Treks into Intuitive Geometry: The World of Polygons and Polyhedra 4431558411, 9784431558415

Table of contents :
Preface
Acknowledgement
Contents
About this Book
About the Authors
Chapter 1 Art From Tiling Patterns
1. Geometric Patterns
2. Tilings
3. Conway Criterion
4. Math Behind Masterpieces
Appendix 1.1.1
Appendix 1.2.1
References
Chapter 2 The Tile-Maker Theorem and Its Applications to Art and Designs
1. How to Draw Repeated Patterns Like Escher’s
2. Tile-Makers
3. Characteristics of a Net of an Isotetrahedron
4. Polyhedral Stampers
5. Dihedral Stampers and Dihedral Tile-Maker Theorem
6. From Shark Heads to Tile-Makers
7. Exhibition of Collage Art
Appendix 2.3.1
Appendix 2.4.1
Appendix 2.5.1
References
Chapter 3 Patchwork
1. Patchwork — Equidecomposable Pairs of Polygons
2. How to Turn a Polygon into a Rectangle
3. Theorem of Equidecomposability
References
Chapter 4 Reversible Pairs of Figures
1. Envelope Magic
2. The Envelope Magic Trick
3. Trapezoidal Envelope Magic
4. Dudeney’s Haberdasher’s Puzzle
5. Reversible Transformation of Figures
6. Tessellability and Shapes of Trunks
7. Miscellaneous Properties of Uni-Trunk Holders
8. Conway Polygons
Appendix 4.4.1
Appendix 4.5.1
Appendix 4.7.1
Appendix 4.8.1 (Continuation of Proof of Theorem 4.8.1)
Appendix 4.8.2
References
Chapter 5 Platonic Solids
1. The Platonic Solids
2. Euler’s Formula for Polyhedra
3. Plato’s Image for Platonic Solids
4. Duality Between a Pair of Polyhedra
5. The Cycle of Platonic Solids
Appendix 5.2.1
Appendix 5.3.1
References
Chapter 6 Cross-Sections of Polyhedra
1. CT Scanners and 3-D Copiers
2. Cross-Sections of a Cube
3. Universal Tilers
4. Mitsubishi and the Star of David
References
Chapter 7 Symmetry of Platonic Solids
1. Diagonal Weights
2. A Soccer Ball and a Fullerene
3. The Platonic Solids through Reflecting Mirrors
Appendix 7.1.1
Appendix 7.1.2
Appendix 7.1.3
References
Chapter 8 Double Duty Solids
1. TetraPak®
2. Double Duty Solids
3. Finding More Double Duty Solids
4. Eleven Types of e-Nets of a Cube
References
Chapter 9 Nets of Small Solids with Minimum Perimeter Lengths
1. Minimum Steiner Tree Problem
2. Melzak’s Algorithm
3. Minimum Perimeter Nets for a Regular Octahedron O
4. Minimum Perimeter Net for a Cube C
5. Soap Experiments for Minimum Steiner Trees
6. Soap Experiments for Nets with Minimum Perimeter Lengths for Small Solids
Appendix 9.1.1
Appendix 9.2.1
References
Chapter 10 Tessellation Polyhedra
1. Regular Faced Polyhedra
2. Tessellation Polyhedra
3. Art
Appendix 10.2.1
References
Chapter 11 Universal Measuring Boxes
1. How to Measure Water Using a 6 dL Box
2. Changing the Shape of the Box
3. Orthogonal Measuring Boxes
4. An Application of the Ternary System
5. Non-Orthogonal Universal Measuring Boxes
Appendix 11.4.1
References
Chapter 12 Wrapping a Box
1. Wrapping Technique of the Experts
2. Point Symmetric Skew Wrapping
3. Patchwork Surgery
4. Investigating Efficient Wrappings
5. Six Remarkable Wrappings Obtained by Patchwork Surgery
6. The Most Saving But Mummy-Like Wrapping
References
Chapter 13 Bees, Pomegranates and Parallelohedra
1. How to Pack Cans Efficiently
2. Honeycombs and Pomegranate Seeds
3. Applications of Sphere Packing
4. Rhombic Dodecahedron
5. Truncated Octahedron
6. Lattices and Voronoi Domains
7. Parallelohedra
References
Chapter 14 Reversible Polyhedra
1. A Pig to a Ham, a Fox to a Snake, and Panda Magic
2. Transformation of a Pig to Ham
3. How to Make a Fox-Snake Solid
4. Make a Fox-Snake Solid with Paper
5. Making Panda Magic!
6. How to Make Reversible Polyhedra
7. Who Will Come Out of the U.F.O.?
8. P1-lattices and (P, Q)-Chimera Superimpositions
9. P1-Reversible Pairs Among Canonical Parallelohedra
10. Reversibility for Parallelohedra and the Double-Reversal-Plates Method
References
Chapter 15 Elements of Polygons and Polyhedra
1. The Power of Triangulation
2. Element Sets and Element Numbers
3. Hilbert’s Third Problem and Dehn Theorem
4. An Element Set and the Element Number for Platonic Solids
5. Another Element Set for the Platonic Solids
Appendix 15.4.1
References
Chapter 16 The Pentadron
1. Atoms for Parallelohedra
2. Another Way to Find a Pair of Pentadra
3. The Chart for Composing Five Parallelohedra with Pentadra
4. Pentadral Complices
5. A Bridge Between Platonic Solids and Parallelohedra
6. Art and Design
Appendix 16.1.1
References
Figure Permissions
© J Art 2015
Copyright
Endnote
Index
Symbol