw three different kinds of geometries—Euclidean flat space, elliptic geometry on a sphere, and hyperbolic geometry on a saddle. In this chapter, we will understand how these geometries are related and what they tell us about the nature of geometry.

19.2

EUCLID’S POSTULATES

At Euclid’s time, around 300 BCE, the common understanding was that the geometry of the plane was the only kind of geometry. Euclid’s presentation in The Elements was so clear, complete, and logical that it was taken to be the model of how knowledge should be presented for over two thousand years. Everyone believed that the geometry Euclid described was the geometry of nature and that Euclid’s geometry was the truth about the world around us. Euclid began his presentation of geometry with postulates, statements about geometry, and common notions, rules about equalities. He chose these to be self-evident statements that anyone could accept as true. Euclid’s postulates about geometry give the fundamental shapes, their properties, and how to construct them. The common notions are simple and straightforward. Quantities equal to the same quantity are equal to one another. If two quantities are equal, one can add or subtract the same quantity from each and the results will be equal. If two shapes coincide, DOI: 10.1201/9781003110972-19

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they are equal. The whole is greater than a part. All mathematicians accept and use these principles today. However, there was one of Euclid’s postulates that mathematicians felt needed further justification before they could accept it completely—and that is Euclid’s parallel postulate. The five postulates are summarized here, with the parallel postulate last: Two Points Determine a Line. This postulate tells us that we can draw a straight line if we are given two points. This means that we can use a straight edge or ruler, although Euclid never used the markings on the ruler. Any Line Can Be Extended Indefinitely. This postulate tells us that the lines are, in effect, infinite, although Euclid only treated lines as potentially infinite, never as actually infinite. Construction of a Circle. A circle can be constructed using any given point as the center with any constant distance as radius. This postulate says that we have a compass that we can use to construct circles of any size. Space Is Homogeneous. This postulate says that the Euclidean plane is everywhere the same—each point is exactly like every other point. This is unlike a cone, for example, where the cone point is very different from other points. Similarly, Euclidean three-dimensional space is everywhere the same. The Parallel Postulate. Euclid defines parallel lines as lines that never meet. This postulate describes the properties of parallel lines. An important property of parallel lines is that for a given line and a given point not on the line, there is one and only one line through the point that is parallel to the given line. This is shown in Figure 19.3, where you can see that any other line except the dashed line will eventually intersect the given solid line. It is the fifth postulate, the parallel postulate, that has inspired much study, debate, and controversy in the two thousand years since Euclid wrote it down. This controversy is justified, as we shall see. The controversy arose because the postulate talks Figure 19.3 Solid line about what happens if lines are extended infinitely—they with point and dashed will never meet. Mathematicians felt comfortable with cirline parallel to the solid cles, triangles, and squares and the idea that a line could line through the given be extended indefinitely; but they didn’t feel comfortable point. with an infinite line. In the nineteenth century, while investigating the parallel postulates, mathematicians discovered that elliptic geometry and hyperbolic geometry both satisfied Euclid’s first four postulates. But, as we saw in the previous chapter, any two geodesics (“lines”) in elliptic geometry intersect. In hyperbolic geometry, many geodesics (“lines”) through a given point not on a given line behave as if they are parallel to the given line. See Figure 19.4.

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Sphere with two intersecting geodesics and saddle with triangle and parallel geodesics. Figure 19.4

19.3

THE VALIDITY OF NON-EUCLIDEAN GEOMETRIES

The existence of the two new geometries— elliptic geometry and hyperbolic geometry— showed that Euclidean geometry was not the only geometry. Further investigations by mathematicians showed that these new geometries were equal in status to Euclidean geometry and that no one of the three geometries was any more real or legitimate than the other two. Euclid geometry could no longer be regarded as the one and only true geometry, but one of three equally valid geometries. This discovery created a revolution in mathematical thought, which was supported by other discoveries in the twentieth century. Mathematics could no longer be regarded as absolute truth, but rather as a way of deriving valid conclusions from a collection of assumptions (postulates). Different assumptions give different conclusions. Truth, and absolute Figure 19.5 El Lissitzky. Untitled, truth, must be something far more encompass- circa 1919–1920. ing than mathematical validity. Perspective bounded and enclosed space, but science has brought about a fundamental revision. The rigidity of Euclidean space has been annihilated by Lobachevski, Gauss, and Riemann. El Lissitzky Russian artist, 1925.

Rather than weakening mathematics, this revolution has strengthened mathematics, opening mathematics up to new ways of thinking and a much wider range of possible theories.

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These new geometries have led to deeper understandings not only in mathematics but also in physical theories of gravity and cosmology and in the philosophy of mathematics. And they have inspired artists such as El Lissitzky, Figure 19.5.

EXERCISES Understand

19.1 Comparing Geometries. Make a table comparing Euclidean geometry, elliptic geometry, and hyperbolic geometry. Include diagrams in your table. Apply

19.2 Declaration of Independence. The second paragraph of the US Declaration of Independence begins with this sentence: We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. Why do you think that the authors used the term “self-evident”? 19.3 Self-Evident Truth. Give a statement that you believe is an example of a selfevident truth. Do you think everyone would agree that the truth is self-evident? Explain. 19.4 Valid Statements. Give a statement that you believe is an example of a something that is valid in some situations but not others. Explain. 19.5 El Lissitzky. How does the painting by El Lissitzky, Figure 19.5, give a feeling of unbounded, unenclosed space? Create

19.6 Euclidean Art. Create a drawing or photograph that depicts mostly flat surfaces. 19.7 Elliptic Art. Create a drawing or photograph that depicts mostly positively curved surfaces. 19.8 Hyperbolic Art. Create a drawing or photograph that depicts mostly negatively curved surfaces. Review

19.9 Summarize. List the two or three most important ideas of this lesson and say why you chose them.

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19.10 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

READINGS • A Long Way from Euclid by Constance Reid • Euclid’s Window by Leonard Mlodinow

Figure 20.1

Salvador Dalí. The Persistence of Memory, 1931.

Figure 20.2

Travis Millard. Man Unraveling.

CHAPTER

20

Topology

EXPLORATIONS • Salvador Dalí (1904–1989) was a surrealist artist who integrated dream-like images with clear-eyed reality. What is dream-like in the painting The Persistence of Memory, shown in Figure 20.1? What is realistic? The title of the painting, The Persistence of Memory, refers to our own personal experience of memory. What is your experience of memory and how memories persist? What do you notice about the clocks? How does the representation of the clocks connect with your subjective experience of memory? • What is happening to the man as he spirals inward toward a black hole in Man Unraveling, Figure 20.2? Do you see any similarity between the changes in the man and the changes in the clocks in Dalí’s picture?

20.1

OVERVIEW

Topology is the most abstract form of geometry—it is the study of shape without considering the measurement of distances, areas, or angles. It may seem like very little is left when measurement is ignored, but in fact topology brings to light important and unexpected properties of shapes and it is surprising how much information there can be about a shape when measurements are ignored. In topology, stretching and shrinking are allowed. Even though stretching and shrinking change measurements, they do not change those properties that topology is interested in—how parts of a shape are connected to one another. Ripping or tearing is not allowed, since it creates a gap between parts that had been close. Similarly, gluing is not allowed, since it brings parts close to one another that were not close before. Travis Millard’s picture Man Unraveling, Figure 20.2, shows the topological transformation of a man going through the event horizon of a black hole, ending with his final disappearance. In topology, we think of shapes as if they are made out of very stretchy and very shrinkable rubber that we can stretch and shrink as much as we like, without tearing or breaking. There is a continuous transformation from the man at the bottom, with arms, legs, and fingers getting stretched and twisted as if made from very stretchy rubber until he is unrecognizable. DOI: 10.1201/9781003110972-20

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In this chapter, we will look at the problem that started topology—the K¨onigsberg bridge problem. Then we’ll look at two famous shapes of topology—the M¨obius band and the Klein bottle—and study their unusual properties.

20.2

¨ THE KONIGSBERG BRIDGE PROBLEM

In the eighteenth century in the city of K¨onigsberg, Prussia, there were seven bridges connecting the banks of the Pregel River and two islands, shown in Figure 20.3. On Sunday afternoons, the city residents enjoyed walking through the city and over the bridges and wondered if it were possible to take a route that would cross each bridge once and only once—but no one was ever able to find such a route. See if you can find a path that works with no repetitions. Now, can you find a path that crosses each bridge if repetitions are allowed? The mathematician Leonhard Euler (1707–1783) became interested in the problem and gave a solution that became a landmark step in the development of topology. (“Euler” is pronounced like “oiler.”) Euler began with an abstract topological representation of the problem. Each landmass—the north and south banks and the two islands—was important in the problem as places to visit, but not for their size or shape. To simplify the problem, Euler replaced the landmasses by points or nodes. Each bridge connected two land masses, but it was unimportant how long or how wide the bridge was—so he replaced each bridge by a line or edge. The resulting diagram has four nodes and seven edges, as shown in Figure 20.4. This is an abstract representation of the problem and many details have been left out, but the essential features—topological features—that were key to the solution remain. Accordingly, Euler could reason about nodes and edges rather than parts of the city and bridges. He wasn’t concerned with how long the bridges were, how large the islands were, or which direction the river flowed. All that mattered was which bridge connected which land masses.

Euler’s map of K¨onigsberg showing its seven bridges. Figure 20.3

An abstract representation of the seven bridges of K¨onigsberg. Figure 20.4

Euler began his solution to the problem by recognizing that you have to start your path somewhere, at one of the nodes, and finish your path somewhere, at either

Topology  203

a different node or possibly at the same node. Also, if you visit a node that was neither your starting node nor your ending node, you would arrive at the node by one edge and leave by a different edge. If you happen to visit that same node again, you would arrive by an edge different from the ones that you had already traveled along and leave by yet another edge. This means that if a node is located somewhere in the middle of the journey, it will have an even number of edges that meet at the node. If you start and end at the same node, there will also be an even number of edges with that node as one endpoint. You leave by one edge, possibly revisit one or more times by entering by an edge that you have not traversed and leaving by yet another edge, and then finally arrive back by a final edge. No matter how many times you revisit the starting/ending node, there must be an even number of edges meeting there. What if your starting node is different from your ending node? In that case, you leave the starting node by one edge, possibly revisiting the node, but in any case, there must be an odd number of edges meeting at the starting node. When you end at the ending node, you may have passed through it before any number of times, traversing an even number of edges meeting that node. But your final trip to the node will always be along an edge that doesn’t have a pairing with another edge of the ending node. So there must also be an odd number of edges meeting at the final node. This means that the starting and ending nodes are different and each of them has an odd number of incident edges while all the other nodes have an even number of incident edges. We can summarize Euler’s observations by saying that an Euler path—a path covering each edge once and only once—has these properties: • Each intermediate node has an even number of edges meeting there. • If the path starts and ends at the same node, that node has an even number of edges that meet there. • If that path starts and ends at different nodes, both nodes must have an odd number of edges meeting there. Once Euler looked at the problem in this light, it was easy to solve. Since each of the four nodes in the K¨onigsberg bridge problem has an odd number of incident nodes, there cannot be a path that includes each edge once and only once. By solving this problem, Euler laid the foundation of topology. He found a solution for a problem that seemed geometric in nature not by using measurements of length and angle, but by looking only at relationships and connections between the parts of the problem.

20.3

TOPOLOGY

In topology, if one shape can be bent, stretched, shrunk, or twisted into another shape without cutting, gluing, or pasting, then the two shapes are regarded to be essentially the same and are said to be topologically equivalent or homeomorphic.

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Topologists call topology “rubber sheet geometry” because stretching and shrinking don’t matter. They also like to claim that they can’t tell the difference between a donut and a coffee cup, since the donut can be stretched into a coffee cup, as shown in Figure 20.5. The shapes that are studied in topology are called topological spaces or simply spaces.

Topological transformation of a coffee cup into a donut (illustration by Emilio Wherry, @y2cakey). Figure 20.5 demonstrates the topological transformation of a donut into a coffee cup. Imagine that the coffee cup is made from very stretchy, shrinkable rubber that can stretch and shrink as much as we like without tearing or breaking. There is a continuous transformation from the coffee cup to the donut, suggested by the steps shown in the figure; the bottom of the cup gets thicker, then the handle gets thicker, and the body of the cup smooths out. Can you think of two other familiar shapes that can be transformed one into the other in a similar fashion? Sketch a representation of the transformation, like this transformation of the coffee cup into the donut. In this way, topology is subtler and more abstract than geometry, studying properties that do not depend on measurements but rather depend on how parts are related or connected to one another. As you saw with the K¨onigsberg bridges problem, the tools of topologists are very different from those of the geometer. Topologists look for properties of a space that don’t change even if the space is bent, stretched, or twisted. They want to be able to tell if a given space can be transformed to a different space—if the given space is topologically equivalent to a different space. Topologists also want tools that can help them determine when two spaces are topologically different. Sometimes a property can distinguish shapes, and sometimes different shapes can share the same property. It is the transformation of one shape into another; however, that is the ultimate test of whether two shapes are topologically equivalent. Topology has applications in areas where stretching, twisting, and bending occur frequently, such as string theory in physics, molecular chemistry, and study of the structure of DNA. In the next sections, we will look at some important topological spaces and their topological properties. Figure 20.5

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¨bius Band The Mo The M¨obius band is a surface with the startling property that it has only one side and only one edge. The best way to understand it is to make one and compare it to a cylinder. To make a cylinder, take a long rectangle of paper and attach the two end edges as shown in Figure 20.6. The arrows on the two end edges show how to attach the edges. The rectangle has two faces, one on each side, and four edges; the cylinder has two edges, one on the top and one on the bottom, and two faces, one inside and one outside. To make the M¨obius band, take a rectangle and twist one end through an angle of 180◦ to make a half twist and then attach the two ends. Notice that the arrows on the ends go in opposite directions; in order to match them, you must make a half twist. This shape has one face and one edge, which you can verify by tracing with a pencil. The cylinder and the M¨obius band are different from each other topologically even though they are both constructed from a rectangle.

Figure 20.6

Construction of a cylinder and a M¨obius band.

Torus

A torus is like an inner tube, an empty donut. To make a torus, take a Slinky toy or flexible rubber tube and join ends together. Or take a rectangle and attach opposite edges, as shown in Figure 20.7. The torus is a two-dimensional surface, but a model of a torus needs three dimensions. The torus is topologically different from a sphere—it is quite clear that you can’t stretch or shrink a torus into a sphere. The torus has an inside and an outside and is very different from another two-dimensional surface, the Klein bottle.

A mathematician named Klein ¨bius band was divine. Thought the Mo Said he, “If you glue The edges of two You get a weird bottle like mine.” Leo Moser, Canadian mathematician

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The Klein bottle

The Klein bottle is one of the most unusual shapes in topology. To construct a Klein bottle, start by making a cylinder, just as you did for a torus. But now you have to twist the ends of the cylinder before joining them, as shown in Figure 20.7. What makes the Klein bottle so special is that you can’t make it in three-dimensional space. Mathematicians can explain, but not show, how it can be made in four dimensions. The best that can be done is to give a two-dimensional representation—but this has self-intersections just as a two-dimensional representation of a M¨obius band has selfintersections. Glass blowers can give a three-dimensional representation, but even this has self-intersections. Another way to make a Klein bottle is to glue two M¨obius bands along their edges. A two-dimensional image of a Klein bottle is shown in Figure 20.7.

Figure 20.7

20.4

Using rectangles to construct the torus and the Klein bottle.

TOPOLOGY IN ART

Using topological transformations, artists can achieve a result that is somewhere between realistic and abstract, as shown in Figures 20.1 and 20.8. Dalí gives us a topologically transformed clock. The realistic aspect of the clock reminds us of time past and memory; the abstract aspect of the distorted clock makes us reflect on the distortions in our memories of the past. Modigliani gives us a topologically transformed woman. The realistic aspect lets us know that this is a portrait of a specific woman, and the abstract aspect emphasizes her grace and gentleness. A dynamic topological transformation is shown by Travis Millard, Figure 20.2. The similarity of different objects—a person’s normal state and his unraveled state— is made clear to the viewer by the sequence of intermediate stages. Swiss sculptor and designer Max Bill uses the M¨obius band to evoke a feeling of infinity and eternity, Figure 20.9.

EXERCISES Understand

20.1 Finding a Path. Find an Euler path that traverses each bridge in Figure 20.10 once and only once. How did you make your choice of where to start your path?

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Figure 20.8

Amedeo Modigliani. Jeanne Hébuterne in yellow sweater, 1918.

c Max Bill. Endless Ribbon, Version IV, 1961–1962. Digital Image CNAC/MNAM, Dist. RMN-Grand Palais / Art Resource, NY.

Figure 20.9

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20.2 Making Maps. Draw two diagrams of a river with islands and bridges, one with an Euler path and one with no Euler path.

Figure 20.10

A river with three islands and twelve bridges.

20.3 Topologically Equivalent Shapes. Of the objects in Figure 20.11, find some that are topologically similar to one another. Choose one of the objects that is not topologically similar to any of the others and draw a shape or object that is topologically similar to it.

Figure 20.11

An assortment of objects.

20.4 Recycling Symbol. Figure 3.17 shows the recycling symbol. Verify that it is a M¨obius band. Using properties of the M¨obius band, explain why it is used for the recycling symbol. Apply

20.5 Persistence of Memory. Describe the topological transformations that create some of the objects in Salvador Dalí’s picture Persistence of Memory, Figure 20.1. 20.6 Topology by Dalí. Find another picture by Salvador Dalí that you think shows topological transformations. What are the topological transformations? Why do you think Dalí used them? 20.7 Topology by MC Escher. Find a topological transformation in the woodcut Day and Night, Figure 7.16, by MC Escher. 20.8 2by2. Look at the main shapes and objects in the two pictures by two artists that you have chosen. If there seem to be topologically transformed shapes, explain why the artist used them.

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20.9 3by1. Look at the main shapes and objects in the three pictures by an artist that you have chosen. If there seem to be topologically transformed shapes, explain why the artist used them. Create

20.10 2D Topological Transformation. Can you think of two familiar shapes that can be transformed one into the other in a fashion similar to the transformation in Figure 20.5? Sketch a representation of the transformation, giving a sequence of intermediate shapes that transforms one shape into the other. 20.11 3D Topological Transformation. Choose two objects that are topologically equivalent and create a sequence of models from clay or other material that transforms one into the other. 20.12 Create Your Own. Integrate a picture of a torus, M¨obius band, or Klein bottle into an imaginative drawing or painting that makes use of the shape’s topological properties. 20.13 Create Your Own Models. Create models of a cylinder and a M¨obius band from paper. Draw a line on the outside of the cylinder and draw a line on the M¨obius band as shown in Figure 20.12. Compare the results. What do you notice? 20.14 Create Your Own. Create models of a cylinder and a M¨obius band from paper. Cut the cylinder and the M¨obius band in half along the line shown in red in Figure 20.12. Compare the results. What do you notice?

Figure 20.12

A cylinder and a M¨obius band with midlines shown in red.

20.15 Create Your Own. Create a picture that integrates the steps of a topological transformation of shapes into the theme of the picture. Use Figure 20.2 as an example. Review

20.16 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 20.17 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

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READINGS • Experiments in Topology by Stephen Barr

ARTISTS • El Greco (1541–1614); topological transformations • Salvador Dalí (1904–1989); topological transformations • Rob Gonsalves (1959–2017); topological transformations

Taylor & Francis Taylor & Francis Group

http://taylorandfrancis.com

George Caleb Bingham. The Jolly Flatboatmen, 1846 (National Gallery of Art. Patrons’ Permanent Fund).

Figure 21.1

Jacob Lawrence, 1964, Street to Mbari (National Gallery of Art. Gift of c 2022 The Jacob and Gwendolyn Knight Lawrence Mr. and Mrs. James T. Dyke Foundation, Seattle / Artists Rights Society (ARS), New York).

Figure 21.2

CHAPTER

21

Pictorial Composition

EXPLORATIONS • Look at the picture The Jolly Flatboatmen by the American artist George Caleb Bingham, Figure 21.1. What are your impressions? What do you think the artist intended to portray? What shapes do you see? What shapes are implied? How did Bingham organize the shapes in the picture? What geometric themes from this course do you see in this picture? • Look at the picture Street to Mbari by the American artist Jacob Lawrence, Figure 21.2. Lawrence created this picture after a trip to Nigeria. What are your impressions? What do you think the artist intended to portray? What shapes do you see? What shapes are implied? What ways did Lawrence use to contrast the people in the marketplace with the shops and goods? How did Lawrence organize the shapes in the picture? What geometric themes from this course do you see in this picture? • Compare Street to Mbari and The Jolly Flatboatmen. How are they different? How are they alike?

21.1

OVERVIEW

In this chapter, we look at how pictures are put together from their parts: how they are structured, designed, and organized—how the artist has composed the picture. The saying “The whole is more than the sum of the parts” applies quite well to a work of art. All parts of the picture should be appropriate; they should belong to the picture. If anything is removed, the wholeness will be impaired. The picture should be complete—nothing lacking and nothing extra. The parts of a picture—whatever they might be—must be connected in a way that establishes the unity or wholeness of the work. In this chapter, we will look at geometric principles that artists use to unify the parts of a picture into a whole. First, we look at how artists use geometric principles to organize the objects in a picture. Then, we consider how the artists use geometry to move the eye around a picture.

DOI: 10.1201/9781003110972-21

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21.2

GEOMETRIC PRINCIPLES OF PICTORIAL COMPOSITION

When we look at the composition of a picture, we are considering how the parts create the wholeness that is the picture. Each picture is unique; each arrangement of objects in a picture is unique. Artists often use triangles, circles, or a line to arrange shapes. They use symmetry, balance, and similarity to connect objects in a painting to one another. All of the geometry in a painting contributes to how the eye of the viewer moves around a painting.

Figure 21.3

Harriet Powers. Pictorial Quilt, 1895–1898.

Symmetry

Bilateral symmetry is one of the most powerful tools that an artist has for creating balance in a picture. Rarely will symmetry be exact in a painting, but even approximate symmetry creates a feeling of balance in the viewer. Both The Jolly Flatboatmen and Street to Mbari have approximate bilateral symmetry in the picture as a whole. My Own Private Ecosystem on page 1 has many shapes that have approximate bilateral symmetry, and the picture as a whole has an approximate bilateral symmetry that echoes the bilateral symmetry of the individual shapes. There are many examples of bilateral symmetry in the small shapes of Pictorial Quilt, Figure 21.3, and the square blocks are arranged in an orderly way. The use of varying orientations for the small shapes, however, gives a greater feeling of dynamism than a picture with overall bilateral symmetry.

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On the other hand, Greetings from a Manhattan Artist, Figure 21.5, does not have an overall feeling of symmetry, even though many of the individual shapes are symmetric. The lack of overall symmetry contributes to the feeling of dynamism and adventure that the viewer associates with Manhattan.

Thomas Hart Benton. Wheat, 1967 (Smithsonian American Art Muc 2022 seum. Gift of Mr. and Mrs. James A. Mitchell and museum purchase. T.H. and R.P. Benton Trusts / licensed by Artists Rights Society (ARS), New York).

Figure 21.4

Balance

Even when an artist wants to convey an uneasy or unsettled feeling, pictures will have balance of some kind to unify the picture. There are many ways to create balance in a picture. We have already seen how bilateral symmetry creates balance, but there are many other ways to create balance in a picture. Artists carefully balance dynamism and silence, even though one may predominate. In The Jolly Flatboatmen, the dynamism of the dancing and playing in the foreground is balanced by the flat river and the peaceful forest in the background. The turbulence of the market in Street to Mbari is balanced by the soft brown tones of the tops of the shops. In Wheat, Figure 21.4, the silence of the fully grown stalks of wheat is balanced by the flowing leaf blades and the curly green cover crop peeking through the stubs in front.

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Ida Abelman. Greetings from a Manhattan Artist, circa 1939 (Smithsonian American Art Museum. Transfer from D.C. Public Library.)

Figure 21.5

As an unusual example, Ida Abelman uses completely empty space on the right to give necessary balance to the disorienting excitement of the Manhattan landscape in Greetings from a Manhattan Artist. In a comparable way, Vincent van Gogh uses vacant areas—the sky above and the street in the foreground—to balance the activity in the midground of The Yellow House. The overall effect here, however, is silence and balance. Triangles

More than any other polygon, the triangle is most used by artists as a way of organizing disparate elements of a picture into a whole. Objects can be arranged in an implied triangle, often with something significant at the top or apex. We see the men on the flatboat in Figure 21.1 arranged in an implied triangle, with the head of the central dancing figure at its apex. In Street to Mbari, we see two implied triangles: one at the top of the picture, as if funneling a mass of people, shops, and trading into the street, and the other at the bottom, spilling out people and their wares. Circles

The circle is symmetric, with no corners and a sense of infinity, and is a very effective way to organize shapes that have about equal importance. Kandinsky’s Circles in a Circle, Figure 13.2, is based on the circle. The major circular objects of My Own

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Private Ecosystem are themselves organized in a large circle, giving them about equal weight. The main objects of the still-life paining, No. 9, Nature Morte Espagnole, Figure 21.6, are more brightly colored and painted with more detail—and are arranged in a circular fashion. Closely related to the circle is the spiral, Section 17.4, which gives a feeling of controlled expansion or contraction. For example, Man Unraveling, Figure 20.2, uses a spiral to show a man gradually falling into a black hole.

Figure 21.6 Diego Rivera. No. 9, Nature Morte Espagnole, 1915 (National Gallery c 2022 Banco de México Diego Rivera Frida of Art. Gift of Katharine Graham. Kahlo Museums Trust, Mexico, D.F. / Artists Rights Society (ARS), New York). Linear Composition

Sometimes, important objects in a picture are lined up horizontally. You see very effective use of linear composition in Wheat, Figure 21.4, and from: (notations): grinding wheels by Mary Jo Toles, Figure 15.2, and in the rows of Pictorial Quilt. Similarity

Artists use congruence and similarity to connect different objects and shapes in a picture. Congruence may be the result of the objects of the same size and shape located in different places in the picture. In My Own Private Ecosystem, for example, we see the same flower or leaf in different locations. The two hunters in Hunters in the Snow, Figure 1.1, are approximately the same size and shape. Similarity may be the result of the objects of the same size and shape located at different distances from the picture plane. So the trees at front left in Hunters

218  Geometry for the Artist

in the Snow are likely to be about the same size and shape—congruent—in life, but they are similar in the picture because they are at different distances from the picture plane. And the water glasses in Painted Water Glasses, Figure 10.1, are similar, again because they are at different distances from the picture plane. However, artists may also use congruence and similarity to make connections between objects that don’t necessarily have the same shape in real life. In St. Anthony Reading by Albrecht D¨ urer, we see that both the town and St. Anthony have approximately similar triangular shapes, but this is due to the choice of pose that D¨ urer gave St. Anthony. D¨ urer is compelling us to compare the two—the secular town with the reclusive and saintly monk.

Figure 21.7

21.3

Albrecht D¨ urer. St. Anthony Reading, 1519.

PORTRAITS

Portraits present unique problems and opportunities to the artist. Generally, the purpose of a portrait is to capture the individuality of the person in the portrait, and there is one main object in a portrait. All of the principles of composition that we have discussed apply to portraits, but for a portrait, the main decision the artist makes is where to place the head of the person portrayed. Will the head be centered or off to one side? If the head is off to one side, how will balance be achieved? Where is the person looking? What story is the artist telling about the person? How should the eye of the viewer move around the picture? Each choice that the artist makes contributes to the wholeness of the picture and tells the viewer something about the person in the portrait. When analyzing the composition of a portrait, first look at the individual in the same way you would look at a person you just met. What might their personality be like? What do you think is important to them? What emotions do they express?

Pictorial Composition  219

Then, see how the geometrical features of the composition help to convey the feelings that you get when looking at the picture.

21.4

HOW THE EYE MOVES AROUND A PICTURE

The geometric organization of a picture contributes to how the eye of the viewer moves around the painting. The artist would like the viewer to enter the picture, move around the picture to take in all the details, and then exit the picture with a feeling of wholeness. To draw the eye into a picture, an artist will use something that catches the eye. This is often by a bright color or a contrasting color, but it can also be a distinctive shape. In Hunters in the Snow, Figure 1.1, one can enter with the hunters in the front left of the picture. The bright yellow in Kandinsky’s Yellow-Red-Blue, Figure 1.2, or the bright blue in Rivera’s still life No. 9, Nature Morte Espagnole draws the eye in very naturally. In Ellen’s Isle, Loch Katrine, Figure 18.2, one enters at the front, where the reflection of the sun captures the eye. Once the viewer has entered the picture, they will move around the picture, following explicit or implicit lines or following the direction given by a pointing finger, the glance of someone in the picture, or some other device created by the artist. A successful composition will move the eye to all parts of the picture in a sequence that lays out the story of the picture but also gives the viewer the freedom to keep exploring and wandering around the picture. You can see how the curves and diagonal lines of Yellow-Red-Blue pull the eye around, making the eye travel the whole picture, but along a jagged path that is balanced by the soft colors. The gentle curves and short line segments of No. 9, Nature Morte Espagnole move the eye around in a more leisurely way, allowing the viewer to enjoy the variety of colors and shapes. To create fulfillment in the viewer, the exit from the picture must be logical and satisfying. The exit should be from an area where there is little activity and little motivation to keep moving around the painting. An exit may be through a door or window in the rear of a scene. In Hunters in the Snow, Figure 1.1, one can exit off through the mountains in the rear. In Ellen’s Isle, Loch Katrine, one moves easily and gently around the landscape, exiting off in the distance. In Circles in a Circle, Figure 13.2, one can leave along the narrow end of the yellow or blue strip. After looking at No. 9, Nature Morte Espagnole, one will probably exit through a darker corner of the picture, feeling complete.

Composition is the art of arranging in a decorative manner the various elements at the painter’s disposal for the expression of his feelings. Henri Matisse “Notes of a Painter”

220  Geometry for the Artist

21.5

UNDERSTANDING PICTORIAL COMPOSTION

There is no formula for understanding how an artist arranged the objects in a picture. You must look at each picture and find patterns and shapes unique to that picture. Look at it for a while. What shapes do you see? How does their geometry connect them? How does your eye move around the picture? Does your eye move gently, following S-curves, flowing from one part of the picture to another? Is your eye jerked around from one place to another? Does your eye move slowly or quickly? What is the sequence that you follow? How does your journey around the painting create a story for you? What attracts your attention? How do you feel? What do you think was the intention of the artist? What can you learn from the artist? As you answer questions like these, you will deepen your understanding of the picture. Studying different pictures by different artists, you will see the features we have studied in this chapter and find new ways of organizing objects.

EXERCISES Understand

21.1 Abstract Compositions. Cut out several circles, triangles, and squares of different sizes. Choose some of them to make a simple composition that shows a. Silence or balance b. Dynamism or tension c. Expansion. Apply

21.2 2by2. For each of the two paintings by two artists that you chose, draw a diagram of the painting, showing the major shapes, curves, and lines. Analyze the composition of each picture and how the composition is related to the rest of the picture. Compare and contrast the two compositions. 21.3 3by1. For each of the three paintings by an artist that you chose, draw a diagram of the painting, showing the major shapes, curves, and lines. Analyze the composition of each picture and how the composition is related to the rest of the picture. Do you see any similarity in the compositions? 21.4 Analysis of a Painting. Choose one of the following pictures and analyze the composition. • Hunters in the Snow by Pieter Bruegel, shown in Figure 1.1 • Yellow-Red-Blue by Wassily Kandinsky, shown in Figure 1.2 • Jeanne Hébuterne in yellow sweater by Amedeo Modigliani, shown in Figure 20.8

Pictorial Composition  221

• The Jolly Flatboatmen by George Caleb Bingham, shown in Figure 21.1 • Street to Mbari by Jacob Lawrence, shown in Figure 21.2 • Wheat by Thomas Hart Benton, shown in Figure 21.4 21.5 Compare and Contrast Compositions. Find another painting that appeals to you by the artist you chose in Exercise 21.4. Compare and contrast the composition of this picture with the picture you studied in that exercise. What differences do you see? What similarities do you see? Create

21.6 Create Your Own. Create your own picture using a composition similar to that of another picture that you have studied. 21.7 Improve Your Own. Analyze the composition of a picture or photograph that you have done in the past. What geometry do you see? What works well in your picture? What could you do to improve the composition? Review

21.8 Summarize. List the two or three most important ideas of this lesson and say why you chose them. 21.9 Important Concept. Choose one important concept from this chapter that appeals to you. Give an example of that concept from a work of art that you have seen or explain how you might use that concept yourself in something new.

FURTHER READINGS • Pictorial Composition: An Introduction by Henry Rankin Poore • Composition: The Anatomy of Picture Making by Harry Sternberg • The Visual Story: Creating the Visual Structure of Film, TV and Digital Media by Bruce Block

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Bibliography [1] Bruce Block. The Visual Story: Creating the Visual Structure of Film, TV and Digital Media. Focal Press/Elsevier, Amsterdam, Boston, MA 2008. [2] Anna Bonshek. Mirror of Consciousness: Art, Creativity, and Veda. Motilal Banarsidass Publishers, Delhi, 2001. [3] Felipe Cucker. Manifold Mirrors: The Crossing Paths of the Arts and Mathematics. Cambridge University Press, Cambridge, 2013. [4] Lewis Day. Pattern Design. Dover Publications, New York, 1999. [5] Ron Eglash. African Fractals: Modern Computing and Indigenous Design. Rutgers University Press, New Brunswick, NJ, 1999. [6] Bruno Ernst. The Magic Mirror of M.C. Escher. Taschen, K¨oln, London, 2007. [7] M.C. Escher. Escher on Escher: Exploring the Infinite. Harry N. Abrams, New York, 1989. [8] Lee Fergusson and Anna Bonshek. The Unmanifest Canvas: Maharishi Mahesh Yogi on the Arts, Creativity and Perception. Maharishi University of Management Press, Fairfield, IA, 2014. [9] Marc Frantz and Annalisa Crannell. Viewpoints: Mathematical Perspective and Fractal Geometry in Art. Princeton University Press, Princeton, NJ, 2011. [10] Lynn Gamwell. Mathematics + Art: A Cultural History. Princeton University Press, Princeton, NJ, and Oxford, 2016. [11] James Gleick. Chaos: Making a New Science. Minerva, London, 1996. [12] István Hargittai and Magdolna Hargittai. Shelter Publications, Bolinas, CA, 1994.

Symmetry: A Unifying Concept.

[13] Linda Henderson. The Fourth Dimension and Non-Euclidean Geometry in Modern Art. The MIT Press, London, England and Cambridge, Massachusetts, 2013. [14] Wassily Kandinsky. Point and Line to Plane. Dover Publications, New York, 1979. [15] Paul Klee. Pedagogical Sketchbook. Frederick A. Praeger, New York, 1960.

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224  Bibliography

[16] Morris Kline. Mathematics in Western Culture. Oxford University Press, Oxford, England, 1969. [17] Stan Lee. How to Draw Comics the Marvel Way. Simon & Schuster, New York, 1978. [18] Allan McRobie. The Seduction of Curves: The Lines of Beauty that Connect Mathematics, Art, and the Nude. Princeton University Press, Princeton, NJ, 2017. [19] Jason Meyer. Vanishing Point: Perspective for Comics from the Ground Up. Impact Books, Cincinnati, OH, 2007. [20] Leonard Mlodinow. Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace. Simon & Schuster, New York, 2002. [21] Robert Osserman. Poetry of the Universe: A Mathematical Exploration of the Cosmos. Anchor Books, New York, 1995. [22] Henry Rankin Poore. Pictorial Composition: An Introduction. Dover Publications, New York, 1976. [23] Constance Reid. A Long Way from Euclid. Dover Publications, Mineola, NY, 2004. [24] Jacques Revault. Designs & Patterns from North African Carpets & Textiles. Dover Publications, Mineola, NY, 1973. [25] Doris Schattschneider. Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M.C. Escher. W.H. Freeman, New York, 1990. [26] Marjorie Senechal and George Fleck. Shaping Space: A Polyhedral Approach. Birkh¨auser, Boston, MA, 1988. [27] Dale Seymour and Jill Britton. Introduction to Tessellations. Dale Seymour Publications, Palo Alto, CA, 1989. [28] Harry Sternberg. Composition: The Anatomy of Picture Making. Dover Publications, Mineola, NY, 2008. [29] Daina Taimin, a. Crocheting Adventures with Hyperbolic Planes. A K Peters, Ltd., Natick, MA, 2009. [30] David Wade. Symmetry: The Ordering Principle. Bloomsbury Publishing USA, 2006. [31] Dorothy Washburn and Donald W. Crowe. Symmetries of Culture: Theory and Practice of Plane Pattern Analysis. Dover Publications, Mineola, NY, 2020. [32] Jeffrey Weeks. The Shape of Space. Chapman & Hall/CRC, Boca Raton, FL, 2020.

APPENDIX

A

Suggestions for Final Projects

A.1

ART RESEARCH PROJECT

Write an essay of about 1000 to 2000 words in length discussing the geometry in one of the paintings that you have studied in this course.

A.2

ART RESEARCH TEAM PROJECT

Pair with another student in your class once you have completed your art research project from A.1. Read your teammate’s essay. Together, write a short essay of about 500 words in length bringing out one important difference between your two chosen paintings and one important way in which they are similar.

A.3

CREATIVE PROJECT

Create an artistic expression using some geometry that was covered in this course. You could create, for example, a drawing, painting, photograph, slideshow, GIF, poem, song, story, or dance. Write one or two sentences describing the geometry that you used.

DOI: 10.1201/9781003110972-A

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APPENDIX

B

Answers to Selected Exercises

This appendix gives answers to selected exercises. 3.1 a. 2 •; b. 5 •; c. ∗ 2 •; d. 2 •; e. ∗ 3 •; f. 3 •; g. 4 •; h. 2 • 3.2 a. 2 •; b. 4 •; c. ∗ 5 •; d. ∗ 2 •; e. 4 •; f. 4 •; g. ∗ 6 •; h. ∗ 2 • 3.4 a. ∗ 1 •; b. ∗ 1 •; c. ∗ 3 •; d. ∗ 4 •; e. ∗ 1 •; f. ∗ 4 •; g. ∗ 3 • 3.5 A rotation of 180◦ ; a rotation of 180◦ 3.6 A rotation of 270◦ clockwise; a rotation of 90◦ clockwise. These answers are different. 3.7 The two answers in 3.5 are the same as each other, and the two answers in 3.6 are different from each other. The result of two performing two symmetries, one after the other, sometimes depends on the order in which they are performed. 3.14 There are many shapes with bilateral symmetry, 2 •; there are crosses with symmetry ∗ 4 •; the suns have many different symmetries, including ∗ 11 •, ∗ 16 •, and ∗ 18 •. 4.1 a. ∗ 2 2 ∞; b. 2 ∗ ∞; c. 2 2 ∞; d. ∞ ∗; e. ∞ ∞; f. ∗ ∞ ∞; g. ∗ 2 2 ∞. 4.2 You can find band ornaments with symmetry classifications ∗ 2 2 ∞, ∞ ×, and ∗ ∞ ∞. DOI: 10.1201/9781003110972-B

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228  Geometry for the Artist

5.2 Figure 5.11 has vertical and horizontal mirror lines, rotocenters of period 2, and glide reflections. All are symmetries of the underlying tiling. Figure 5.12 has rotocenters of period 2 and glide reflections; there are no mirror lines. All are symmetries of the underlying tiling. Figure 5.13 has rotocenters of degree 2, mirror lines, and glide reflections. All are symmetries of the underlying tiling. Figure 5.14 has rotocenters of period 3, mirror lines, and glide reflections. All are symmetries of the underlying tiling. 6.1 a. finite design; b. tiling; c. tiling; d. no possible pattern; e. band ornament; f. tiling; g. no possible pattern; h. band ornament 6.2 a. 6 3 2; b. × ×; c. 3 ∗ 3; d. 4 4 2; e. ∗ 6 3 2; f. 4 4 2; g. ∗ 3 3 3; h. 4 ∗ 2 6.3 a. 2 ∗ 2 2, different; b. 2 2 ×, different; c. 2 ∗ 2 2, different; d. 3 ∗ 3, different 6.4 Each square of new tiling has mirror lines, so the signature must have a star (∗). The signature of the new tiling is 2 ∗ 2 2. 7.1 2 2 ∗ 7.2 3 3 3 8.3 The central vanishing point is to the right of Saint Jerome’s head. 11.3 a. three-point perspective, from above; b. one-point perspective; c. two-point perspective; d. three-point perspective, from below; e. any; f. three-point perspective, from below 12.1 All vanishing points are outside the picture. 12.2 All vanishing points are outside the picture. 12.3 Vanishing point is in the center of the picture. 13.2 girl, 4; teddy bear, 2; clown, 4; laughing man, 7; frog, 3; giraffe, 9

Answers to Selected Exercises  229

13.3 Approximately 4 heads tall 14.2 Scaling factor 4: point, 1; interval, 4; square, 16; cube, 64. Scaling factor 5: point, 1; interval, 5; square, 25; cube, 125 14.3 Scaling factor, 2; number of pieces, 3. Dimension is approximately 1.585 14.4 Scaling factor, 3; number of pieces, 8. Dimension is approximately 1.893 15.1 0 7→ 0 7→ 0 7→ 0 7→ 0 . . . 1 7→ −1 7→ 1 7→ −1 . . . −1 7→ 1 7→ −1 7→ 1 . . . 0.5 7→ −0.5 7→ 0.5 7→ −0.5 7→ 0.5 . . . −0.5 7→ 0.5 7→ −0.5 7→ 0.5 7→ −0.5 . . . 2 7→ −2 7→ 2 7→ −2 7→ 2 . . . −2 7→ 2 7→ −2 7→ 2 7→ −2 . . .

Figure B.1 The phase line of the dynamical system x 7→ −x. All numbers except 1 have bounded orbits and are shown in red. The number 0 is a fixed point and is shown in green.

15.2 1 7→ 1 7→ 1 7→ 1 . . . −1 7→ −1 7→ −1 7→ −1 . . . 0.5 7→ 2 7→ 0.5 7→ 2 7→ 0.5 7→ 2 . . . −0.5 7→ −2 7→ −0.5 7→ −2 7→ −0.5 7→ −2 . . . 2 7→ 0.5 7→ 2 7→ 0.5 7→ 2 7→ 0.5 . . . −2 7→ −0.5 7→ −2 7→ −0.5 7→ −2 . . .

The phase line of the dynamical system x 7→ 1/x. All numbers except 0 have bounded orbits and are shown in red. This dynamical system is not defined for x = 0, and the number 0 is shown in white.

Figure B.2

230  Geometry for the Artist

15.3 0 7→ 0 7→ 0 7→ 0 7→ 0 . . . 1 7→ 2 7→ 6 7→ 42 . . . −1 7→ 0 7→ 0 7→ 0 . . . 0.5 7→ 0.75 7→ 1.3125 7→ 3.03515625 7→ 12.24732971 . . . −0.5 7→ −0.25 7→ −0.1875 7→ −0.15234375 7→ −0.129135132 . . . 2 7→ 6 7→ 42 7→ 1806 . . . −2 7→ 2 7→ 6 7→ 42 7→ 1806 . . .

The phase line of the dynamical system x 7→ x2 + x. Both 0 and −1 are fixed points and shown in green. Positive numbers have unbounded orbits and are shown in blue. Numbers between −1 and 0 have bounded orbits and are shown in red. The numbers less than −1 have unbounded orbits and are shown in blue. Figure B.3

15.4 0 7→ 0 7→ 0 7→ 0 7→ 0 . . . 1 7→ 0 7→ 0 7→ 0 7→ 0 . . . −1 7→ 2 7→ 2 7→ 2 . . . 0.5 7→ −0.25 7→ 0.3125 7→ −0.21484375 7→ 0.261001587 . . . −0.5 7→ 0.75 7→ −0.1875 7→ 0.22265625 7→ −0.173080444 . . . 2 7→ 2 7→ 2 7→ 2 7→ 2 . . . −2 7→ 6 7→ 30 7→ 870 . . .

The phase line of the dynamical system x 7→ x2 − x. Both 0 and 2 are fixed points and are shown in green. Numbers between 0 and 2 have bounded orbits and are shown in red. Numbers larger than 2 have unbounded orbits and are shown in blue. Numbers less than 0 have unbounded orbits and are shown in blue.

Figure B.4

16.1 a. 11i; b. 5i; c. −15 ; d. 4 + 8i; e. 12 − i; f. 21 + 20i 16.2 (3 − i)2 = (3 − i)(3 − i) = 9 − 1 − 3i − 3i = 8 − 6i 16.3 0.5 7→ 0.75 7→ 1.0625 7→ 1.62890625 7→ 3.153335571 . . . 16.4 −0.5 7→ −0.25 7→ −0.4375 7→ −0.30859375 7→ −0.404769897 . . .

Answers to Selected Exercises  231

17.1 C has the largest curvature, then A, and B has the smallest curvature 17.2 E, B, A, D, C, and F in order of the size of the curvature, from largest to smallest 18.1 a. hyperbolic geometry; b. elliptic geometry; c. flat geometry; d. flat geometry; e. elliptic geometry; f. hyperbolic geometry; g. hyperbolic geometry; h. elliptic geometry; i. elliptic geometry; j. elliptic geometry 20.1 Start at the lower bank and end on the center island or start at the center island and end on the lower bank. 20.3 Shapes a and d are topologically the same; c, h, and i are topologically the same; f and g are topologically the same; b and e are not topologically the same as any of the other shapes.

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Index Abelman, Ida, 38, 177, 216 Alberti, Leon Battista, 5, 90, 92 An Artist Drawing a Seated Man, 77, 83 Ascending and Descending, 123, 124 asymmetry, 19 attractor, 159 axis imaginary, 166 real, 166, 169 band classification, 36 M¨obius, 202, 205, 206, 208, 209 ornament, 4, 12, 31, 33, 37, 43, 51, 131 Belvedere, 120, 122, 123 Bill, Max, 206 Bingham, George Caleb, 212, 213, 221 body-to-head ratio, 135–137 Botero Angulo, Fernando, 193 Botticelli, Sandro, 162 broccoli, 144, 145 Bruegel, Pieter, 2, 3, 220 butterfly effect, 161 Canaletto, 74, 75, 80, 82–84 Cantor dust, 147 Cantor set, 146, 147, 151 construction of, 147 Cantor, George, 147 catenary, 181, 184 caustic, 181, 182 center, see also rotocenter, 102 of circle, 100 of rotation, 16 chance, 161 chaos, 156, 159–161 checkerboard, 37, 59, see also grid chevron, 54, 59

circle, 15, 24, 99, 100, 102, 103, 169, 178, 188, 216 center, 100 circumference of, 134 construction, 196 diameter of, 134 great, 189 in perspective, 103, 105 of curvature, 178–180 similar, 134 Circles in a Circle, 130, 131, 134, 216, 219 circumference, 100, 102 coffee cup, 204 common notion, 195 Composition (No. 1) Gray-Red, 176 cone, 99–101, 196 visual, 100, 101 congruence, 131, 133 congruent, 131 conic section, 99, 188 consciousness, 3, 146, 147 Conversation Piece, 177, 178 corresponding, 131 cube impossible, see impossible cube Necker, see Necker cube curvature, 178, 180 of a surface, 191 Cycle, 64–66, 70, 111, 112 D¨ urer, Albrecht, 77, 80, 82, 83, 85, 87, 94, 135, 218 Dalí, Salvador, xiii, 153, 200, 201, 206, 208, 210 Day and Night, 66, 69, 208 design finite, 15 Japanese, 15, 25 233

234  Index

symmetric, 4, 19, 131 construction, 26 diagonal line, 177 dimension, 149, 188 fractal, 150 directrix, 102, 104 donut, 192, 204 Duncanson, Robert S., 186, 187, 192 dynamical system, 167

construction of, 146 dimension, 149–151 geometry, 4 natural, 145, 146, 148, 151 stage of, 147 from: (notations): grinding wheels, 154, 217

general relativity, 189 geodesic, 188, 196 GeoGebra, 8 Eakins, Thomas, 81, 91, 95 The Geometer’s Sketchpad, 8 edge, 43 geometry Einstein, Albert, 189 elliptic, 5, 188, 189, 195–197 El Greco, xiii, 210 application, 189 Euclidean, 4, 188–190, 195 The Elements, 188 Ellen’s Isle, Loch Katrine, 186, 187, 192, fractal, 145 219 hyperbolic, 5, 190, 195–197 ellipse, 99–102, 105 non-Euclidean, 5, 197 construction of, 102 of nature, 145 rubber sheet, 5 focus of, 102 elliptic geometry, see geometry, elliptic sacred, 25 Endless Ribbon, Version IV, 206 glide reflection, 31, 34, 35, 37, 45, 59 equator, 189 Gogh, Vincent van, 108, 109, 113, 130, equidistant, 94 131, 133, 216 Escher, MC, 4, 64–66, 118–120, 125, 127, golden ratio, 138 208 golden rectangle Euclid, 195 construction, 138, 140 Euclidean geometry, see geometry, definition, 138 Euclidean properties, 138 golden spiral Euler path, 203 Euler, Leonard, 202 construction, 139 Evans, Minnie, 185 definition, 139 Gonsalves, Rob, 210 Fish, Janet, 98, 181 The Great Wave off Kanagawa, 154, 155 flag, 25 Greetings from a Manhattan Artist, 38, focus 177, 215, 216 of ellipse, 101, 102 grid, 91, 92, 94 of parabola, 102–104 in perspective, 96 footprint, 59 square, 92 footprints, 37 gyration, 18–20, 25 foreshorten, 91, 95, 96, 121 fountain, 99, 100, 102 Høeg, Peter, 3 fractal, 129, 145, 150, 151, 155, 161, 163, heads tall, 135–137 hemisphere, 189 169, 188

Index  235

hexagon, regular, 22, 43 construction of, 22 tiling, 46 Hibiscus with Plumeria, 173, 192 Hockney, David, 181, 185 Hogarth, William, 96, 97, 180 Hokusai, Katsushika, 154, 155 homeomorphic, 203, 204 Hunters in the Snow, 2, 3, 87, 217, 219, 220 hyperbola, 99, 100, 103 hyperbolic geometry, see geometry, hyperbolic The importance of knowing perspective, 96, 97 impossible cube, 122, 123 impossible world, 4, 119, 120 In a Medici Garden, 74, 75, 106 initial condition, 156, 158–161, 167 Isle of Man flag of, 25 iterate, 158 Jeanne Hébuterne in yellow sweater, 183, 207, 220 The Jolly Flatboatmen, 212–215, 221 Jones, Owen, 10 The Judgement of Paris, 162 K¨onigsberg bridge problem, 202 kale, 190, 192 kaleidoscope, 17, 19, 20, 25 Kandinsky, Wassily, 2, 3, 130, 131, 134, 185, 216 Klein bottle, 202, 205, 206, 209 Koch snowflake, see snowflake, Koch La condition humaine, 73, 87 Lawrence, Jacob, 212, 213, 221 Leonardo da Vinci, 18 Leonardo’s theorem, 18, 19 line, 176, 188, 217 horizon, 78, 80, 81, 83, 84, 92, 94, 119 horizontal, 177

infinite, 196 mirror, 17, 33–35 of beauty, 180 parallel, 79–82, 188, 196 phase, 158 real number, 157 receding, 81, 120 straight, 176 tangent, 178, 179 vertical, 177 line segment, see segment linear composition, 217 Lissitzky, El, 197 logarithm, 150 logo, corporate, 27 Lorenz, Edward, 161 Magritte, René, 73, 87 Maharishi Mahesh Yogi, xiii, 6 Man Unraveling, 200, 201, 206, 209, 217 Mandelbrot set, 164, 165, 167, 169 Mandelbrot, Benoit, 145, 148, 165, 167 Marc, Franz, 174, 175 Masquerade Ball, 94 Matisse, Henri, 219 The Milkmaid, 135 Millard, Travis, 200, 201, 206 miracle, 34–37, 52–55, 59 Modigliani, Amedeo, 183, 206, 207 Mondrian, Piet, 176, 185 motif, 32–35, 39 My Own Private Ecosystem, 1, 214, 217 nadir, 112 Navajo rug, 9, 28, 40 Necker cube, 122, 123 No. 9, Nature Morte Espagnole, 217, 219 nuclear power plant, 103 number complex, 165–167 computation, 166 graphing, 166 real, 157 O’Keefe, Georgia, 173, 192 Oceania, 10, 11

236  Index

orbit, 158–160 bounded, 158, 160 periodic, 159, 160 unbounded, 158–160 orthogonal, 80–83, 94 Painted Water Glasses, 98, 99, 181, 218 The Pair-Oared Shell, 95 Palladio, 135 parabola, 99, 100, 102, 104, 181 path, Euler, 203 pattern frieze, 4, 12, 31 periodic, 32 strip, 12 wallpaper, 4, 12, 43 pentagon, regular, 21 construction of , 21 signature of, 21 pentagram, 21 signature of, 21 period of a rotation, 16 Persian rug, 9, 28, 40 The Persistence of Memory, 200, 201, 206, 208 perspective, 4, 75, 81, 96 aerial, 85, 120 atmospheric, 85 linear, 76–78, 119 one-point, 81 three-point, 112 two-point, 111 pictorial composition, 213, 214, 220 Pictorial Quilt, 28, 192, 214, 217 pinwheel, 17, 53–56, 62 plane complex, 166 picture, 77, 79, 119 tangent, 191 Poe, Edgar Allan, 12 Poem Scroll with Deer, 133, 183 point antipodal, 189 attracting fixed, 159

central vanishing, 78–85, 93, 94, 119 diagonal vanishing, 92, 94 distance, 92 fixed, 158 gyration, 18 inflection, 180 kaleidoscopic, 17 repelling fixed, 159 station, 77–79, 93, 94, 99, 110, 119 vanishing, 80, 91, 120 point symmetry, 22, 23 Pollock, Jackson, 153 polygon regular, 19, 22, 43 portrait, 218 postulate Euclid’s, 196 parallel, 196 Powers, Harriet, 28, 214 proportion, 188 quilt, 28, 70, 192, 214, 217 radius, 100, 102, 188, 196 Raphael, 90, 91 rattan stool, 103 reciprocal, 179 Red Deer II, 174, 175 reflection, 4, 17, 19, 31, 36, 59 horizontal, 32 vertical, 32 Relativity, 118–121, 127 relativity, general, 189 repellor, 159 rhombus, 22, 23 rigid motion, 16 rise, 177 Romanesco broccoli, 144, 145 rosette, 15 rotation, 4, 16, 19, 31–33, 36 rotocenter, 16, 35 ruffle, 5 run, 177 S-curve, 180, 183, 220 saddle, 5, 190–192, 195, 197

Index  237

Saint Jerome in His Study, 85, 87, 106 Sargent, John Singer, 74, 75, 82, 106 satellite dish, 103 The School of Athens, 87, 90, 91 seed, 158, 159 segment, 176 self-similarity, 145 set Cantor, see Cantor set Mandelbrot, see Mandelbrot set shadow, 100 Sheeler, Charles, 177, 178 Sierpiński carpet, 152 triangle, 148, 152 Sierpiński, Waclaw, 148 signature, 19, 63 band ornament, 33–36 design, 19, 20 of regular pentagon, 21 of square, 21 of triangle, 21 tiling, 51, 58, 60, 61 similar, 131, 133 similarity, 131, 188 Sloan, John, 98 slope, 177, 178 Smilla’s Sense of Snow, 3 snowflake curve, Koch, 147, 148 snowflake, Koch, 147, 148, 150, 151 dimension, 150 solar cooker, 103 Sotatsu, Tawaraya, 133 space topological, 204 sphere, 5, 101, 188 spiral, 182 Spring Turning, 186, 187 square, 21, 43, 44 construction of, 21 symmetry of, 16, 17, 21 tiling, 44 St. Anthony Reading, 218 Starry Night over the Rhône, 130, 131, 133

Stella, Frank, 4 Still Life: Peaches and Grapes, 194 Street to Mbari, 212–216, 221 Stubbs, George, 194 surface, 188 symbol recycling, 24, 208 yin-yang, 22 symmetry, 11, 15, 16, 20, 63, 131, 214 bilateral, 23, 33, 214 point, 22, 23 radial, 24 transformation, 4, 16, 31, 32 trivial, 19 tessellation, 43 TesselManiac, 63, 67, 71, 72 theorem, Leonardo’s, 18 Tiger, 174, 175 tiling, 4, 12, 43, 45, 50, 131 Archimedean, 63 Egyptian, 50 regular, 43, 46 semiregular, 63 topological space, 204 topologically equivalent, 203 topology, 5, 201, 203, 204 torus, 205 tracing paper, 25 translation, 4, 31–36 transversal, 80, 82, 83 Travelling Carnival, Santa Fe, 98–100 Trees in Hawai’i, 144 triangle, 188, 216 equilateral, 20, 43, 45 construction, 21 symmetry of, 20 tiling, 45 isosceles, 93 Sierpiński, 148 tumbling blocks, 70 unit cell, 33, 35 unstable, 159 validity, mathematical, 197

238  Index

van Gogh, Vincent See Gogh, Vincent van, 130 Venice: The Doge’s Palace and the Riva degli Schiavoni, 74, 75 Vermeer, Johannes, 134, 135 vertex, 43 visual cone, 100 visual ray, 77, 78, 93, 99 Vitruvian man, 137, 138, 142 Vitruvius, 137, 138, 142 Web Sketchpad, 8 Wheat, 215, 217, 221 Whistlejacket, 191, 194 White, Susan Chrysler, 1 Wood, Grant, 186, 187 world, impossible, see impossible world The Yellow House, 108, 109, 113, 216 Yellow-Red-Blue, 2, 3, 219, 220 yin-yang, 22 Young Woman with a Water Pitcher, 135 zenith, 112 zigzag, 59