Case Studies in Parametric Design. A Guide to Visual Scripting in Architecture 9781032289755, 9781032289717, 9781003299417

Citation preview

Case Studies in Parametric Design

Case Studies in Parametric Design is a guide to scripting digital models for architects, designers, and builders. The use of parametric design in architecture has afforded the realization of incredible built work; modelling software can resolve complex geometry and aid in the development of stunning creations. Methods for creating the digital models to achieve these results, however, can be perplexing. Learning curves are steep, and benefits garnered from adapting existing workflows to incorporate new tools may appear trivial. This book describes programming techniques for a variety of buildings and provides novices an understanding of language and processes, challenges intermediate users with rigor and intentionality, and offers proficient practitioners objectives beyond novel form-making. The case studies consist of six mass topologies and six facade topologies; each includes sample topology models and scripts, descriptions of the steps for generating customizable parametric models, and suggestions for additional modelling inquiries. This is essential reading for students and practitioners interested in harnessing the full potential of parametric design. Jeffrey Collins is a Registered Architect and Associate Professor of Architecture at Kennesaw State University. He earned his B.S.Arch and M.Arch degrees from The Ohio State University and his doctorate with a concentration in design computation from the Georgia Institute of Technology. Dr. Collins has been honored by AIA Atlanta with the Emerging Voices Citation and by the National Council of Architectural Registration Boards (NCARB) as a Scholar in Professional Practice.

“Case Studies in Parametric Design provides a welcoming entry point for architecture students and professionals interested in learning visual scripting. Collins presents clear diagrams, straightforward text, and accessible terminology to describe the timeless logics of geometric relationships in architecture.” –Shelby Elizabeth Doyle, AIA, LEED AP, Associate Professor, Department of Architecture, Iowa State University, USA “At a critical time while we are experiencing a paradigm shift in design practice, Case Studies in Parametric Design highlights the importance of metric-based solutions, preparing designers for a more performative future.” –Arash Soleimani, Ph.D., Chair, Design Computation and Applied Computer Science – Media Arts Associate Professor, School of Architecture, Woodbury University, USA

Case Studies in Parametric Design

A Guide to Visual Scripting in Architecture Jeffrey Collins

Designed cover image: © Jeffrey Collins First published 2024 by Routledge 4 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 605 Third Avenue, New York, NY 10158 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2024 Jeffrey Collins The right of Jeffrey Collins to be identified as author of this work has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Collins, Jeffrey (Jeffrey M.), 1978- author. Title: Case studies in parametric design : a guide to visual scripting in architecture / Jeffrey Collins. Description: Abingdon, Oxon : Routledge, 2024. | Includes bibliographical references and index. Identifiers: LCCN 2023011109 | ISBN 9781032289755 (hardback) | ISBN 9781032289717 (paperback) | ISBN 9781003299417 (ebook) Subjects: LCSH: Architectural design–Data processing–Case studies. | Architecture– Mathematics–Case studies. | Parametric modeling–Case studies. | Structural optimization–Case studies. | Computer-aided design–Case studies. Classification: LCC NA2728 .C627 2024 | DDC 720.285—dc23/eng/20230324 LC record available at https://lccn.loc.gov/2023011109 ISBN: 978-1-032-28975-5 (hbk) ISBN: 978-1-032-28971-7 (pbk) ISBN: 978-1-003-29941-7 (ebk) DOI: 10.4324/9781003299417 Typeset in Corbel by Apex CoVantage, LLC

CONTENTS

Image credits

vii

Acknowledgments

viii

Foreword

ix

INTRODUCTION

3

1 EXTRUDE

36

2 LOFT

57

3 STACK

85

4 CARVE

114

5 NEST

139

6 TRIM

164

7 QUAD

191

8 DIAMOND

214

9 TESSELLATE

238

Carabenchel Housing

Absolute Towers

Gates Center for Computer Science and Hillman Center for Future Generation Technologies

The Orange Cube

Ftown Building

Centre Pompidou-Metz

Manifold House

Prada Aoyama

Bergeron Center for Engineering Excellence

vi

CONTENTS

10 ATTRACTOR

268

11 IRREGULAR

293

12 LAYER

322

Conclusion

346

Referenced built works

347

Index

351

City View Garage

Sugar Hill Development

Gardner Neuroscience Institute

IMAGE CREDITS

EXTRUDE

LOFT STACK

Carabenchel Housing

38

Absolute Towers

62

Gates Center for Computer Science and Hillman Center for Future Generation Technologies

88

Photo by Duccio Malagamba Fotografia de Arquitectura Photo by Jeffrey Collins

Photo by Jeffrey Collins

CARVE NEST TRIM QUAD

DIAMOND

The Orange Cube

116

Ftown Building

142

Centre Pompidou-Metz

166

Manifold House

196

Prada Aoyama

218

Photo by Roland Halbe Fotografie Photo by Daici Ano Photo by Ken Lee

Photo by Paul Warchol Photography Inc Photo by Johannes Marburg, Geneva

TESSELLATE Bergeron Center for Engineering Excellence

252

ATTRACTOR City View Garage

272

IRREGULAR

Sugar Hill Development

300

Gardner Neuroscience Institute

324

Photo by Jeffrey Collins

Photo by IwamotoScott Architecture

LAYER

Photo by Rubi Xu

Photo by Jeffrey Collins

ACKNOWLEDGMENTS

I am grateful to many mentors, peers, students, publishers, and family members who have helped me in innumerable ways to write this book. I recognize advisors Russell Gentry, Professor, School of Architecture, Georgia Institute of Technology, and Dennis Shelden, Director, CASE Center for Architecture Science and Ecology and Associate Professor, School of Architecture, Rensselaer Polytechnic Institute, for their guidance and support. I thank colleagues Margaret Fletcher, Professor, School of Architecture Planning and Landscape Architecture, Auburn University, and Arash Soleimani, Chair, Design Computation and Applied Computer Science – Media Arts and Associate Professor, School of Architecture, Woodbury University, for their significant feedback on early manuscript drafts. I appreciate my students – past, present, and future – for inspiring me with rigor and curiosity. I value all efforts by Jake Millicheap, Lydia Kessell, and Fran Ford at Routledge. Finally, my love and gratitude to my amazing partner and wife Jennifer and our extraordinary daughters Lida and Edie for their unwavering encouragement during this undertaking (including an epic road trip to see several of these buildings!) and to whom I dedicate this book.

FOREWORD

Professor Jeffrey Collins’ Case Studies in Parametric Design is a significant and worthy contribution to the historical arc of computational architecture and design. I had the privilege of working with Collins during his time at Georgia Tech in the School of Architecture and the Digital Building Lab – the building information modelling, data science and ontology program founded by the great Chuck Eastman. I had recently arrived from Gehry Technologies (GT) as the new director of the lab on Chuck’s retirement. Georgia Tech was – as it still is – at the forefront of computational design and applications to practice, and a central contributor to the history of computational design and building information modelling. While he was conducting his research, Collins was also leading instruction in the geometric and parametric modelling courses taken by all levels of students in the program from undergraduate through M.Arch and Ph.D. The methodologies and exercises discussed in this book emerged from Collins’ work as a leader of this program. Parametric design has taken on a critical role in the theory and practice of architecture. For some, it is part of an automation process that can accelerate design iteration and reduce the labor of knowledge work and rework. For others, it is an enabler of a generative aesthetic that breaks the mold of the tyranny of Euclidean constructs of antiquity by making new algorithmic geometries accessible to adept digital practitioners. Patrick Schumacher, in his extensive treatise The Autopoiesis of Architecture, positions parametrics as both a formal and sociological construct that can connect interventions of built form to societal agendas taking place in the public in a formal and deterministic way. Collins’ contribution is part of this history but is seen from a specific vantage that is not simply about the automation of the documentation or generation of form but rather of what might be described as an archeological investigation of form – an inquiry into the hidden forces underlying the architectural agenda in general and the built projects that stand as a record of these intentions. Buried under and throughout the visible end result of the architectural design and construction processes is an architectural, geometric, and logically founded language. This position was articulated in Bill Mitchell’s foundational book The Logic of Architecture, a treatise highly influenced by a generation of computational theorists working as part of the shape grammar community. As with

x

FOREWORD

written and spoken language, the grammatical structures that emerge through an evolution of communication, through infinite mimesis, repetition, and evolution, form a sociological series of structural agreements across interlocutors. Design languages undergo a similar historical arc of consensual development, but it is one of spatial rather than spoken or written words, phrases, and texts, that ultimately concerns the meanings of architecture and of building. The elements of these languages are not simply signifiers, but they carry the weight of geometric, spatial, and physical forces in their constructions. Shape grammars and, in similar ways, parametrics provide the frameworks for this archaeological investigation. Parametrics allows us to capture, formally encode, and explicitly play back the layers of this geometric logic of form. Collins’ work applies this archeological agenda to the reconstruction of the underlying tectonic logic of form and construability; this agenda is also very much the focus of Gehry and GT’s approach to parametrics and is now in much of the architectural building information modelling practice. For Collins, the beams, columns, slabs, and facades of buildings spring from underlying scaffolds, the regular partitioning of space through orthogonal subspaces of planes, lines, and points in a regularized boundary graph. These scaffolds form the parametric drivers on which the ensuing language of construction is expressed. The duality of the constructed, physical components of building systems and the system of spaces and their programmatic organizations are both generated from this unifying scaffold. Herein, Collins expands this foundational framework to present how this methodology is expressed across formal building topologies and provides even the most entry-level architectural designer access to these processes in a constructible, practical, and operationalizable way. As such, Case Studies in Parametric Design is simultaneously a theoretical manifesto, an archaeological expedition, and a guide book for designers willing to undertake this shared journey of languages of the logic of architecture in an educational and professionally meaningful way. Dennis Shelden, Ph.D. Director, CASE Center for Architecture Science and Ecology Associate Professor, School of Architecture, Rensselaer Polytechnic Institute

0.01

INTRODUCTION

The use of parametric design in architecture has afforded the realization of incredible built work; modelling software can resolve complex geometry and aid in the development of stunning creations. The methods for creating the digital models to achieve these results, however, can be perplexing. This book aims to empower the reader with the knowledge and confidence to generate their own customizable parametric models. For those with some parametric modelling experience already, this book offers strategies for increasing rigor and intentionality.

Opposite page: Image of a digital model based on Trabeculae Pavilion designed by ACTLAB. DOI: 10.4324/9781003299417-1

0.02

INTRODUCTION

What, why, and how of parametric design In order to generate parametric models, we must first understand geometry or more particularly, how to define geometry. Geometry is concerned with the shapes, sizes, angles, locations, and relationships between parts of objects. Whether deliberately or not, when we design forms, we are defining sets of geometrical rules. When we are more explicit about these formal relationships – which developing a parametric model requires us to be – we gain more control over form. Therein, we gain more control over our designs. Let us think about a circle. What dimensions are required to define a circle? Although there may be multiple ways to define a circle, one of the simplest is with two factors: a centerpoint and a radius. In fact, this is the way that any CAD program helps us to draw a circle. When you initiate the “circle” command in your software, you are first prompted to define a location for the centerpoint of the circle and then a dimension for the radius of the circle. At this stage, you could be done with this circle and move on to drawing other things. But this circle is fixed. It will always be at this location and this size. (Yes, we could move it or scale it, but that is cumbersome and not the point.) If instead we want to make this circle parametric – wherein we can directly and on the fly adjust its centerpoint and radius – we need to combine geometry with algebra. The key to algebra is the use of variables. Variables are placeholders. Instead of assigning one specific number, a variable stands in for any possible number. So, rather than a fixed location and size, if we can instead consider the location and size of our circle as variable, then we have defined an abstract model for all possible circles. Although this is a very simple example, conceptually, such a circle is a parametric model. This concept is very powerful. When we create more complex geometries, develop more intricate relationships using similar logic, and incorporate these methods of defining form into our design exploration processes, this is parametric design.

Opposite page: Image of a digital model based on the Lightmos building designed by architectkidd.

5

6

INTRODUCTION

The benefits of parametric design include: 1. Enhanced geometrical control Because we consider how forms are defined in order to model them, we have increased control over these forms 2. Ability to produce design variations Because we incorporate variables and the relationships between variables, we can test different values and assess the results 3. Potential of emergent design possibilities Because we are abstractly modelling all geometrical possibilities rather than specific geometrical instances, we may happen upon designs that may not have otherwise been considered We can begin thinking parametrically by listing the variables that define geometry. A point is defined by its location in a coordinate system: x, y, and z. A line connects two such points. A circle is defined by a centerpoint and a radius. Further examples that pair geometry with lists of variables – beginning with 2D shapes and extending to 3D forms – are shown on the following pages. Note how the lists build on one another; the cylinder begins with a circle that is then extruded, the cube and array both begin with a square, and so forth.

8

point coordinates:

INTRODUCTION

x, y, z

0.03

line control point: control point:

x, y, z x, y, z

0.04

circle location: radius:

x, y, z r

0.05

surface (square) location: x, y, z length: x width: y

0.06

INTRODUCTION

9

extrude (cylinder) circle location: x, y, z radius: r extrude shape: height:

circle z 0.07

loft (square, circle) square location: x, y, z length: x width: y circle location: radius:

x, y, z r 0.08

Boolean difference (cube, sphere) cube location: x, y, z length: x width: y height: z sphere location: radius:

x, y, z r

0.09

linear array (squares) square location: x, y, z length: x width: y array shape: height: number:

square z n

0.10

INTRODUCTION

What, why, and how of visual scripting Let us return for a moment to the process of drawing a circle in a CAD program, but let us think about it from the program’s point of view. You say you want to draw a circle. But this is not enough data. So, the program asks, “Where should the circle be located?” and “How big should the circle be?” When you provide this information, the program can then draw your circle. From this perspective, we can see that the program already considers the input data that you provide as variables. The program knows that a circle can be anywhere and any size. Similar steps occur every time that we draw or model any object. In fact, the list of variables that we would develop in order to explicitly describe geometry – like those on the previous pages – comprise the same data that a program needs in order to draw or model these forms. One strategy to be able to communicate even more effectively with our software is to learn to speak its language: code. Learning to code – even just a little – can help us understand how the computers that we rely on work. The following pages show a code for each form illustrated in the previous section. The code language used is RhinoScript (Robert McNeel & Associates). The main elements of the code include commands and inputs. For a point, the command is “AddPoint.” The inputs are the desired coordinates for this point. Rather than defining specific numbers, inputs can also be defined as variables. For example, in the code for a line, the coordinates for two points are defined. Then, the command “AddLine” refers to each point for the location data. Similar to the previous lists of variables, note how the codes build upon one another by reusing elements or lines from simple codes as part of more complex ones.

11

12

INTRODUCTION

point Coordinates in the x, y, and z directions are defined.

rs.AddPoint(1, 1, 0)

line Two control points are defined, and a line is drawn between them.

point1 = (2, -6, 0) point2 = (8, 8, 0) rs.AddLine(point1, point2)

circle A centerpoint and a radius are defined.

point1 = (0, 0, 0)

radius = 5.0

rs.AddCircle(point1, radius)

surface (square) One corner of a rectangle and the length and width of the rectangle are defined. This profile is converted to a surface.

point1 = (-5, -5, 0)

length = 10.0

width = 10.0

shape = rs.AddRectangle(point1,

length, width)

rs.AddPlanarSrf(shape)

extrude (cylinder) A centerpoint and a radius for a circle and an extrusion height are defined. “Path” describes a line, which is the direction and length of the extrusion.

point1 = (0, 0, 0)

radius = 5.0

height = 10.0

point2 = (0, 0, height)

shape = rs.AddCircle(point1, radius)

path = rs.AddLine(point1, point2)

rs.ExtrudeCurve(shape, path)

loft (square, circle) One corner of a rectangle and the length and width of the rectangle are defined. A centerpoint and a radius for a circle are defined. “Curves” creates a list of shapes to

be lofted.

point1 = (-5, -5, 0)

length = 10.0

width = 10.0

shape1 = rs.AddRectangle(point1,

length, width)

point2 = (0,0,10)

radius = 3.0

shape2 = rs.AddCircle(point2, radius)

curves = (shape1, shape3)

rs.AddLoftSrf(curves)

INTRODUCTION

Boolean difference (cube, sphere) One corner of a rectangle, the length and width of the rectangle, and an extrusion height are defined. “Path” describes a line, which is the direction and length of the extrusion to create a cube. A centerpoint and a radius for a sphere are defined. The sphere is removed from the

cube using a Boolean difference.

13

point1 = (-5, -5, 0)

length = 10.0

width = 10.0

height = 10.0

point2 = (-5, -5, height)

shape = rs.AddRectangle(point1,

length, width)

path = rs.AddLine(point1, point2)

cube = rs.ExtrudeCurve(shape, path)

rs.CapPlanarHoles(box)

centerpoint = (5, -5, 10)

radius = 5.0

sphere = rs.AddSphere(centerpoint,

radius)

rs.BooleanDifference(cube, sphere)

linear array (squares) The length and width of a rectangle

are defined. The height of an array is defined. “NumLevels” represents the

number of copies of the surface to make. For each level, a corner is defined. Level 1 is on the XY plane (the z coordinate is 0). For level 2, the z coordinate is equal to “height” divided by the number of levels. For level 3, the z coordinate is equal to “height” divided by the number of levels and then multiplied by 2. For level 4, the z coordinate is equal to “height” divided by the number of levels and then multiplied by 3. Each profile is converted to a surface.

length = 10.0

width = 10.0

height = 10.0

numLevels = 4

point1 = (-5, -5, 0)

shape1 = rs.AddRectangle(point1,

length, width)

rs.AddPlanarSrf(shape1)

point2 = (-5, -5, height/numLevels)

shape2 = rs.AddRectangle(point2,

length, width)

rs.AddPlanarSrf(shape2)

point3 = (-5, -5, height/numLevels*2)

shape3 = rs.AddRectangle(point3,

length, width)

rs.AddPlanarSrf(shape3)

point4 = (-5, -5, height/numLevels*3)

shape4 = rs.AddRectangle(point4,

length, width)

rs.AddPlanarSrf(shape4)

14

INTRODUCTION

Even with these relatively simple geometries, we can already tell that as codes become more complex, understanding how they operate will be more challenging. For many of us, coding is not intuitive. Computers are really good at tracking numbers and relationships between assorted inputs and lists of data. For people, writing, running, debugging, editing, and re-running code can be painstaking. This is especially true if we did not write the code ourselves (or we wrote it some time ago), which is common practice. It is more intuitive for us to organize functions graphically. Representing our thinking visually helps us organize information and identify relationships (Brand). The following diagrams represent how the previous codes operate. This is visual scripting; the diagram is called a script. In a script, we see graphically how individual elements in a code (which are symbolized as nodes) interact to produce the desired output. Again, note how the scripts build upon one another by reusing elements or combinations of elements from simple scripts to develop more complex ones.

INTRODUCTION

15

point Parameters (black nodes) are connected to variables (white nodes) within functions (grey nodes) with black dashed lines.

0.11

line Coordinate data are embedded within control points.

0.12

circle The inputs for a circle are a centerpoint and radius.

0.13

surface (square) Some output can be connected to additional functions with grey dashed lines.

0.14

16

INTRODUCTION

extrude (cylinder) The output for a circle is the input for an extrusion.

0.15

loft (square, circle) The output for both a rectangle and circle is the input for loft.

0.16

INTRODUCTION

17

Boolean difference (cube, sphere) For a cube, the same parameter value is used for a rectangle’s length and width and extrusion height.

0.17

linear array (squares) In this case, a rectangle’s length and width are kept distinct for flexibility; therefore, their values could be the same or different.

0.18

18

INTRODUCTION

It is useful to think of each node as carrying data. When developing a script (just like writing a code), consider what input each node requires. Consider the geometrical possibilities of each corresponding output. Some output data may be individual elements, such as a cylinder. Other output may be a series of items – for example, each surface of the linear array – arranged in a list. Subsequent operations can be applied to all such items, specific items, or filtered sets of items. In addition to the benefits discussed for parametric design, the benefits of visual scripting include: 1. Graphic depiction of code Because nodes representing variables and functions are organized visually, the relationships between them are clearer. 2. Real time model variations Because scripts link variations in code to digital models, graphic updates to both are simultaneous. 3. Effective use of variables Because we can see repeating or redundant variables, we can strategize whether to intentionally share (or not) variables between distinct functions. With these basic approaches for parametric design and visual scripting in mind, the next step is to extend these practices to models that represent buildings. We can begin by considering the shapes and forms previously discussed as parts of buildings. The extruded and lofted shapes illustrated could represent conceptual building masses. The arrayed rectangles shown could represent building floors. In fact, most buildings are composed of similar elements – an overall shape, an organizational grid pattern, floor levels, and exterior wall surfaces. If we explicitly define the parameters of and relationships between each of these components, then we will develop a conceptual model for a building. This concept is what I call a scaffold model.

Opposite page: Image of a digital model based on Aqua Tower designed by Studio Gang.

0.19

20

INTRODUCTION

Scaffold models Scaffold models are digital parametric concept models that represent buildings. Such models can serve as a framework with which additional components can be associated. The case studies in this book are organized into two scaffold model types: mass models and facade models. Mass scaffold models are structured by four interrelated components:

0.20

SHAPE

GRID

Shape defines a basic boundary.

Grid defines an organizational system.

rectangle length x width y

regular grid shape number of divisions X number of divisions Y

[100] [50]

[rectangle] [6] [3]

INTRODUCTION

21

0.21

LEVELS

SURFACES

Levels define each floor.

Surfaces define each building facade.

linear array shape number of floors n floor-to-floor height z

[rectangle] [3] [19]

extrude shape height z

[rectangle] [num of flrs * flr-to-flr height]

22

INTRODUCTION

0.22

MASS SCAFFOLD The combination of the previously modelled shape, grid, levels, and surfaces defines a scaffold for a building mass. The script includes six parameters: • • • • • •

rectangle length x rectangle width y grid number of divisions X grid number of divisions Y number of upper floors n floor-to-floor height z

INTRODUCTION

0.23

23

24

INTRODUCTION

Similarly, most facades are composed of key elements – a surface, an organizational grid, various shapes, and patterns. Facade scaffold models are therefore also structured by four interrelated components:

0.24

SURFACE

GRID

Surface is a selected face of the building mass.

Grid defines an organizational system.

rectangle length x height z

[100] [57]

regular grid shape number of divisions X number of divisions Z

[surface] [12] [3]

INTRODUCTION

25

0.25

SHAPE

PANELIZATION

Shape defines an initial pattern.

Panelization defines individual pieces.

offset curve distance s

regular grid shape number of divisions X number of divisions Z

[panels] [0.30]

[surface] [same as grid] [1]

26

INTRODUCTION

0.26

FACADE SCAFFOLD The combination of the previously modelled surface, grid, shape, and panelization defines a scaffold for a building facade. The script includes six parameters: • • • • • •

surface length x surface height z grid number of divisions X grid number of divisions Z panel offset distance s panelization number of divisions Z

INTRODUCTION

0.27

27

28

INTRODUCTION

Using these basic conceptual scaffolds as a starting point, more complex building forms and facade patterns can be developed. In addition to those discussed for parametric design and visual scripting, the benefits of scaffold models include: 1. Breaking down model-making into steps Because digital modelling is complex, defining elements in a step-bystep process increases rigor and intentionality. 2. Ability to link further elements Because scaffold models serve as a conceptual framework, additional building components can be added to and associated with this predefined structure. 3. Opportunity to define topologies Because scripts produce variations of similar forms, models can be organized into types.

INTRODUCTION

29

Symbols, terms, abbreviations, and definitions θ

Theta; Greek symbol used to denote angle

algorithm

A set of instructions used to solve a problem

array

A function wherein an object is repeated n number of times

Boolean difference

Trimming of distinct solid objects; one or more objects are used to carve away from one or more other objects

Boolean intersect

Finding the overlapping portions of more than one solid object

Boolean union

A combination of distinct solid objects into one mass

CAD

Computer Aided Design

ctrl

Control “Ctrl pt” designates a control point, a point that defines an aspect of another geometry

computation

The act of computing; using a computer

code

Program instructions

coordinate

A numerical location in a coordinate system

coordinate system

A method for identifying a location in relation to a fixed origin

curves

Lines Although “curve” typically refers to lines that are not straight, some software identifies straight and not straight lines as curves

data

Information

design computation

The use of computers to aid in design processes

diagrid

A spatial organization based on parallel diagonal lines

difference

In digital modelling, trimming of distinct objects; one or more objects are used to carve away from one or more other objects

digital model

A three-dimensional representation of a building developed by using computer software (Continued)

30

INTRODUCTION

div

Divisions, typically used in the phrase “number of divisions”

extrude

Extending a shape in one direction by a set amount

f

Factor

facade scaffold

Parametric model defining the relationships between key elements of a building facade: surface, grid, shapes, and panelization

flr

Floor

flrs

Floors An activity performed by a computer program that involves one or more variables

glide-reflect

A function used in tessellated patterns; to move and mirror a copy of an object

grd

Ground

grid

A spatial organizational system Grid typically refers to “regular grid;” however, there are many types of organizational systems

hex grid

A spatial organization based on hexagons

ht

Height

index

The numerical location of data in a list (Note computer lists begin with “0” rather than “1”)

input

Data put into a computer function

intersect

Finding the overlapping portions of one of more objects

levels

Type of geometry within the mass scaffold concept representing the floors of a building

linear array

A function wherein an object is repeated n number of times in a defined direction

list

An ordered set of data

INTRODUCTION

31

loft

Generating surfaces between shapes to create form

mass scaffold

Parametric model that defines the relationships between key elements of a building mass: shape, grid, levels, and surfaces

model

A representation, typically three-dimensional; in this book, “model” generally refers to a digital model of a building

n

Number

node

In a script, an individual element In geometry, an intersection point

num

Number

output

Data resulting from a computer function

panelization

The pattern that defines individual pieces of a facade One of a group of numerical values set in relation to one another to define an object

parametric

Flexible through the use of variables

parametric design

The creation and use of parametric models during design processes

parametric model

A geometrical representation defined by using variables

polar array

A function wherein an object is repeated n number of times around a defined base point

polyline

A multi-segment line composed of straight elements that join control points

program

Computer software or the function of a building

programming

Writing computer software or describing the use of space

pt

Point

pts

Points

r

Radius (Continued)

32

INTRODUCTION

range

Values in between a defined minimum and maximum for a parameter

reflect

When referring to geometry, to mirror across a defined axis

region

A defined portion of a surface

regular grid

A spatial organization based on parallel orthogonal lines

rotate

To turn an object around a base point

s

Distance Parametric model that defines the relationships between key building systems and with which additional components can be associated

scale

The size of an object or to change the size of an object by a certain factor about a base point

script

A graphic representation of a code

spline

A curved (not straight) line formed via control points

syntax

The arrangement of words, phrases, and symbols that create a language

tessellate

Patterns composed of repeated tiled shapes that are copied, rotated, mirrored, and/or arrayed in various combinations The classification of geometrical objects based on common features that remain unchanged under deformation such as stretching

translate

When referring to geometry, to move

triangular grid

A spatial organization based on diagonal lines and parallel lines that form triangular shapes

union

The combination of distinct objects into one

value

A number

INTRODUCTION

33

A feature of an object that can be modified

visual scripting

The creation and use of scripts to develop parametric models

x

Length; dimension in the x direction of a coordinate system

y

Width; dimension in the y direction of a coordinate system

z

Height; dimension in the z direction of a coordinate system

34

INTRODUCTION

Organization of this book As we create and collect different parametric models, we may note relationships between the way that we have written a script and its possible formal outcomes; how the geometry is formed, the tools that we have used to define it, and the ways that variations behave are interrelated. Similarly, as we investigate and model different works of architecture, we find repeating formal types. It is helpful to classify scripts into topologies. Then, we can strategize and predict which modelling techniques will produce the desired parametric variations. The case studies in parametric design included in this book consist of six mass topologies and six facade topologies. Mass models include forms that are extruded, lofted, stacked, carved, nested, and trimmed. Facade models include patterns that are quadrilateral, composed of diamonds, tessellated, effected by attractors, composed of irregularities, and layered. Each case study section includes: • • • • • • • • •

sample topology model(s) sample topology script(s) reference to example topologically similar built works description of the steps for generating an “initial scaffold” parametric model of the case study project description of the steps for generating a “developed scaffold” parametric model of the case study project frequent “further discussion on” pages that explore additional modelling procedures scaffold script for the case study project design variations of the case study project achieved via the scaffold model prompts for additional discussion and modelling exploration

The projects included herein are curated to represent variety; they intentionally exhibit assorted shapes, forms, scales, and materials, serve various purposes, were designed by diverse size and notoriety of architecture firms, and are in locations around the world. The buildings were selected to demonstrate digital modelling techniques. It is not important whether the design teams for each project incorporated parametric design into their design exploration processes. Instead, the building models are used to demonstrate step-by-step instructions to inspire the reader to create their own scripts of these and – more importantly – other parametric designs. Then, these models can be used to optimize design decisions.

INTRODUCTION

Additional Recommended Resources Architectural Geometry by Helmut Pottmann, Andreas Asperl, Michael Hofer, and Axel Kilian. (Bentley Institute Press. 2007.) Textbook illustrating the use of geometry to create digital models of buildings. Computational Design Thinking edited by Achim Menges and Sean Ahlquist (John Wiley & Sons. 2011.) A sample of writings by seminal and influential leaders in the field of design computation, both technically and philosophically. Code: The Hidden Language of Computer Hardware and Software by Charles Petzold. (Microsoft Press. 2000.) Describes how computers work and have evolved from simpler, easier-tounderstand technologies. Visual Thinking: Empowering People and Organizations through Visual Collaboration by Willemien Brand. (Laurence King Publishing. 2017.) Discusses both the importance of visual thinking and steps for incorporating visual thinking strategies into your work. Operative Design: A Catalog of Spatial Verbs by Anthony di Mari. (Laurence King Publishing. 2013.) A “visual dictionary” of abstract mass forms. Facades: Design, Construction &Technology by Lara Menzel. (Braun Publishing. 2012.) A collection of built works with unique skins, organized by facade pattern type.

Notes Brand W. Visual Thinking: Empowering People and Organizations Through Visual Collaboration. Laurence King Publishing, 2017. Robert McNeel & Associates. RhinoDeveloper. “RhinoScriptSyntax in Python.” https://developer.rhino3d.com/guides/rhinopython/python-rhinoscriptsyntaxintroduction/

35

01

EXTRUDE

Extending a shape to create form

1.1

We begin with extrude as it is the simplest method for creating a three-dimensional form. The mass scaffold model discussed in the Introduction is extruded. When extruding, a defined shape is extended in one direction by a set amount. But do not let this simple description fool you. Extruded buildings can be very beautiful, their surface expression enhanced by quiet forms (see SHoP Architects’s Mulberry House or Ensamble Studio’s Ensamble Fabrica). Similarly, this simple transformation can generate form from complex shape profiles (such as Cobe’s Forfatterhuset Kindergarten). Although shapes are typically extruded vertically – extending from plan – they can also be extruded horizontally – extending in section (such as MVRDV’s Markthal). Furthermore, extrude can be used at multiple scales to generate an overall form or individual elements (for example, the fins of POLO Architect’s Red Cross-Flanders or the balconies of Motta’s Lomo Cubes).

DOI: 10.4324/9781003299417-2

EXTRUDE

Scripting extrusions

1.2

Input data for the extrude typology are shape and height. For a simple extruded rectangle, there are therefore three parameters: length, width, and height. Other extruded geometry may have more or less parameters depending on the shape.

37

1.3

CASE STUDY 01

Carabenchel Housing building type architect location size shape material year built

housing Foreign Office Architects Madrid, Spain 122,000 sf rectangle bamboo screens 2008

Although regulations set the number and type of units and the maximum height for the project, the design team for Carabenchel Housing set out to break other affordable housing norms. Each residence spans the width of the thin building, providing views, daylight, and airflow. From the selection of cladding material, incorporation of balconies, and arrangement of units, a lot is going on in this extruded form.

Opposite page: Photo of Carabenchel Housing by Duccio Malagamba Fotografia de Arquitectura.

40

EXTRUDE

1.4

SHAPE rectangle length x width y

[330] [55]

Scripting a model of the Carabenchel Housing building begins with a basic rectangle – 330' by 55' – defined on the XY plane.

EX TRUDE

41

1.5

GRID regular grid shape number of divisions X number of divisions Y

[rectangle] [16] [1]

A regular grid is applied to the rectangular boundary. To represent the units that span the full width of the building, the number of divisions in the Y direction is set to one.

42

EXTRUDE

1.6

LEVELS

linear array

shape number of floors n floor-to-floor height z

[rectangle] [6] [9.5]

The previously defined rectangular profile is converted to a surface and arrayed vertically seven times in order to model six floor levels. (The seventh surface represents the roof.) The spacing between each surface copy is nine-and-a-half feet, which is the placeholder floor-to-floor height.

EXTRUDE

43

1.7

SURFACES

extrude shape height z

[rectangle] [number of floors * floor-to-floor height]

The rectangular profile is extruded vertically by an amount equal to the

product of the previously defined number of floors (six) and the floor-tofloor height (nine-and-a-half): 57'.

44

EXTRUDE

1.8

INITIAL SCAFFOLD The combination of the previously modelled shape, grid, levels, and surfaces defines an initial mass scaffold for the Carabenchel Housing building. The following are noted as limitations of this model, whose solutions will be discussed in the next section: • the grid for Carabenchel Housing has irregular spacing • the floor-to-floor heights in Carabenchel Housing vary • the surfaces of Carabenchel Housing extend to produce a parapet

EXTRUDE

45

1.9

REGIONS Instead of the consistent spacing that a regular grid affords, the grid spacing for Carabenchel Housing is irregular. This can be achieved through the application of regions to the rectangular boundary. Each region then has a pattern applied (or not) to it as needed. The above diagram applies seven regions to the defined rectangular boundary as follows:

regions region 1 region 2 region 3 region 4

region 5 region 6 region 7

regular grid number of divisions X number of divisions Y

[10] [1]

FURTHER DISCUSSION ON REGIONS The following steps create regions within a shape. Different patterns can then be applied to individual regions.

01 rectangle length x width y

02 point on curve

[330] [55]

curve factor f

03 closest point

04 line

point curve

point point

05 loft

06 loft

curve curve

curve curve

07 regular grid

08 regular grid

region number of divisions X number of divisions Y

[6] [1]

Different patterns can be applied to individual regions. 1.10

region number of divisions X number of divisions Y

[9] [4]

09 patterns in regions In an alternative method, regions can be created using sorted patterns. Here, patterns are applied within patterns:

01 rectangle length x width y

[330] [55]

03 sort

A list of panels from a regular grid is sorted to select every other panel.

05 patterns in regions 1.11

02 regular grid

rectangle number of divisions X number of divisions Y

[4] [2]

04 triangular grid

panels number of divisions X number of divisions Y

[3] [1]

48

EXTRUDE

1.12

VARYING FLOOR-TO-FLOOR HEIGHTS Instead of the consistent spacing that a linear array affords, the floorto-floor heights for Carabenchel Housing are irregular. Specifically, the ground floor height is ten feet, whereas the upper level floor heights are nine-and-a-half feet. This can be achieved by moving a copy of the ground floor level vertically ten feet and then arraying this level for the remainder of the floors.

EXTRUDE

49

1.13

PARAPET Instead of simply extruding the exterior walls by an amount equal to the product of the number of floors and the floor-to-floor height, the surfaces of Carabenchel Housing extend to produce a parapet. This can be achieved by extruding the walls to a height equal to the sum of each floor level plus the height of the parapet.

extrude shape height z

[rectangle] [ground floor height + (number of upper floors * upper floors’ floor-to-floor height) + parapet height]

50

EX TRUDE

1.14

DEVELOPED SCAFFOLD The script for this developed Carabenchel Housing mass scaffold includes seven* parameters: • • • • • • •

rectangle length x rectangle width y regions and patterns to define the grid ground floor height number of upper floors n upper floors’ floor-to-floor height z parapet height

* This list conflates the grid – which is defined by a collection of curves generated through combinations of regions and patterns – into one parameter for simplicity.

EX TRUDE

1.15

51

52

EX TRUDE

Rectangle length x and width y define the scaffold SHAPE. Associated regions and patterns define the GRID. A copy of the rectangular profile is moved vertically by the ground floor height. This profile is then arrayed vertically to represent the number of upper floors. These surfaces comprise the scaffold LEVELS. SURFACES are created by extruding the original rectangle by a height equal to the sum of each floor level plus the height of the parapet. Given a set number of units and maximum height, the key parameters for the Carabenchel Housing building appear to be shape width, number of levels, and floor-to-floor heights. As shown in the following design variations – achieved via the developed scaffold – other variables are kept constant, while these key parameters are modified.

EX TRUDE

53

1.16

base model

rectangle width y number of upper floors n upper floors’ floor-to-floor height z

[55] [5] [9.5]

54

EXTRUDE

1.17

variation 01

rectangle width y number of upper floors n upper floors’ floor-to-floor height z

[90] [4] [12.0]

Variation 01 is wider than the base model and has a higher floor-to-floor height but fewer floors.

EXTRUDE

55

1.18

variation 02

rectangle width y number of upper floors n upper floors’ floor-to-floor height z

[45] [6] [9.0]

Variation 02 is narrower than the base model and has a lower floor-to-floor height but more floors.

56

EX TRUDE

Additional Modelling Inquiries 1. Replace the rectangle with other shapes, such as a circle, polygon, spline, and more. 2. Arrange regions to optimize the sizes of housing units. 3. Is extrude good for housing? 4. Does the grid need to be the same on each level? 5. Link the size of regions and the size of facade panels; which controls the other? 6. Adapt the script to model other extruded buildings.

02

LOFT

Generating surfaces between shapes to create form

2.1

Like extrusions, lofted building masses may also begin with simple geometry and gain complexity in form by combining two or more different shapes. Modelling these shapes and the relationship between them parametrically offers maximum form exploration. Rather than using dissimilar shapes, another option is to loft between variations of the same shape. In these cases, other transformations – including rotate or scale – may be used. When thinking of parametric design, it is likely that lofted forms first come to mind. This is due to the misconception that parametric design is always curvy. This is of course not true; benefits of parametric modelling are not limited to curvy buildings, and lofted forms are not limited to curvy shapes. Examples of lofted masses include Snohetta’s Le Monde Group Headquarters, de Architekten Cie’s Menzis office building, and Kengo Kuma and Associates’ Victoria & Albert Museum. Like extrude, shapes can be lofted vertically – between plan profiles – or horizontally – between section profiles (such as The Smile by Alison Brooks Architect). Furthermore, loft can be used to generate form (see Peach Hut by Atelier XI) and facade elements (see Xiangcheng District Planning Exhibition Hall by Lacime Architects).

DOI: 10.4324/9781003299417-3

58

LOFT

Scripting lofts

2.2

Input data for the loft typology are two (or more) shapes. In the model shown on the previous page, an initial shape is copied, moved vertically, and rotated to develop a second shape. This results in a TWISTED form with four parameters: length, width, height, and rotation angle.

LOF T

2.3

2.4

Replacing the move + rotate function with move + move results in a SHEARED form with four parameters: length, width, height, and distance (dimension in the x and y directions).

59

60

LOFT

2.5

2.6

Using move + scale results in a TAPERED form with four parameters: length, width, height, and scale factor. Further combinations of functions – move + rotate + scale, move + move + rotate, move + move + scale, etc. – produce even more complex forms. Of course, rather than using an edited copy of the initial shape, the second shape (or third or more) could be generated independently and used to create lofted forms. Additionally, lofts can be developed from dissimilar shapes, such as rectangle and circle, circle and polygon, etc.

2.7

CASE STUDY 02

Absolute Towers building type architect location size shape material year built

housing MAD Architects Mississauga, Canada Tower A: 455,000 sf, Tower B: 430,000 sf ellipse glass, concrete 2012

Challenging the boxy nature of typical skyscrapers, Absolute Towers features curvaceous forms generated by lofting rotating ellipses. Continuous balconies offer residents uninterrupted views and connection to the neighborhood and to one another.

Opposite page: Photo of Absolute Towers by Jeffrey Collins.

64

LOFT

2.8

SHAPE

rotated ellipse radius 1 radius 2 angle θ

[45] [60] [138]

Scripting a model of the Absolute Towers building begins with a rotated ellipse defined on the XY plane.

LOF T

65

2.9

GRID

rotated grid

shape number of divisions X number of divisions Y angle θ

[rotated ellipse] [8] [8] [64]

A rotated grid is applied to the elliptical boundary. As the building rises and rotates, this grid remains constant. To accomplish this, the grid is projected onto each level.

66

LOFT

2.10

LEVELS

polar array

shape number of floors n floor-to-floor height z angle θ

[rotated ellipse] [50] [9.5] [3.5]

The previously defined elliptical profile is converted to a surface and arrayed both radially and vertically – fifty times – and spaced nine-andone-half feet apart. A placeholder rotation angle of 3.5 is used.

LOF T

67

2.11

SURFACE

loft

shapes

[array]

The boundaries of each arrayed ellipse are lofted to model the outer surface of the building.

68

LOFT

2.12

INITIAL SCAFFOLD The combination of the previously modelled shape, grid, levels, and surfaces defines an initial mass scaffold for the Absolute Towers building. The following are noted as limitations of this model, whose solutions will be discussed in the next section: • the balconies extend from Absolute Towers and are specially shaped • the core of Absolute Towers is extruded and extends beyond the level of the roof • the amount of rotation from level to level in Absolute Towers varies • the glass walls on the exterior of Absolute Towers are extruded, not lofted

LOF T

2.13

BALCONY SHAPE Balconies extend beyond the lofted exterior surface of Absolute Towers. This profile shape originates from an ellipse but is “pinched” at both ends. When arrayed radially and vertically, this pinched ellipse exaggerates the twisting form of the building.

69

FURTHER DISCUSSION ON ABSOLUTE TOWERS’ BALCONY SHAPE

The following are the steps to create a custom shape that originates with an ellipse.

01 ellipse

radius 1 radius 2

03 offset

shape distance s

[44] [60]

[rotated ellipse] [4]

This first offset establishes the width of the balcony. 2.14

02 rotate

shape angle θ

04 offset

shape distance s

[ellipse] [138]

[rotated ellipse] [8]

This second offset defines the edges of the pinch points on both ends of the ellipse.

05 divide

shape number s

[rotated ellipse] [12]

06 closest point points curves

The points on the offset ellipses closest to the defined divisions points on the rotated ellipse are selected.

07 spline points 2.15

08 pinched ellipse

72

LOFT

2.16

EXTRUDED CORE Similar to other structural elements, the core of Absolute Towers is an extruded form, not rotated like the glass walls and slab edges.

extrude shape length x width y angle θ height z

[rotated rectangle] [8] [8] [64] [ground floor height + (number of interstitial floors * interstitial floors’ floor-to-floor height) + top floor height]

LOF T

2.17

GRAPHS Instead of rotating each level equally, the amount of rotation from level to level in Absolute Towers varies. This can be achieved through the use of graphs to apply varying rotation amounts to each level.

73

FURTHER DISCUSSION ON GRAPHS Graphs bring in and modify data based on mathematical curves. Output data are controlled by variations represented in graphed profiles. The following are examples of the forms generated by lofting radially and vertically arrayed profiles wherein the rotation angle is linked to the graph’s shape.

graph sinc

2.18

graph sinc

graph sinc

graph bezier

graph conic

graph square root

2.19

76

LOFT

2.20

EXTRUDED GLASS WALLS Sets of both the ellipse and the offset pinched ellipse that defines the balconies are arrayed radially and vertically by the same amounts. The original ellipses define the glass walls at each level of Absolute Towers. Rather than lofted, these profiles are extruded. In addition, the ground floor and the top floor are extruded by a larger amount than that of the interstitial floors.

LOF T

2.21

DEVELOPED SCAFFOLD The script for this developed Absolute Towers mass scaffold includes thirteen* parameters: • • • • • • • • • • • • •

ellipse radius 1 ellipse radius 2 ground floor rotation angle θ balcony width s pinch point width s grid number of divisions X grid number of divisions Y grid rotation angle θ ground floor height number of interstitial floors n interstitial floors’ floor-to-floor height z top floor height graph for form rotation

* This list excludes core parameters for simplicity.

77

78

2.22

LOFT

LOF T

79

80

LOFT

A rotated and offset pinched ellipse defines the scaffold SHAPE. A rotated regular grid is applied to this boundary and projected onto each level to define the GRID. A copy of the offset pinched elliptical profile – and the original rotated ellipse – are moved vertically by the ground floor height. Both of these profiles are then arrayed radially and vertically – controlled via graph shape – to represent the number of interstitial floors. The arrayed offset pinched elliptical surfaces comprise the scaffold LEVELS. Three sets of extrusions create the SURFACES: the original rotated ellipse is extruded for the ground floor, the arrayed rotated ellipses are extruded for the interstitial floors, and the uppermost arrayed rotated ellipse is extruded for the top floor. Given the interest in exploring the skyscraper form, the key parameters for the Absolute Towers building appear to be shape and rotation. As shown in the following design variations – achieved via the developed scaffold – other variables are kept constant, while these key parameters are modified. In addition, these images depict two towers. Models for both buildings are generated from the same script and vary only in the rotation angle. The images demonstrate that the rotating form of each tower is as important – and seductive – as the tenuous relationship between the towers.

LOF T

81

2.23

base model

Tower A [45] [60] [138] [4] [8]

ellipse radius 1 ellipse radius 2 ground floor rotation angle θ balcony width s pinch point width s graph for form rotation

Tower B same as A same as A [238] same as A same as A

sinc

2.24

sine

2.25

82

LOFT

2.26

variation 01

Tower A [40] [65] [120] [6] [6]

ellipse radius 1 ellipse radius 2 ground floor rotation angle θ balcony width s pinch point width s graph for form rotation

Tower B same as A same as A [310] same as A same as A

bezier

2.27

sinc

2.28

LOF T

83

2.29

variation 02

Tower A [42] [62] [175] [4] [10]

ellipse radius 1 ellipse radius 2 ground floor rotation angle θ balcony width s pinch point width s graph for form rotation

Tower B same as A same as A [295] same as A same as A

conic

2.30

sine

2.31

84

LOFT

Additional Modelling Inquiries 1. Replace the ellipse with other shapes. 2. Loft between dissimilar shapes. 3. The described process for modelling Absolute Towers used linear and polar arrays to rotate the floor plates. The series of floor plates was then lofted to create the mass. Another approach is to loft a form first and then intersect the floor plate levels. Which method is best? 4. Why is it beneficial to extrude – rather than loft – the glass walls? 5. What effects do the Absolute Towers balcony guardrails add to the building? 6. Adapt the script to model other lofted buildings. 7. Graphs can be used to vary input data for many different functions. Use graphs to create other forms.

03

STACK

Layering multiple objects on top of one another to create form

3.1

Forms created by stacking geometry benefit from the use of extruded shapes; playful massing is achieved through conceiving a collection of solids as large-scale building blocks. When placed atop one another, stacked forms may be shifted (Mecanoo’s Library of Birmingham) or rotated (Bengo Studio’s QiyunshanTreehouse), use repeating or differing shapes, and intentionally align (Matsuyama Architect and Associates’ Deta Heisei Eye Clinic) or misalign (Zalewski Architecture Groups’ Dom Cube-2-Box) the edges of levels. Stacked forms result in excess exterior surfaces; floors are often used as exterior terraces (as in Sanjay Puri Architect’s Aria Resort & Spa), while soffits offer new faces for further design expression (see Metaform’s Apartment building in Cessange).

DOI: 10.4324/9781003299417-4

86

STACK

Scripting stacks

3.2

Input data for the stack typology depend on the stacked shape. In the model shown on the previous page, an initial shape is extruded by height z and then copied and moved vertically by the same height z. This higher “stacked” shape is rotated and extruded (again by height z). Such stacking can be repeated n number of times to result in a ROTATING stack.

STACK

3.3

3.4

Replacing the move + rotate function with move + move results in a SHIFTING stack. Of course, stacked forms can also be created with more complex shapes and have varying degrees of rotation and/or amounts of shift.

87

3.5

CASE STUDY 03

Gates Center for Computer Science and Hillman Center for Future Generation Technologies building type architect location size shape material year built

academic Mack Scogin Merrill Elam Architects Pittsburgh, Pennsylvania 208,000 sf irregular metal panel, glass 2009

The Gates Center for Computer Science and Hillman Center for Future Generation Technologies (Gates Center) completes a computer science quad on Carnegie Mellon University’s campus. The buildings’ stacked irregular form is a response to external and interior factors: the site is steeply sloped, bridges to two other buildings, and houses offices, conference rooms, collaborative spaces, project rooms, a reading room for more than one hundred twenty faculty, three hundred fifty graduate students, one hundred researchers or postdoctoral fellows, and fifty administrative staff, ten classrooms, two computer clusters, a two hundred fifty seat auditorium, and a café. The first two levels of the building contain a parking garage (not modelled).

Opposite page: Photo of Gates Center by Jeffrey Collins.

90

STACK

level 3

level 4

level 5

levels 6 & 7

levels 8 & 9 3.6

SHAPES Scripting a model of the Gates Center building begins by defining each irregular floor level shape.

STACK

91

level 3

level 4

level 5

levels 6 & 7

levels 8 & 9 3.7

GRIDS The levels are divided into two “zones” to which a placeholder regular grid is applied.

92

STACK

3.8

LEVELS

move

shape height z

level 3 level 4 level 5 level 6 level 7 level 8 level 9

[polyline] [0] [20] [35] [50] [65] [80] [95]

Each defined floor profile is converted to a surface and moved vertically to the correct elevation.

STACK

93

3.9

SURFACES

extrude shape height z

level 3 level 4 level 5 level 6 level 7 level 8 level 9

[polyline] [level 4 elevation−level 3 elevation] [level 5 elevation−level 4 elevation] [level 6 elevation−level 5 elevation] [level 7 elevation−level 6 elevation] [level 8 elevation−level 7 elevation] [level 9 elevation−level 8 elevation] [level 9 elevation−level 8 elevation]

Each floor profile is then extruded vertically by an amount equal to the difference of the elevation of that level and the level below it; this is the floor-to-floor height for each level. The floor-to-floor height for level nine is kept the same as level eight.

94

STACK

3.10

INITIAL SCAFFOLD The combination of the previously modelled shapes, grids, levels, and surfaces defines an initial mass scaffold for the Gates Center building. The following are noted as limitations of this model, whose solutions will be discussed in the next section: • the irregular floor level shapes previously defined are not yet parametric • the floor levels are planar with roofs of levels below • the grids for Gates Center are tailored to floor shapes • the top-most roofs need grids

2D POPULATE Rather than defining coordinates for individual vertices, points can be distributed across a boundary:

01 rectangle

02 2D populate

These points can be used to generate shapes via open polyline:

03 open polyline

04 offset

variation 01

variation 02

3.11

05 loft

Or open spline (originates with steps 01 and 02 on the previous page):

03 open spline

04 offset

variation 01

variation 02

05 loft

Closed polyline:

03 closed polyline

variation 02 3.12

04 surface

variation 01

Or closed spline:

03 closed spline

04 surface

variation 01

variation 02 Shapes may also be generated through combination by adding closed polyline surfaces:

01 shape 1

02 shape 2

03 shape 1 + shape 2

04 union [shape 1, shape 2]

variation 01

variation 02

3.13

Or adding closed spline surfaces:

01 shape 1

02 shape 2

03 shape 1 + shape 2

04 union

variation 01

variation 02

[shape 1, shape 2]

Or subtracting closed polyline surfaces:

01 shape 1

02 shape 2

03 shape 1 + shape 2

04 difference

variation 01

variation 02

[shape 1, shape 2] 3.14

Or subtracting closed spline surfaces:

01 shape 1

02 shape 2

03 shape 1 + shape 2

04 difference

variation 01

variation 02

[shape 1, shape 2] 3.15

GAINING CONTROL The previous shapes can be customized through the following steps:

01 rectangle

02 2D populate

03 closed polyline

04 isolate

05 rotate

06 offset

07 iterate

08 intersect

09 closed polyline

10 surface 3.16

11 shape 1

12 shape 2

13 shape 3

14 shape 1 + shape 2 + shape 3

15 union

16 shape 4

17 shape 5

18 union + shape 4 + shape 5

19 difference

3.17

[shape 1, shape 2, shape 3]

[union, shape 4, shape 5]

102

STACK

level 3

level 4

level 5

level 6

level 7

level 8

3.18

STACK

level 9 3.19

CUSTOM LEVELS Parametric surfaces are developed for each level.

103

roof

104

STACK

level 3

level 4

level 5

level 6

level 7

level 8

3.20

STACK

level 9 3.21

CUSTOM GRIDS Parametric grids are developed for each level.

105

roof

106

STACK

3.22

DEVELOPED SCAFFOLD The script for this developed Gate Center mass scaffold includes six sets of parameters: • • • • • •

shape polyline control points shape polyline angles θ shape polyline offset distances s grid curve angles θ grid curve offset distances s level elevations s (plus height of top floor)

STACK

3.23

107

108

STACK

Custom closed polyline profiles define the scaffold SHAPES. The edges of these boundaries inform each level’s GRID. These profiles are moved vertically to the correct elevation to define the LEVELS. Each level is then extruded vertically by an amount equal to the difference between the floor level and the one above to define the SURFACES. Given the flexibility of the shape profiles to produce a multitude of stacked configurations, the key parameters for the Gates Center building appear to be polyline angles and offset distances. As shown in the following design variations – achieved via the developed scaffold – other variables are kept constant, while these key parameters are modified.

STACK

3.24

Base model In the base model for the Gates Center building, the mass on the left is mostly extruded, whereas the mass on the right is staggered into three main chunks: base, middle, and top.

109

110

STACK

3.25

Variation 01 In variation 01, the top floors of the mass on the left are offset to mimic the stacked nature of the mass on the right. The mass on the right is modified by pushing and pulling the edges of the extruded floor level shapes; the base levels extend toward the camera, and the top levels recess away and to the right.

STACK

3.26

Variation 02 In variation 02, the top floors of the mass on the left are rotated and push into the courtyard space. The edges of the base and middle levels of the mass on the right are aligned, while the top levels extend dramatically towards the camera.

111

112

STACK

Additional Modelling Inquiries 1. Replace the shapes; try regular shapes, composite shapes, and combinations. 2. Consider which edges align or misalign from floor-to-floor. 3. Link shapes and grids; which controls the other? 4. Although stacking may allow the arrangement of each floor level to be optimized, threading building systems (structural, mechanical, and vertical circulation) through can be challenging. What are some strategies for overcoming this? 5. Is stack good for academic buildings? 6. What effects do the choice and allocation of cladding materials on Gates Center have on the perception of the building mass? 7. Model other stacked buildings. 8. Use 2D populate to create other forms.

04

CARVE

Cutting with objects to create form

4.1

Forms created by carving geometry also benefit from the use of extruded shapes, the often-organic void spaces (such as Zaha Hadid Architects’ Opus) enhanced by the simplicity of the surrounding orthogonal building mass. Void spaces may express differences in surface material (the windows on Steven Holl Architects’ Hunters Point Library) or literally remove material from the building (the front face of TRU Architects’ Simone 0914 Flagship Store). Such removal may designate entry (Takato Tamagami Architectural Design’s Sunwell Muse Kitasando), provide exterior space (the courtyard in Lorcan O’Herlihy Architects’ Mariposa1038), or allow visual connection (the slab openings through the podium of Toyo Ito & Associates, Architects’ Porta Fira Towers).

DOI: 10.4324/9781003299417-5

CARVE

Scripting carving

4.2

Input data for the carve typology concern a solid mass and masses that are removed from it; namely, voids. In the model shown on the previous page, a shape is extruded to form a solid. Then, a series of spheres – defined by centerpoints pt and radii r – are carved from the solid. Both solid and void forms can be alternate typologies, for example, lofts.

115

4.3

CASE STUDY 04

The Orange Cube building type architect location size shape material year built

office Jakob + Macfarlane Architects Lyon, France 68,000 sf rectangle metal panel screens 2011

The Orange Cube is conceived of as a simple orthogonal cube into which giant holes are carved. These voids respond to the site conditions of light, airflow, and views.

Opposite page: Photo of The Orange Cube by Roland Halbe Fotografie.

118

CARVE

4.4

SHAPE rectangle length x width y

[100] [112]

Scripting a model of The Orange Cube building begins with a basic rectangle – 100' by 112' – defined on the XY plane.

CARVE

119

4.5

GRID regular grid shape number of divisions X number of divisions Y

[rectangle] [5] [5]

A regular grid is applied to the rectangular boundary.

120

CARVE

4.6

SOLID extrude shape height z

[rectangle] [80]

The rectangular profile is extruded vertically by the full height of the building to create a solid mass.

CARVE

121

4.7

VOIDS loft

void 1

void 2

void 3

shape

[circle a]

location angle diameter

[45,-66,0] [90,0,0] [30.5]

[-5,-9,94] [0,0,0] [13.7]

[-48,-75,37] [90,0,145] [29.6]

shape

[circle b]

location angle diameter

[45,-35,0] [90,0,0] [14.0]

[-5,-9,42] [0,0,0] [10.8]

[-4,-6,48] [90,0,145] [14.5]

Three conic-shaped masses are modelled, each by lofting between two circles. These represent the voids that will be subtracted from the solid.

122

CARVE

4.8

SURFACES difference brep brep

[solid] [voids]

The conic masses are removed from the extruded rectangular mass to define the carved scaffold surfaces.

FURTHER DISCUSSION ON CONICS The void shapes employed in The Orange Cube mass scaffold are truncated cones. Variations of conic shapes can be made in a similar manner to the following steps for creating a double cone:

01 circle

02 move

03 divide curve

radius [10]

height z [40]

circles number of points [18]

04 line

05 shift

06 loft

points points Vertical lines from one circle to the other 4.9

point list The list of points on one circle can be shifted so that the lines connect diagonally

lines

Conic sections are curves that result from an intersection of a double cone and various planes:

When a horizontal plane intersects one of the cones, the result is a circle

When the plane is rotated slightly, the intersection defines an ellipse

When the plane is rotated further so that it intersects the base of the cone, this defines a parabola

When a vertical plane intersects both of the cones, this defines a hyperbola

4.10

126

CARVE

4.11

LEVELS intersection brep brep

[scaffold surfaces] [linear array]

shape num of flrs n flr to flr height z

[rectangle] [7] [building height/num of flrs]

The previously defined rectangular profile is converted to a surface and arrayed vertically eight times in order to model seven floor levels. (The eighth surface represents the roof.) A placeholder floor-to-floor height is derived by dividing the full building height by the number of floors. Each arrayed plane is then intersected with the carved surfaces in order to obtain the profile of each level.

CARVE

4.12

INITIAL SCAFFOLD The combination of the previously modelled shape, grid, solid, voids, surfaces, and levels defines an initial mass scaffold for The Orange Cube building. The following are noted as limitations of this model, whose solutions will be discussed in the next section: • the column grid is pulled in from The Orange Cube facade • the floor-to-floor heights in The Orange Cube vary • the uppermost level of The Orange Cube has a trimmed roof

127

128

CARVE

4.13

SCALED GRID As the column grid is pulled in from The Orange Cube facade, the grid is applied to a scaled copy of the rectangular boundary. regular grid shape number of divisions X number of divisions Y

[scaled rectangle] [5] [5]

CARVE

129

4.14

VARYING FLOOR-TO-FLOOR HEIGHTS Instead of the consistent spacing that a linear array affords, the floor-to-floor heights for The Orange Cube are irregular. This can be achieved by moving copies of the rectangular surface vertically to the correct elevations and then intersecting these planes with the carved surfaces. move shape height z

intersection brep brep

level 1 level 2 level 3 level 4 level 5 level 6 level 7

[rectangular surface] [0] [11] [21] [32.5] [44] [55.5] [67]

[scaffold surfaces] [rectangular surfaces]

130

CARVE

4.15

TRIMMED ROOF The top level of The Orange Cube has an open air roof terrace. This is modelled by scaling a copy of the original rectangular boundary and then projecting this new shape onto the upper two levels. A surface is lofted between these profiles, and the uppermost profile is used to trim the roof surface.

CARVE

4.16

DEVELOPED SCAFFOLD The script for this developed The Orange Cube mass scaffold includes nine* parameters: • • • • • • • • •

rectangle length x rectangle width y building height grid boundary scale factor grid number of divisions X grid number of divisions Y voids (3 voids x 6 parameters each) floor level elevations (7 levels) trimmed roof scale factor

* The number of parameters used to define the voids and floor level elevations are conflated for simplicity.

131

132

4.17

CARVE

CARVE

133

134

CARVE

Rectangle length x and width y define the scaffold SHAPE. A scaled copy of this rectangle bounds a regular grid, which is projected onto each level to define the GRID. The original rectangle is extruded vertically by the full height of the building to create a solid mass. Three conic-shaped lofted voids are subtracted from the solid extrusion to define the SURFACES. Copies of the original rectangle are also copied vertically to the elevations of each level and then intersected with the carved surfaces to define profiles of the LEVELS. The top level is trimmed in order to model the open air roof terrace using another projected copy of the original rectangle. Given the goal of the carved voids to improve access to light, air flow, and views, the key parameters for The Orange Cube building appear to be the control points for these lofted shapes. As shown in the following design variations – achieved via the developed scaffold – other variables are kept constant, while these key parameters are modified.

CARVE

135

4.18

base model

void 1

void 2

void 3

shape

[circle a]

location angle diameter

[45,-66,0] [90,0,0] [30.5]

[-6,-9,94] [0,0,0] [13.7]

[-48,-75,37] [90,0,145] [29.6]

shape

[circle b]

location angle diameter

[45,-36,0] [90,0,0] [14.0]

[-6,-9,42] [0,0,0] [10.8]

[-4,-6,48] [90,0,145] [14.5]

136

CARVE

4.19

variation 01 shape

[circle a]

location angle diameter

void 1 [45,-66,0] [90,0,0] [27.6]

void 2 [-6,-9,94] [0,0,0] [13.7]

void 3 [-16,-75,37] [90,0,145] [24.6]

shape

[circle b]

location angle diameter

[45,65,0] [90,0,0] [27.6]

[-6,-9,42] [0,0,0] [13.7]

[8,-6,48] [90,0,145] [14.5]

In variation 01, void 1 is extended through the whole building as a tunnel. Void 2 has a consistent rather than a tapered diameter. Void 3 is moved to the front elevation rather than the corner and is reduced in size.

CARVE

137

4.20

variation 02 shape

[circle a]

location angle diameter

void 1 [45,-66,0] [90,0,0] [50.0]

void 2 [-33,-23,94] [0,0,0] [36.0]

shape

[circle b]

location angle diameter

[45,-51,0] [90,0,0] [12.8]

[-6,-9,11] [0,0,0] [19.2]

In variation 02, void 1 is very shallow. Void 2 is enlarged in diameter and depth and is rotated to intersect with the left facade of the building. Void 3 is not used.

138

CARVE

Additional Modelling Inquiries 1. Replace the rectangle with other shapes, such as circle, polygon, spline, and more. 2. Create voids using alternate forms. 3. Arrange voids to optimize light, airflow, and views. 4. Adapt the script to model other carved buildings. 5. Is carve good for office buildings? 6. Consider cladding materials. Do voids determine material allocation? Can facade panels also be perforated to contribute to light, airflow, and view optimization?

05

NEST

Embedding objects within another to create form

5.1

Nesting occurs when a series of objects are overlapped. Nested forms can be additive or a combination of additive and subtractive. Additive nesting results in layered solid objects (such as Ming Architects’ Cube House). When combining additive and subtractive nestings, void objects are carved away from solid objects (such as Studio Zhu Pei’s Shou County Culture and Art Center). Nesting is related to both stacking and carving. Nested forms benefit from repetitive use of the same or similar geometry. Like stacking, intentional aligning or misaligning of geometry is involved. Nesting is distinguished from stacking as nesting incorporates intersecting geometry, whereas stacking implies volumes that are unable to intersect. When incorporating void objects, nesting is distinguished from carving in that nested solid and void geometries are similar (both are cubes), whereas carved voids are distinct in shape from their solid counterparts. Nested geometry can occur on the exterior of a building (Lewis Tsurumaki Lewis’ Bornhuetter Hall) and on the interior of a building (Office of Mcfarlane Biggar Architects + Designers’ Telus Garden), be expressed on the facade (Victoria Acebo & Angel Alonso’s National Museum of Science and Technology), or extend through the facade (Tabanlioglu Architects’ Dogan Media Center).

DOI: 10.4324/9781003299417-6

140

NEST

Scripting nests

5.2

Input data for the nest typology include a boundary and a set of solid masses. In the model shown on the previous page, a rectangle is extruded to form the boundary. A series of cubes – defined by centerpoints pt and side lengths x – are populated within the boundary.

NEST

5.3

5.4

Nested forms can also incorporate voids. In this example, a series of void cubes are subtracted from the series of solid cubes.

141

5.5

CASE STUDY 05

Ftown Building building type architect location size shape material year built

commercial Atelier Hitoshi Abe Sendai, Japan 21,000 sf rectangle fiberglass reinforced plastic (FRP) 2008

The design of Ftown Building includes spaces with varying floor heights and multiple circulation routes, achieved via voids in-between and around overlapping boxes.

Opposite page: Photo of Ftown Building by Daici Ano.

144

NEST

5.6

SHAPE

rectangle length x width y

[54] [52]

Scripting a model of Ftown Building begins with a basic rectangle – 54' by 52' – defined on the XY plane.

NEST

145

5.7

GRID

regular grid

shape number of divisions X number of divisions Y

[rectangle] [4] [3]

A regular grid is applied to the rectangular boundary.

146

NEST

5.8

SOLIDS cube location size

solid 1 [-9,-11,28] [26]

solid 2 [7,20,106] [28]

solid 3 [17,-21,74] [24]

solid 4 [-11,21,63] [23]

solid 5 [21,11,2] [19]

Six cubes are populated in a volume extruded from the rectangular boundary.

solid 6 [-24,26,18] [20]

NEST

147

5.9

SURFACES

union brep

[solids]

The cubes are joined together to define the nested scaffold surfaces.

148

NEST

5.10

LEVELS

intersection brep brep

[scaffold surfaces] [linear array]

shape num of flrs n flr to flr height z

[rectangle] [7] [building height/num of flrs]

The previously defined rectangular profile is converted to a surface and arrayed vertically eight times in order to model seven floor levels. (The eighth surface represents the roof.) A placeholder floor-to-floor height is derived by dividing the full building height by the number of floors. Each arrayed plane is then intersected with the nested surfaces in order to obtain the profile of each level.

NEST

149

5.11

INITIAL SCAFFOLD The combination of the previously modelled shape, grid, solids, surfaces, and levels defines an initial mass scaffold for Ftown Building. The following are noted as limitations of this model, whose solutions will be discussed in the next section: • the locations and sizes of nested cubes in Ftown Building require further control • the column grid is pulled in from the Ftown Building facade • the floor-to-floor heights in Ftown Building vary

3D POPULATE Like 2D populate, points can be distributed through a three-dimensional space:

01 box

02 3D populate

These points can be used to generate form with – among other methods – solid cubes (originates with steps 01 and 02):

03 cubes 5.12

variation 1

variation 2

Solid and void cubes:

03 cubes 1

04 3D populate

05 cubes 2

06 difference

variation 1

variation 2

variation 1

variation 2

[cubes 1, cubes 2] Solid spheres:

03 spheres 5.13

Spheres trimmed at boundary:

03 spheres

04 intersect [box, spheres]

variation 1

variation 1

variation 2

variation 2

Cells for Voronoi:

03 voronoi 5.14

Control points for faceted geometry:

03 lines

variation 1

variation 2

Or control points for curved geometry:

03 spline 1

04 3D populate

05 spline 2

06 loft

[spline 1, spline 2]

variation 1

variation 2

5.15

154

NEST

5.16

CUSTOM SOLIDS The locations and sizes of six boxes – which have differing lengths, widths, and heights – are defined. box

solid 1

solid 2

solid 3

solid 4

solid 5

solid 6

location size

[14,9,7] [45,15,15]

[-15,-13,30] [45, 43,40]

[1,-6,43] [45,41,42]

[11,9,60] [46,40,38]

[-14,-6,90] [45,43,42]

[1,2,103] [61,41,33]

NEST

155

5.17

SCALED GRID As the column grid is pulled in from the Ftown Building facade, the grid is applied to a scaled copy of the rectangular boundary.

regular grid

shape number of divisions X number of divisions Y

[scaled rectangle] [4] [3]

156

NEST

5.18

VARYING FLOOR-TO-FLOOR HEIGHTS Instead of the consistent spacing that a linear array affords, the floor-to-floor heights for Ftown Building are irregular. This can be achieved by moving copies of the rectangular surface vertically to the correct elevations and then intersecting these planes with the solid boxes.

move

shape height z

intersection brep brep

level 1 level 2 level 3 level 4 level 5 level 6 level 7 roof

[rectangular surface] [0] [15.3] [29.5] [46.9] [61.0] [75.2] [92.6] [106.9]

[solids] [rectangular surfaces]

NEST

5.19

DEVELOPED SCAFFOLD The script for this developed Ftown Building mass scaffold includes eight* parameters: • • • • • • • •

rectangle length x rectangle width y solid box locations solid box sizes grid boundary scale factor grid number of divisions X grid number of divisions Y floor level elevations (8 levels)

* The number of parameters used to define the solids and floor level elevations are conflated for simplicity.

157

158

5.20

NEST

NEST

Rectangle length x and width y define the scaffold SHAPE. A scaled copy of this rectangle bounds a regular grid that is projected onto each level to define the GRID. Six boxes define the SURFACES. Copies of the original rectangle are moved vertically to the elevations of each level and then intersected with the boxes to define profiles of the LEVELS. Given the design concept of varying floor heights and multiple circulation routes, the key parameters for Ftown Building appear to be the locations and sizes of the nested boxes. As shown in the following design variations – achieved via the developed scaffold – other variables are kept constant, while these key parameters are modified.

159

160

NEST

5.21

base model location size

solid 1 [14,9,7] [45,15,15]

solid 2 [-15,-13,30] [45, 43,40]

solid 3 [1,-6,43] [45,41,42]

solid 4 [11,9,60] [46,40,38]

solid 5 [-14,-6,90] [45,43,42]

solid 6 [1,2,103] [61,41,33]

NEST

161

5.22

variation 01 location size

solid 1 [11,1,116] [16]

solid 2 [-13,-5,1] [16]

solid 3 [-7,7,58] [11]

solid 4 [9,2,24] [17]

solid 5 [-8,-6,89] [14]

location size

solid 6 [14,-5,67] [16]

solid 7 [-13,5,34] [23]

solid 8 [-14,2,115] [23]

solid 9 [6,-3,44] [11]

solid 10 [-5,9,12] [18]

Variation 01 increases the number of solid boxes.

162

NEST

5.23

variation 02 location size

solid 1 [2,1,77] [26]

solid 2 [3,2,3] [27]

solid 3 [-2,1,43] [27]

solid 4

[1,0,99] [24]

location size

void 1 [27,15,76] [20]

void 2 [-22,-11,0] [17]

void 3 [-29,-8,99] [17]

void 4 [-29,22,41] [17]

location size

void 6 [-28,-28,53] [18]

void 7 [25,-28,41] [18]

void 8

[5,-27,80] [18]

Variation 02 decreases the number of solid boxes. Void boxes are incorporated.

void 5

[26,8,18] [19]

NEST

163

Additional Modelling Inquiries 1. Arrange solids and voids to optimize interior and exterior spaces. 2. Link the size of boxes, grids, and facade panels; which controls the others? 3. Adapt the script to model other nested buildings. 4. Revisit intersecting arrayed planes and nested solids. What further operations could be explored with these new profiles? 5. Use 3D populate to create other forms.

06

TRIM

Defining boundaries to create form

6.1

Trimmed forms comprise a base geometry and a cutting geometry. The trim boundary is often indifferent to the trimmed object. In fact, this juxtaposition is frequently a goal. As such, base geometry is commonly extruded. Forms are most often trimmed in section; at the roof. Examples of trimmed masses include BIG’s Via 57 West and Toshiko Mori Architect’s Thread Artists’ Residences & Cultural Center. Roofs of trimmed buildings – now highlighted – offer space for further investigation (such as the solar panels atop Miller Hull Partnership’s Bullitt Center or the green roof atop Centerbrook’s Biomass Heating Facility). Trim boundaries can be planar, curvy, or angular. Trim boundaries can be expressed only as void (as seen connecting the disparate geometries of Safdie Architect’s Exploration Place) or can manifest as an extension of the base geometry (as seen overhanging dramatically in MPH Architects’ Thebarton Community Centre).

DOI: 10.4324/9781003299417-7

TRIM

165

Scripting trimming

6.2

Input data for the trim typology concern a solid mass and boundary that cuts it. In the model shown on the opposite page, a shape is extruded to form a solid. Then, a surface – defined by a series of control points – trims the solid.

166

6.3

TRIM

CASE STUDY 06

Centre Pompidou-Metz building type architect location size shape material year built

cultural Shigeru Ban Architects Metz, France 122,000 sf hexagon timber, fiberglass 2010

The form of Centre Pompidou-Metz is conceived of as a stack of volumes under a large undulating roof. This roof extends beyond the envelope, blurs the interior and exterior, and invites the public into the museum. The roof trims the atrium space below. The roof is trimmed by a scaled copy of the base shape.

Opposite page: Photo of Centre Pompidou-Metz by Ken Lee.

168

TRIM

6.4

SHAPE

rotated hexagon radius r angle θ

[130] [30]

Scripting a model of Centre Pompidou-Metz begins with a rotated hexagon defined on the XY plane.

TRIM

169

6.5

GRID

grid

shape number of div X number of div Y

rotated grid shape num of div X num of div Y angle θ

[rotated hexagon] [1] [47]

rotated grid shape num of div X num of div Y angle θ

[rotated hexagon] [1] [47] [-30]

Three grids are layered to create a hexagonal pattern.

[rotated hexagon] [1] [47] [30]

170

TRIM

6.6

LEVELS

linear array

shape number of floors n floor-to-floor height z

[rotated hexagon] [4] [23.5]

The previously defined rectangular profile is converted to a surface and

arrayed vertically five times in order to model four floor placeholder levels. (The fifth surface represents the roof.) The spacing between each

surface copy is twenty-three-and-a-half feet, which is the floor-to-floor

height.

TRIM

171

6.7

SURFACES

extrude shape height z

[rotated hexagon] [number of floors * floor-to-floor height]

The hexagonal profile is extruded vertically by an amount equal to the

product of the previously defined number of floors (four) and the floor-

to-floor height (twenty-three-and-a-half): 94'.

172

TRIM

6.8

TRIM BOUNDARY

surface patch

[control points]

FURTHER DISCUSSION ON PATCH An irregular surface can be created using control points:

01 rectangle

6.9

02 2D populate

6.10

03 move

Each point is moved vertically by an amount defined by a graph.

6.11

04 patch

6.12

A similar process translates contour lines into a topography:

01 contour lines

6.13

02 move

Each contour line is moved vertically to its correct elevation.

6.14

03 control points

6.15

04 patch

6.16

176

TRIM

6.17

TRIMMED ROOF A scaled copy of the original rotated hexagon trims the roof edge.

TRIM

6.18

INITIAL SCAFFOLD The combination of the previously modelled shape, grid, levels, surfaces, and trim boundary defines an initial mass scaffold for Centre Pompidou-Metz. The following are noted as limitations of this model, whose solutions will be discussed in the next section: • the floor level shapes for Centre Pompidou-Metz are irregular • the locations of control points for the Centre Pompidou-Metz roof require further control

177

178

TRIM

atrium

ground level

level 2

level 3

level 4 6.19

CUSTOM LEVELS A parametric profile is developed for each level. The atrium profile is extruded and later trimmed by the roof surface. Each level profile is extruded and stacked.

TRIM

6.20

TRIM BOUNDARY A new roof surface is defined.

179

180

TRIM

6.21

TRIMMED ROOF A scaled copy of the original rotated hexagon trims the roof edge. Additional shapes cut holes to allow the stacked galleries to protrude through the roof surface.

TRIM

6.22

TRIMMED EXTRUSION The atrium profile is extruded and trimmed by the roof surface.

181

182

TRIM

6.23

DEVELOPED SCAFFOLD The script for this developed Centre Pompidou-Metz mass scaffold includes eleven* parameters: • • • • • • • • • • •

hexagon radius hexagon rotation angle θ hexagon scale factor grid number of divisions X grid number of divisions Y grid rotation angle θ shape polyline control points shape polyline angles θ shape polyline offset distances s floor level elevations roof surface control points and trim shapes

* The number of parameters used to define the level profile, roof surface control points, and trim shapes are conflated for simplicity.

FURTHER DISCUSSION ON THE CENTRE POMPIDOU-METZ ROOF STRUCTURE The roof structure of Centre Pompidou-Metz is generated by projecting the hexagonal grid pattern onto the roof surface:

01 regular grid

02 rotated grid

03 rotated grid

04 layered grids

05 trimmed roof

06 project [grids, trimmed roof]

6.24

184

6.25

TRIM

TRIM

185

186

TRIM

A rotated hexagon defines the scaffold SHAPE. Rotated regular grids are layered to create a custom hexagonal GRID. Custom closed polyline profiles are moved vertically to the correct elevation to define the LEVELS. Three sets of geometry create the SURFACES: each level profile is extruded and stacked, the roof surfaces are defined by sets of control points, and the atrium profile is extruded and trimmed by the roof surface. A scaled copy of the original rotated hexagon trims the roof edge. Additional shapes cut holes to allow the stacked galleries to protrude through the roof surface. Given the function of the large undulating roof to unify stacked volumes and blur interior and exterior spaces, the key parameters for Centre Pompidou-Metz appear to be the roof surface control points and trim profile. As shown in the following design variations – achieved via the developed scaffold – other variables are kept constant, while these key parameters are modified.

TRIM

6.26

Base model In the base model for Centre Pompidou-Metz, the roof creates a high interior atrium that slopes downward and beyond the extruded glass walls.

187

188

TRIM

6.27

Variation 01 In variation 01, the slope of the roof surface is reduced and rises towards the exterior. Vertical columns, rather than blending into the roof surface, are incorporated into the extruded glass walls.

TRIM

6.28

Variation 02 In variation 02, the roof surface blends with the columns; however, they are moved inside the atrium.

189

190

TRIM

Additional Modelling Inquiries 1. Replace the hexagon with other shapes. 2. Replace the grid with other patterns. What is the relationship between the shape and grid? 3. How many trim operations occur in the described process of modelling Centre Pompidou-Metz? 4. Extend the described modelling steps to blur the columns and roof. 5. What kinds of buildings benefit from a large expressive roof? 6. Model other trimmed buildings. 7. Use patch to create other forms.

07

QUAD

Facades organized by orthogonal lines

7.1

Similar to extrude, we begin this second half of case studies with quad as it is the simplest method for creating a facade pattern; it is the spacing of orthogonal lines across a surface. The facade scaffold model discussed in the Introduction is a quad pattern. However, again, like extrude, this simplicity can be deceiving. Unassuming quad patterns can allow us to appreciate complex massing or mask internal intricacy. Although the term “quadrilateral” refers to any four-sided shape, for simplicity, this section references quads with ninety-degree corners: rectangles and squares. Quads can express verticality (such as the operable louvers of Ernst Giselbrecht + Partner’s Kiefer Technic Showroom), horizontality (such as the glass panels of Degelo Architekten’s Freiburg University Library), or neutrality (such as the sun shading system of INNOCAD’s C&P Corporate Headquarters). Other buildings with quad facades include Domiuque Perrault Architecture’s Fukoku Life Osaka Project, Digsau’s 150 Rouse Boulevard, and Pasel Kunzel Architects’ V36K0809 House. Proportions that define quad patterns may typically be associated with floor-to-floor heights and structural bay widths, but this is not always the case.

DOI: 10.4324/9781003299417-8

192

QUAD

Scripting quads

7.2

Input data for the quad pattern are the surface length and height and number of divisions horizontally and vertically.

FURTHER QUAD TYPES

A variety of patterns are organized by orthogonal lines. Regular grids, which have consistent spacing horizontally and vertically:

variation 01 num div X num div Z

[4] [4]

variation 02

num div X num div Z

[4] [8]

variation 03

num div X num div Z

Irregular grids, which have inconsistent spacing horizontally and/or vertically:

variation 01 7.3

variation 02

variation 03

[3] [3]

Horizontal running bond:

variation 01 num div X num div Z

[4] [4]

variation 02 num div X num div Z

[4] [20]

variation 03 num div X num div Z

[10] [3]

Vertical running bond:

variation 01 num div X num div Z 7.4

[3] [10]

variation 02 num div X num div Z

[10] [3]

variation 03 num div X num div Z

[4] [4]

Quads can also stagger horizontally:

variation 01 num div X num div Z

[20] [10]

variation 02 num div X num div Z

[50] [3]

variation 03 num div X num div Z

[100] [3]

Or stagger vertically:

variation 01 num div X num div Z 7.5

[3] [50]

variation 02

num div X num div Z

[10] [10]

variation 03

num div X num div Z

[2] [60]

7.6

CASE STUDY 07

Manifold House building type architect location size shape material year built

single family residence David Jameson Architect Arlington, Virginia 3,000 sf rectangle weathered steel 2018

The exterior walls of Manifold House consist of a regular grid frame and panels rotated at various angles; both are made of weathered steel. In addition to creating a dynamic facade, the rotated panels provide varying degrees of privacy and daylight exposure.

Opposite page: Photo of Manifold House by Paul Warchol Photography Inc.

198

QUAD

7.7

SURFACE

rectangle length x height z

[63] [30]

Scripting a model of the Manifold House facade begins with a basic rectangle – 63' by 30' – defined on the XZ plane.

QUAD

199

7.8

GRID

regular grid

shape number of divisions X number of divisions Z

[surface] [24] [4]

A regular grid is applied to the rectangular boundary.

200

QUAD

7.9

SHAPE

offset

curve distance s

[panels] [0.08]

Lines from the defined grid are offset to represent individual facade panels and framing.

QUAD

201

7.10

PANELIZATION

regular grid

shape number of divisions X number of divisions Z

[surface] [48] [1]

Another regular grid is applied to the surface. Every other vertical line is hidden. The remaining lines split panels in half.

202

QUAD

7.11

INITIAL SCAFFOLD The combination of the previously modelled surface, grid, shape, and panelization defines an initial facade scaffold for Manifold House. The following are noted as limitations of this model, whose solutions will be discussed in the next section: • the horizontal lines in the Manifold House facade are indifferent to floor levels and vary • the corner panels in Manifold House are half-size • the panels in the Manifold House facade rotate outward at various angles

QUAD

7.12

HORIZONTAL REGIONS Instead of the consistent spacing that a regular grid affords, the grid spacing for Manifold House is irregular. This can be achieved through the application of regions to the rectangular boundary. Each region then has a pattern applied (or not) to it as needed. Four horizontal regions are defined on the rectangular boundary:

regions region 1 region 2 region 3 region 4

203

204

QUAD

7.13

VERTICAL REGIONS Three vertical regions – with a regular grid applied to region 2 – are defined on the rectangular boundary:

regions region 1 region 2 region 3

regular grid number of divisions X number of divisions Z

[22] [1]

QUAD

205

7.14

ROTATED PANELS Lines defined by the horizontal regions and the vertical regions (and grid) define boundaries for the new facade panels. The panels are randomly rotated outward at a collection of angles:

rotate

shape angle θ

[panels] [-90,-45,0,45,90]

206

QUAD

7.15

DEVELOPED SCAFFOLD The script for this developed Manifold House facade scaffold includes seven* parameters: • • • • • • •

rectangle length x rectangle height z regions and patterns to define the grid panelization number of divisions X panelization number of divisions Z frame width s panel rotation θ

* This list conflates the grid – defined by a collection of curves generated through combinations of regions and patterns – into one parameter for simplicity.

QUAD

Rectangle length x and height z define the scaffold SURFACE. Associated regions and patterns define the GRID. Lines from this grid are offset to represent facade SHAPES, which are split in half by a PANELIZATION grid. These panels are rotated outward at various angles. Given the playful composition, a key parameter for the Manifold House facade appears to be panel rotation angles. As shown in the following design variations – achieved via the developed scaffold – other variables are kept constant, while this key parameter is modified.

207

208

7.16

QUAD

QUAD

209

210

QUAD

7.17

Base model In the base model for Manifold House, the facade panels rotate randomly at -90, -45, 0, 45, and 90 degrees.

QUAD

7.18

Variation 01 In variation 01, the facade panels are rotated only -90, 0, or 90 degrees.

211

212

QUAD

7.19

Variation 02 In variation 02, the facade panels are rotated only -45 or 45 degrees.

QUAD

Additional Modelling Inquiries 1. Correlate the rotation of panels to the desired interior conditions (privacy and daylight). 2. Adapt the script to model other quad patterns. 3. Apply quad patterns to other mass models. 4. Experiment with different combinations of scaling and rotating quad panels on the surface plane. 5. Apply quad panels to the nodes or centerpoints of other grids. Combine with the previous scaling and rotating. 6. Link the size of quad panels to the grid and floor-to-floor heights; which controls the other? What are the implications of having these linked (or not)? 7. Explore the thickness of the facade design.

213

08

DIAMOND

Facades organized by diagonal lines

8.1

Also composed of continuous lines across a surface, diamond patterns are diagonal rather than orthogonal lines. Diamond patterns are further distinguished from quad patterns in that the organizational lines for diamond patterns do not intersect at ninety degrees. (Those that do are classified as a rotated quad pattern.) Diamonds can express verticality (Foster + Partners’ 30 St Mary Axe) or horizontality (Twelve Architects’ The Diamond) or inform – or be informed by – building massing strategies (such as the thickened veil of Diller Scofidio + Renfro’s The Broad or the faceted form of OMA’s Seattle Central Library). Other buildings with diamond facades include SHAU’s Microlibrary and Woods Bagot’s South Australian Health and Medical Research Institute.

DOI: 10.4324/9781003299417-9

DIAMOND

Scripting diamonds

8.2

Like quads, input data for the diamond pattern are the surface length and height and number of divisions horizontally and vertically.

215

FURTHER DISCUSSION ON APPLYING SHAPES TO PATTERNS Grids have different control points that can be used as insertion points for additional shapes and the development of further patterns.

01 diagrid

02 nodes

03 centerpoints

04 circles on nodes

05 circles on centerpoints

06 circles on nodes + circles on centerpoints

04 triangles on nodes

05 triangles on nodes rotated

06 triangles on nodes rotated

intersection points

angle [-67.4] 8.3

center of each panel

angle [random]

04 hexagons on nodes

05 hexagons on nodes scaled

06 hexagons on nodes scaled

factor [1.58]

factor

04 hexagons on nodes

05 circles on centerpoints

06 difference

04 rotated squares on nodes

05 rotated squares on centerpoints

8.4

[random]

[hexagons, circles]

06 union

[squares, squares]

8.5

CASE STUDY 08

Prada Aoyama building type architect location size shape material year built

commercial Herzog & de Meuron Tokyo, Japan 10,000 sf irregular glass 2003

The faceted form and diamond-shaped glass panels give the Prada Aoyama building a jewel-like quality. Not limited to the facade pattern, diamond shapes also extend to the buildings’ structure and interior spaces.

Opposite page: Photo of Prada Aoyama by Johannes Marburg, Geneva.

220

DIAMOND

8.6

SURFACE The mass of the Prada Aoyama building is defined by ten surfaces. One is selected.

DIAMOND

221

8.7

GRID

diagrid

shape number of divisions X number of divisions Z

[surface] [10] [34]

A diagrid is applied to the surface.

222

DIAMOND

8.8

SHAPE

offset

curve distance s

[panels] [0.08]

Lines from the defined grid are offset to represent individual facade panels.

DIAMOND

223

8.9

PANELIZATION

diagrid

shape number of divisions X number of divisions Z

[surface] [6] [17]

Another diagrid is applied to the surface. These panels group smaller panels together to define entrances and interior spaces.

FURTHER DISCUSSION ON PRADA AOYAMA SURFACE PATTERN Applying a grid to an irregularly shaped surface does not result as intended.

01 surface

02 diagrid

The following steps resolve this problem and allow the origin of the pattern to be relocated. For the Prada Aoyama facade, diagonal lines stem from the corners.

03 move line 8.10

04 extrude curve

05 diagrid

06 trim

This process also reveals how the control points of the Prada Aoyama building mass are synchronized with nodes of the diamond pattern. 8.11

226

DIAMOND

8.12

INITIAL SCAFFOLD The combination of the previously modelled surface, grid, shape, and panelization defines an initial facade scaffold for the Prada Aoyama building. The following are noted as limitations of this model, whose solutions will be discussed in the next section: • the panelization pattern groups sections of the Prada Aoyama facade together • some panels of the Prada Aoyama facade bulge outward

DIAMOND

8.13

REDUCED PANELS The panelization grid is used to reduce the number of facade panels. Shown in the image above, the removed area defines an entrance.

227

228

DIAMOND

8.14

PATTERNS IN PATTERNS The grid is reapplied over the reduced panels.

DIAMOND

8.15

BULGED PANELS Some of the panels are bulged outward. This is accomplished by patching a surface between the boundary of the panel and a copy of the centerpoint of the panel offset from the building surface.

229

230

DIAMOND

8.16

DEVELOPED SCAFFOLD The script for this developed Prada Aoyama facade scaffold includes eight* parameters: • • • • • • • •

surface boundary grid number of divisions X grid number of divisions Z panelization number of divisions X panelization number of divisions Z frame width s panel reduction factor panel bulge factor

* The number of parameters used to define the surface boundary are conflated for simplicity.

DIAMOND

The faceted geometry of the building mass defines the scaffold SURFACE. A diagrid is applied to the surface to define the GRID. Lines from this grid are offset to represent facade SHAPES. Another PANELIZATION diagrid defines entrances and interior spaces. Given their proportional relationship and effect on the overall mass, the key parameters for the Prada Aoyama facade appear to be the number of grid divisions and the number of panelization divisions. As shown in the following design variations – achieved via the developed scaffold – other variables are kept constant, while these key parameters are modified.

231

232

8.17

DIAMOND

DIAMOND

233

234

DIAMOND

8.18

base model

grid number of divisions X grid number of divisions Z panelization number of divisions X panelization number of divisions Z

[10] [34] [6] [17]

In the base model for Prada Aoyama, there is a proportional relationship between the number of grid divisions and the number of panelization divisions, wherein: • panelization number of divisions X = (grid number of divisions X / 2) + 1 • panelization number of divisions Z = grid number of divisions Z / 2

DIAMOND

235

8.19

variation 01

grid number of divisions X grid number of divisions Z panelization number of divisions X panelization number of divisions Z

[16] [16] [9] [8]

In variation 01, the proportional relationships between the number of grid divisions and the number of panelization divisions are kept the same, but the variable inputs result in a vertical rather than horizontal diamond pattern.

236

DIAMOND

8.20

variation 02

grid number of divisions X grid number of divisions Z panelization number of divisions X panelization number of divisions Z

[10] [34] [6] [17]

select panels grid number of divisions X grid number of divisions Z

[4] [4]

In variation 02, the relationships from the original facade are maintained. An additional grid is applied to some of the panels, reducing the size of these windows. Some panels are rendered as solid rather than glass, adding further variation to the facade.

DIAMOND

Additional Modelling Inquiries 1. Adapt the script to model other diamond patterns. 2. Apply diamond patterns to other mass models. 3. Filter lines to create alternative diagonal patterns. 4. Experiment with different combinations of scaling and rotating diamond panels on the surface plane. 5. Apply diamond panels to the nodes or centerpoints of other grids. Combine with the previous scaling and rotating. 6. Adjust the Prada Aoyama mass model. How does this effect the diamond pattern? Which controls the other? 7. Link the size of diamond panels to the grid and floor-to-floor heights; which controls the other? What are the implications of having these linked (or not)? 8. Study the interior of Prada Aoyama. Note the relationship between interior elements (walls, floors, ceilings, and stairs) and the facade pattern. Extend your facade patterns into the interior to achieve the same effects or should the interior elements extend to effect the facade pattern? Which controls the other? 9. Explore solid and void (opaque and clear and flat and bulged) compositions of facade design.

237

09

TESSELLATE

Facades composed of repeating tiled shapes

9.1

Tessellated patterns create a surface through repetition. Series of repeated operations – such as copy, rotate, mirror, and array – are applied to a shape or groups of shapes to develop a surface. This is conceptually a different task than previous surface patterns that were applied to a surface. In tessellations, the individual shape and surface pattern become indistinguishable. Tessellation patterns occurring “periodically” – or repeating across a surface – can be categorized into seventeen wallpaper groups (Joyce). Buildings with tessellated facades include the arrayed panels of Wulf Architekten’s P22a Multi Story Car Park, the reflected panels of Amit Khanna Design Associates’ Tri Tessellate, the horizontally and vertically reflected panels of Carey Jones Chapman Tolcher’s University of Leeds Multi Story Car Park, the intricate screens of Ateliers Jean Nouvel’s Doha High Rise Office Tower, the rotating diamond panels of UN Studio’s Lane 189, and the operable skin of AHR’s Al Bahr Towers. David Eck, Professor of Mathematics and Computer Science at Hobart and William Smith Colleges, developed an online interactive wallpaper exploration tool (Eck).

DOI: 10.4324/9781003299417-10

TESSELL ATE

239

Scripting tesselations

9.2

The script above describes the tessellation pattern of WALLPAPER GROUP 05, wherein a copy of a shape (in this case, a triangle) is reflected horizontally. This pair is then translated horizontally and vertically (for example, in a diagrid pattern).

240

TESSELL ATE

9.3

9.4

Other tessellation patterns use operations in addition to mirroring and patterns beyond a diagrid. The script above describes the tessellation pattern of WALLPAPER GROUP 13, wherein a copy of a shape is rotated 120 and 240 degrees. These three shapes are then copied in a triangular pattern.

TESSELL ATE

241

9.5

9.6

The script above describes the tessellation pattern of WALLPAPER GROUP 07, wherein a copy of a shape is reflected horizontally, and then, this pair is glide-reflected vertically. These four shapes are then arrayed in a regular grid pattern.

FURTHER WALLPAPER GROUPS

In the following pattern diagrams, the same triangular shape is applied to each wallpaper group for comparison. Each has a description of how the shape is repeated, lists the symmetry group, and references the IUC (International Union of Crystallography) notation (Aroyo).

9.7

01

p1

A copy of a shape is arrayed horizontally and vertically, at any angle.

p2

A copy of a shape is rotated 180 degrees. This pair is arrayed horizontally and vertically, at any angle.

9.8

02

9.9

03

pm

A copy of a shape is reflected horizontally (as above) or vertically. This pair is arrayed in a quad pattern.

pg

A copy of a shape is glide-reflected horizontally (as above) or vertically. This pair is arrayed in a quad pattern.

9.10

04

9.11

05

cm

A copy of a shape is reflected horizontally (as above) or vertically. This pair is arrayed horizontally and vertically, at any angle.

pmm

A copy of a shape is reflected horizontally. This pair is reflected vertically. These four shapes are arrayed in a quad pattern.

9.12

06

9.13

07

pmg

A copy of a shape is reflected horizontally (as above) or vertically. This pair is glide-reflected in the opposite direction (if the shape is initially reflected horizontally, then the pair is glide-reflected vertically; if the shape is initially reflected vertically, then the pair is glide-reflected horizontally). These four shapes are arrayed in a quad pattern.

pgg

A copy of a shape is rotated 180 degrees. This pair is glide-reflected horizontally (as above) or vertically. These four shapes are arrayed in a quad pattern.

9.14

08

9.15

09

cmm A copy of a shape is rotated 180 degrees. This pair is reflected vertically and horizontally and rotated 180 degrees. These eight shapes are arrayed in a quad pattern.

9.16

10

p4

A copy of a shape is rotated 90, 180, and 270 degrees. These four shapes are arrayed in a square quad pattern.

9.17

11

p4m

A copy of a shape is rotated 90, 180, and 270 degrees. A copy of a shape is reflected at 45 degrees (bisecting the square) and then also rotated 90, 180, and 270 degrees. These eights shapes are arrayed in a square quad pattern.

p4g

A copy of a shape is reflected horizontally. This pair is reflected vertically. These four shapes are rotated 90, 180, and 270 degrees. These sixteen shapes are arrayed in a square quad pattern.

9.18

12

9.19

13

p3

A copy of a shape is rotated 120 and 240 degrees. These three shapes are arrayed in a hexagonal pattern divided into equilateral triangles.

9.20

14

p31m

A copy of a shape is rotated 120 and 240 degrees. A copy of a shape is reflected (bisecting the equilateral triangle) and then also rotated 120 and 240 degrees. These six shapes are arrayed in a hexagonal pattern divided into equilateral triangles.

9.21

15

p3m1

A copy of a shape is rotated 120 and 240 degrees. A copy of a shape is reflected (using an edge of the equilateral triangle) and then also rotated 120 and 240 degrees. These six shapes are arrayed in a hexagonal pattern divided into equilateral triangles.

p6

A copy of a shape is rotated 60, 120, 180, 240, and 300 degrees. These six shapes are arrayed in a hexagonal pattern divided into equilateral triangles.

9.22

16

9.23

17

p6m

A copy of a shape is rotated 60, 120, 180, 240, and 300 degrees. A copy of a shape is reflected at 60 degrees (bisecting the equilateral triangle) and then also rotated 60, 120, 180, 240, and 300 degrees. These twelve shapes are arrayed in a hexagonal pattern divided into equilateral triangles. (The same pattern would result if the shape was reflected using the edge of the equilateral triangle.)

9.24

CASE STUDY 09

Bergeron Center for Engineering Excellence building type architect location size shape material year built

academic ZAS Architects Toronto, Canada 170,000 sf irregular metal panel, glass 2015

The design of the Bergeron Center for Engineering Excellence (Bergeron Center) imagines new ways of learning, discovering, and interacting in higher education. Behind the tessellated facade, typical lecture halls are replaced with multiple scales and types of active learning spaces.

Opposite page: Photo of Bergeron Center for Engineering Excellence by Jeffrey Collins.

254

TESSELL ATE

9.25

SURFACE The mass of the Bergeron Center building is extruded from a closed spline.

TESSELL ATE

255

9.26

GRID

diagrid

shape number of divisions X number of divisions Z

[surface] [158] [12]

The tesselation grid is a diagrid applied to the surface.

256

TESSELL ATE

9.27

SHAPE

hexagon

location [centerpoints] The tessellation shape is a hexagon applied to centerpoints of the diamond grid panels.

TESSELL ATE

257

9.28

PANELIZATION

line

point point

[vertices] [moved centerpoints]

A pattern is produced on each tessellation shape by connecting a line from each vertex of the hexagon to a slightly displaced centerpoint of the diamond grid panel. This procedure is applied to each hexagon.

258

TESSELL ATE

9.29

INITIAL SCAFFOLD The combination of the previously modelled surface, grid, shape, and panelization defines an initial facade scaffold for the Bergeron Center building. The following are noted as limitations of this model, whose solutions will be discussed in the next section: • the tiled surface of the Bergeron Center facade is trimmed, making the upper levels appear to float above the ground level • some panels of the Bergeron Center facade are metal – of different finishes – and others are glass

TESSELL ATE

9.30

TRIMMED SURFACE The original spline defining the Bergeron Center mass is moved vertically using a sine graph to create an irregular line. This line trims the tiles above.

259

260

TESSELL ATE

9.31

PANEL MATERIALITY Panels are filtered into several different groups representing different cladding materials.

TESSELL ATE

9.32

DEVELOPED SCAFFOLD The script for this developed Bergeron Center facade scaffold includes seven parameters: • • • • • • •

control points for surface extrusion height of surface extrusion height of curve to trim surface grid number of divisions X grid number of divisions Z distance for relocated diagrid centerpoint panel materiality filter

The building mass is created from an extruded closed polyline. This closed polyline is moved vertically using a sine graph to create an irregular profile, which trims the SURFACE. A diagrid is distributed across the surface to define the GRID. Hexagon SHAPES are applied to centerpoints of the diamond grid panels. A PANELIZATION pattern is created by connecting a line from each vertex of the hexagon to slightly displaced centerpoints of the diamond grid panels. The resulting triangular panels are then filtered into several groups representing different cladding materials and finishes. The key to the Bergeron Center facade is the selected tessellation type. As shown in the following design variations – achieved via the developed scaffold – other variables are kept constant, while the utilized wallpaper group is changed.

261

262

9.33

TESSELL ATE

TESSELLATE

263

264

TESSELL ATE

9.34

Base model The base model for the Bergeron Center facade employs wallpaper group 01. The repeated shape is a hexagon. Each hexagon is split into triangles by connecting lines from each vertex of the hexagon to a slightly displaced centerpoint. The hexagons and associated triangular panels are then arrayed in a diagrid pattern.

TESSELL ATE

9.35

Variation 01 Variation 01 employs wallpaper group 13. The repeated shape is a diamond split into two quadrilaterals. Each shape is rotated 120 and 240 degrees. The diamonds and associated quadrilaterals are then copied in a triangular pattern.

265

266

TESSELL ATE

9.36

Variation 02 Variation 02 employs wallpaper group 07. The repeated shape is a triangle that is reflected horizontally. This pair is glide-reflected vertically. Then, these four shapes are arrayed in a regular grid pattern.

TESSELL ATE

Additional Modelling Inquiries 1. Apply tessellation patterns to other mass models. 2. Experiment with different wallpaper groups; apply the same shape to each for comparison. 3. Filter and scale panels. Explore solid and void compositions. Develop alternative tessellation-inspired patterns. 4. The Bergeron Center facade pattern is created by applying a pattern within a pattern. What other combinations of patterns are beneficial? 5. The Bergeron Center facade employs wallpaper group 01 on a mass formed by an extruded closed spline. Do certain combinations of wallpaper groups and mass topologies work better together than others? 6. How does the trimmed surface (lifting the patterned surface above the ground level) enhance the approach for the Bergeron Center building mass and facade pattern? 7. What shapes or kinds of buildings benefit from tessellated facades?

Notes Aroyo, M. (ed.). International Tables for Crystallography, Volume A: Space-Group Symmetry. Wiley, 2016. https://it.iucr.org/Ac/ Eck, D. Wallpaper Symmetry. Department of Mathematics and Computer Science, Hobart and William Smith Colleges, https://math.hws.edu/eck/js/ symmetry/wallpaper.html Joyce, D. The 17 Plane Symmetry Groups. Department of Mathematics and Computer Science, Clark University, https://www2.clarku.edu/faculty/djoyce/ wallpaper/seventeen.htm

267

10

ATTRACTOR

Facade patterns that vary based on location

10.1

Attractor geometry functions by modifying a variable of a facade component – such as location, size, and rotation – by a factor. This factor is often based on the effected component’s proximity to control points or curves. The effect often resembles a natural occurrence, such as the wind fluttering elements of Koning Eizenberg Architecture’s Children’s Museum of Pittsburgh facade. The additional factor is referred to as a “disruption.” The effect is a “resultant.” Building facade components effected by a disruption include the type of brick on Behet Bondzio Lin Architekten’s Administration Building Textilverband, the size of window openings on KAAN Architecten’s Crematorium Heimolen, the angle of louvers on Ferrer Arquitectos’ North Mediterranean Health Center and Studio for Architecture’s Lafayette 148, and the locations of perforations on Lacime Architects’ Suzhou Financial Center Exhibition Hall. Often, such geometrical morphing is associated with project objectives such as air flow or daylight filtration.

DOI: 10.4324/9781003299417-11

AT TRACTOR

Scripting attractors

10.2

Input data for attractor scripts include the surface, pattern, effected shape, and disruption; these elements cause change to the norm. In the model shown on the opposite page, the radii of the circles vary based on the distance from the centerpoint of the diagrid pattern to a control point.

269

270

AT TRACTOR

10.3

Replacing the point with a line, the radii of the circles here vary based on the distance from the centerpoint of the diagrid pattern to the line.

AT TRACTOR

10.4

271

10.5

CASE STUDY 10

City View Garage building type architect location size shape material year built

parking garage IwamotoScott Architecture Miami, Florida 15,000 sf L aluminum panel screen 2015

Facade panels for City View Garage create a screen, encouraging natural ventilation. A gestural spline defines locations for panel types with varying aperture sizes. Panel types are also tinted in a gradient of shades to exaggerate the effect.

Opposite page: Photo of City View Garage by IwamotoScott Architecture.

274

AT TRACTOR

10.6

SURFACE The parking garage for the City View Garage project has a radiused corner that blends one side with another. The ground floor of the parking garage is retail.

AT TRACTOR

275

10.7

GRID diagrid shape number of divisions X number of divisions Z

[surface] [106] [19]

A diagrid is applied to the surface.

276

AT TRACTOR

10.8

SHAPE scale curve factor f

[panels] [0.42]

difference brep brep

[surface] [shapes]

Each diamond shape from the defined grid is scaled about the centerpoint of each panel. These shapes are cut from the surface to create a screen.

AT TRACTOR

10.9

DISRUPTION A gestural spline is drawn across the surface.

277

278

AT TRACTOR

10.10

RESULTANT scale curve factor f

[panels] [varies]

difference brep brep

[surface] [shapes]

The scale factor for the width of each diamond varies based on its proximity to the spline. The height of the diamond opening is constant.

AT TRACTOR

279

10.11

PANELIZATION hexagon grid shape number of divisions X number of divisions Z

[surface] [106] [19]

regular grid shape number of divisions X number of divisions Z

[rectangle] [106] [1]

Both a hexagon grid and a regular grid are applied to the surface. These lines split the screen into panels.

280

AT TRACTOR

10.12

INITIAL SCAFFOLD The combination of the previously modelled surface, grid, shape, disruption, resultant, and panelization defines an initial facade scaffold for City View Garage. The following are noted as limitations of this model, whose solutions will be discussed in the next section: • the City View Garage screen is not flat; panels are folded to create depth • the City View Garage panels are painted in a gradient of color across the facade

AT TRACTOR

10.13

FOLDED PANELS A series of triangles are formed by connecting vertices of the diamonds and half-hexagons. Again, the scale factor for the width of each diamond varies based on its proximity to the disruptor spline.

281

282

AT TRACTOR

10.14

GRADIENT A gradient is applied to the surface. Shades are based on proximity to the disruptor spline. Panels are painted in a similar manner to emphasize the attractor effect.

USING IMAGES TO CREATE PATTERNS In addition to using geometry to modify patterns, images can also be used. In the following example, the radii of circles in a regular grid pattern vary based on the tonal values occurring in corresponding locations on the referenced picture.

01 circles on nodes

10.15

02 reference image

10.16

03 resultant varying radii

10.17

284

AT TRACTOR

10.18

DEVELOPED SCAFFOLD The script for this developed City View Garage facade scaffold includes seven parameters: • • • • • • •

control points for surface extrusion height of ground floor height of surface grids number of divisions X grids number of divisions Z control points for spline panel color filter

AT TRACTOR

Control points for a spline, height above the ground, and extrusion height define the scaffold SURFACE. A diagrid is applied to the surface to define the GRID. Panels of this grid are scaled based on their proximity to a gestural spline drawn across the surface, which represents facade SHAPES. Vertices from overlaid hex and regular grids connect to vertices of the scaled diamonds, to form the PANELIZATION pattern. The key to the City View Garage facade is clearly the effect of the disruptor spline. This is not only aesthetic; proximity to this spline effects the panel opening size and, therefore, the amount of air and daylight allowed into the garage. As shown in the following design variations – achieved via the developed scaffold – other variables are kept constant, while the disruptor geometry is changed.

285

286

10.19

AT TRACTOR

AT TRACTOR

287

288

AT TRACTOR

10.20

Base model Disruptor geometry for the base model of the City View Garage facade is a gestural spline, creating a wave-like effect across the surface.

AT TRACTOR

10.21

Variation 01 Variation 01 employs two horizontal lines to vary the size of the diamond-shaped openings and colors of the panels, creating a banded facade.

289

290

AT TRACTOR

10.22

Variation 02 Variation 02 employs a series of control points to vary the size of the diamond-shaped openings and colors of the panels, creating hot spots on the facade.

AT TRACTOR

Additional Modelling Inquiries 1. Apply attractor patterns to other mass models. 2. Experiment with adjusting further component variables – location, size, and rotation – through attractors. 3. Correlate disruptor geometry to other project objectives (air flow or daylight filtration). 4. The City View Garage facade modifies the width of diamond-shaped openings based on their proximity to a spline. Do certain combinations of patterns, adjusted component variables, and disruptors work better together than others?

291

11

IRREGULAR

Facade patterns that embrace informal relationships

11.1

Random. Unplanned. Impulsive. These may sound like concepts contrary to the rigorous processes described thus far in this book. However, practical parametric models of irregular patterns can be developed and – paradoxically – organized. In fact, the reasons to attempt organizing irregular patterns are the same as the benefits of parametric design listed in the Introduction: enhanced geometrical control, the ability to produce design variations, and the potential of emergent design possibilities. Projects that display irregular facade patterns include Perot Museum of Nature and Science by Morphosis, U.S. Census Bureau Headquarters by SOM, Times Eureka Pavilion by Nex, Diagonal ZeroZero Telefonica Tower by Estudi Massip-Bosch Architects, Faculty of Engineering, University of Southern Denmark by C.F. Møller Architects, and Pierhead Street Multi-Story Car Park by Scott Brownrigg Architects.

DOI: 10.4324/9781003299417-12

IRREGULAR PATTERNS The following diagrams capture a selection of irregular pattern types.

hatching

lines between points

lines from point on surface to points on edges of surface

lines vertical or horizontal connecting points on edges of surface

lines creating a maze

lines connecting points on boundaries of regions

11.2

meandering

curves between points

single curve connecting points on surface

curves vertical or horizontal from points on edges of surface

series of offset curves similar to a contour map

curves from points on boundaries of regions

11.3

packing

shapes on points

shapes on points across surface

rotating and scaling shapes

shapes packed to reduce negative space

voronoi

11.4

branching

iterative placement of lines or shapes

lines from points on lines

network

ice ray

fractal

11.5

298

IRREGUL AR

Scripting irregular patterns

11.6

Scripts for irregular patterns will vary largely based on the type of input data. In the model shown on page 293, lines are drawn from control points on opposite edges of a surface. These lines are then offset to give thickness.

11.7

CASE STUDY 11

Sugar Hill Development building type architect location size shape material year built

housing David Adjaye Associates New York, New York 172,000 sf rectangle precast concrete 2014

Sugar Hill Development is a mixed-use building including affordable housing units and educational programs. Exterior walls are precast concrete panels dappled with an inset abstract rose motif and square windows. Multiple irregular patterns occur and overlap; panel size, window location, and inset size and location.

Opposite page: Photo of Sugar Hill Development by Rubi Xu.

302

IRREGUL AR

11.8

SURFACE The mass of the Sugar Hill Development building is formed from three extruded stacked profiles. One surface is selected.

IRREGULAR

11.9

GRID Points are populated across the surface.

303

304

IRREGUL AR

11.10

SHAPES

circles

location radii

squares location radii

difference brep brep

[points] [varies] [points] [varies] [surface] [squares]

Points are filtered; placeholder circles are applied to some of the points, and squares are applied to others. The square shapes are subtracted from the surface to create windows.

IRREGULAR

305

11.11

PANELIZATION

horizontal running bond shape number of divisions X number of divisions Z

[surface] [8] [12]

A horizontal running bond quad pattern is applied to the surface, dividing it into individual panels.

306

IRREGUL AR

11.12

INITIAL SCAFFOLD The combination of the previously modelled surface, grid, shapes, and panelization defines an initial facade scaffold for the Sugar Hill Development building. The following are noted as limitations of this model, whose solutions will be discussed in the next section: • Sugar Hill Development uses a rose motif rather than circles • the rose motif is carved into the precast facade panels • the location of Sugar Hill Development windows requires further control • the Sugar Hill Development panelization pattern varies across several regions of the facade

IRREGULAR

11.13

CUSTOM SHAPE Instead of circles of varying radii, roses of varying scales are distributed across the surface.

307

308

IRREGUL AR

11.14

PRECAST CARVING As described on the pages 310–311, the roses are abstracted into a pattern that is carved into the facade panels.

FURTHER DISCUSSION ON SUGAR HILL FACADE TEXTURE The followings steps describe a process for transforming a rose shape into an abstract design carved into the facade panels.

01 rose shaped surfaces

02 regular grid

03 divide

04 intersection

shape number s 11.15

[lines] [10]

number of divisions X number of divisions Z

[10] [1]

points that overlap rose shape surfaces

05 intersection

06 circles

07 loft

08 difference

points that do not overlap rose shape surfaces

vertical rows of circles 11.16

larger diameter circles on points on the XY plane that overlap rose shape surfaces, smaller diameter circles on points on the XY plane that do not overlap rose shape surfaces

brep brep

[surface] [lofts]

312

11.17

WINDOW LOCATIONS Insertion points for the square windows are more evenly and precisely defined.

IRREGUL AR

IRREGULAR

313

11.18

PANELIZATION REGIONS A series of regions are defined across the surface. Region 1

region 2

region 3

regular grid number of divisions X number of divisions Z

[3] [1]

region 5

regular grid number of divisions X number of divisions Z

[5] [1]

region 6

horizontal running bond number of divisions X [5] number of divisions Z [5]

region 7

horizontal running bond number of divisions X number of divisions Z

[5] [2]

horizontal running bond number of divisions X number of divisions Z

[5] [3]

regular grid* number of divisions X number of divisions Z

region 4 * For region 7, every other vertical line is hidden to produce a running bond pattern.

[10] [1]

314

IRREGUL AR

11.19

DEVELOPED SCAFFOLD The script for this developed Sugar Hill Development facade scaffold includes ten parameters: • • • • • • • • • •

control points for surface control points for locations of windows and roses size of windows scale of roses texture grid number of divisions X texture grid number of divisions Z texture loft radii control points for regions regions patterns number of divisions X regions patterns number of divisions Z

IRREGULAR

One side of the extruded stacked building mass defines the scaffold SURFACE. Points are populated across this surface as a GRID for locating square and rose SHAPES. Squares are subtracted from the surface as windows. Circles of varying diameters are applied to points along a regular grid pattern that intersect and do not intersect the rose shapes. These circles are lofted, and the lofts are subtracted from the surface as texture. A series of regions and grids define the surface PANELIZATION. The key parameters effecting the irregular pattern of the Sugar Hill Development façade are shape distribution and surface panelization. As shown in the following design variations – achieved via the developed scaffold – other variables are kept constant, while these key parameters are modified.

315

316

11.20

IRREGUL AR

IRREGULAR

317

318

IRREGUL AR

11.21

Base model In the base model for Sugar Hill Development, the facade has regions to define various horizontal running bond panelization patterns. Windows – while somewhat coordinated with panel joints – are randomly distributed across the surface. Texture is similarly stretched across the surface. Each panel is unique.

IRREGULAR

11.22

Variation 01 In variation 01, a consistent horizontal running bond panelization pattern is applied to the surface. Windows and textures are copied to each panel. All panels are the same.

319

320

IRREGUL AR

11.23

Variation 02 In variation 02, four different panels are developed, each with a unique size, window placement, and texture. These panels are distributed across the surface.

IRREGULAR

Additional Modelling Inquiries 1. Can you further classify irregular pattern types? 2. Apply irregular patterns to other mass models. 3. How does the horizontal running bond panelization pattern enhance the approach for the Sugar Hill Development building mass? 4. Study the precast panels on the Sugar Hill Development building. Are there (or could there be) repeating panel types?

321

12

LAYER

Facades with overlapping organizations

12.1

In Case Study 08 Diamonds, we discussed layering patterns of shapes to create composite geometries. Some facades also incorporate physical separation between patterns. These layers – and the space in between – are very often used to enhance buildings’ environmental performance. Facade layers can be composed of a variety of materials including perforated metal (as seen on Henning Larsen’s University of Southern Denmark – Campus Kolding, ZGF’s University of Arizona Cancer Center, and Yazdani Studio of Cannon Design’s CJ Blossom Park), ETFT (ethylene tetrafluoroethylene, as seen on KieranTimberlake’s U.S. Embassy in London, and Behnisch Architekten’s Unilever Headquarters), or fiberglass (as seen on Gerber Architekten’s King Fahad National Library).

DOI: 10.4324/9781003299417-13

LAYER

Scripting layers

12.2

Input data for layered scripts will vary based on the types of patterns used. In the example on the previous page, a regular grid and a diagrid are applied to the same surface. Additional functions can also be added, such as offset, rotate, or scale.

323

12.3

CASE STUDY 12

Gardner Neuroscience Institute building type architect location size shape material year built

healthcare Perkins & Will Cincinnati, Ohio 194,000 sf rectangle polyester fiber mesh 2019

In contrast to the rectangular glass box underneath, the second skin of the Gardner neuroscience Institute building is a light faceted mesh. Creating a dynamic facade, this additional layer controls heat gain and prevents glare while maintaining visual connections to the exterior.

Opposite page: Photo of Gardner Neuroscience Institute by Jeffrey Collins.

326

L AYER

12.4

SURFACE 1 Two surfaces are selected on the upper portion of a rectangular stacked mass. rectangle length x width y

[103] [221]

extrude shape height z

[rectangle] [40]

LAYER

327

12.5

SURFACE 2 The surfaces are offset and joined together at the corner. offset geometry distance s

[surfaces] [4.5]

328

L AYER

12.6

GRID 1

regular grid

shape number of divisions X number of divisions Z

regular grid

shape number of divisions X number of divisions Z

[front surface] [23] [3]

[side surface] [50] [3]

LAYER

329

12.7

GRID 2

regular grid

shape number of divisions X number of divisions Z

regular grid

shape number of divisions Y number of divisions Z

[front surface] [12] [1]

[side surface] [26] [1]

330

L AYER

12.8

SHAPE 1

offset

curve distance s

[panels] [0.08]

Lines from the defined grid are offset to represent individual facade panels and framing.

LAYER

12.9

SHAPE 2 Sets of control points within each grid panel define a spaceframe.

331

332

L AYER

12.10

PANELIZATION 1 The panelization pattern for surface 1 is the same as the grid.

LAYER

12.11

PANELIZATION 2 Surfaces are applied to the spaceframe structure.

333

334

L AYER

12.12

INITIAL SCAFFOLD The combination of the previously modelled surfaces, grids, shapes, and panelization patterns define an initial facade scaffold for the Gardner Neuroscience Institute facade. The following are noted as limitations of this model, whose solutions will be discussed in the next section: • the Gardner Neuroscience Institute grid (and the resulting shapes of the glass panels) are irregular • the Gardner Neuroscience Institute surface offset is based on panel width • the Gardner Neuroscience Institute offset surfaces have struts to connect them back to the building

LAYER

12.13

IRREGULAR GRID An irregular grid is developed using regular spacing in the X or Y directions along with the heights of lines in the Z direction.

335

336

L AYER

12.14

IRREGULAR SHAPES

offset

curve distance s

[panels] [0.08]

Lines from the defined grid are offset to represent individual facade panels and framing.

LAYER

12.15

CALCULATED OFFSET SURFACES The offset dimension of the surfaces is set to the width of the panels on the adjacent.

337

338

L AYER

12.16

STRUTS Horizontal lines connect the uprights of surface 2 panels to surface 1.

12.17

LAYER

DEVELOPED SCAFFOLD The script for this developed Gardner Neuroscience Institute facade scaffold includes ten parameters: • • • • • • • • • •

mass width x mass length y mass height z surface 1 grid number of divisions X or Y surface 1 grid number of divisions Z height of surface 1 grid lines in Z direction surface 1 frame width s surface 2 grid number of divisions X or Y surface 2 grid number of divisions Z control points for struts and mesh surface

There are two layers to the Gardner Neuroscience Institute facade. The width, length, and height of the rectangular mass define SURFACE 1. An irregular GRID is developed using regular spacing in the X or Y directions along with the heights of lines in the Z direction. Lines from this grid are offset to represent facade SHAPES. The grid is repeated as surface 1 PANELIZATION. Surface 1 is offset by a factor equivalent to the grid width to create SURFACE 2. A regular GRID is applied. Control points are defined within each grid panel to create faceted SHAPES. Mesh surfaces are stretched across this structure as PANELIZATION. Given the proportional relationship between the surface grids and the faceted structure, the key parameters for the Gardner Neuroscience Institute facade appear to be the surface 1 grid number of divisions X or Y and surface 2 grid number of divisions X or Y. As shown in the following design variations – achieved via the developed scaffold – other variables are kept constant, while these key parameters are modified.

339

340

12.18

L AYER

LAYER

341

342

L AYER

12.19

base model number of divisions X number of divisions Y

surface 1 [23] [50]

surface 2 [12] [26]

LAYER

343

12.20

variation 01 number of divisions X number of divisions Y

surface 1 [23] [50]

surface 2 [6] [13]

344

L AYER

12.21

variation 02 number of divisions X number of divisions Y

surface 1 [12] [24]

surface 2 [26] [52]

LAYER

Additional Modelling Inquiries 1. Use other grids to create layered patterns. 2. The Gardner Neuroscience Institute facade employs an irregular grid and a regular grid; both are quad patterns. Do certain combinations of grid types layer better together than others? 3. Consider the relationship between surface patterns and surface offset distance. 4. Optimize layers to improve other project objectives (for example, environmental performance).

345

CONCLUSION This book has described processes for creating digital parametric models for a variety of buildings. These case studies were affiliated with common mass or facade topologies. Modelling procedures for each followed the same series of deliberate steps. Visual scripts, likewise, helped to organize modelling functions graphically. These techniques are intended to empower the reader with the knowledge and confidence to create their own customizable parametric models and to overcome any previous apprehension or confusion. Look back through these pages. Construct your own models and variations of the scripts. Engage each prompt for additional discussion and exploration. Incrementally incorporate these steps into your own design processes. You are a parametric designer.

REFERENCED BUILT WORKS

INTRODUCTION ACTLAB. Trabeculae Pavilion. Milano, 2018, www.act-lab.net/trabeculae-pavilion. html Architectkidd.Lightmos. Bangkok, 2008, www.architectkidd.com/index.php/2010 /04/speckled-superstructures/ Studio Gang. Aqua Tower. Chicago, IL, 2010, https://studiogang.com/project/ aqua-tower

EXTRUDE Cobe. Forfatterhuset Kindergarten. Copenhagen, 2014, www.cobe.dk/place/ forfatterhuset-kindergarten EnsambleStudio. Ensamble Fabrica. Madrid, 2019, www.ensamble.info/ensamblefabrica Foreign Office Architects. Carabenchel Housing. Madrid, 2008, https://azpml. com/#/projects/carabanchel/373 Motta. Lomo Cubes. Lugano, 2012, www.gmotta.it/project/lomo-cubes-building/ MVRDV. Markthal. Rotterdam, 2014, www.mvrdv.nl/projects/115/markthal POLO Architect. Red Cross-Flanders. Zuienkerke, 2018, https://polo-architects. be/nl/projecten/rode-kruis-zuienkerke/ SHoP Architects. Mulberry House. New York, NY, 2013, www.shoparc.com/projects/ mulberry-house/

LOFT Alison Brooks Architect. The Smile. London, 2016, www.alisonbrooksarchitects. com/project/the-smile/ Atelier XI. Peach Hut. Jiaozuo, 2020, www.atelierxi.com/peach-hut/ de Architekten Cie. Menzis. Groningen, 2005, https://cie.nl/page/787/menzis Kengo Kuma andAssociates. Victoria& Albert Museum. Dundee, 2019, https://kkaa. co.jp/works/architecture/va-dundee/ Lacime Architects. Xiangcheng District Planning Exhibition Hall. Suzhou, 2017, www.lacime-sh.com/index/index/Caseinfo.html?id=28 MAD Architects. Absolute Towers. Mississauga, 2012, www.i-mad.com/work/ absolute-towers/?cid=4 Snohetta. Le Monde Group Headquarters. Paris, 2020, https://snohetta.com/ projects/526-le-monde-group-headquarters

STACK Bengo Studio. QiyunshanTree House. Xiuning, 2016, www.chinese-architects.com/ en/bengo-studio-xuhui-district-shanghai/project/qiyunshan-tree-house Mack Scogin Merrill Elam Architects. Gates Center for Computer Science and Hillman Center for Future Generation Technologies. Pittsburgh, 2009, http://msmearch. com/type/academic/carnegie-mellon-university-gates-center-for-computerscience-and-hillman-center-for-future-generation-technologies Matsuyama Architect and Associates. Deta Heisei Eye Clinic. Kumamoto, 2019, www.matsuyama-a.co.jp/project/idetaheiseiganka/

348

REFERENCED BUILT WORKS

Mecanoo. Library of Birmingham. Birmingham, 2013, www.mecanoo.nl/Projects/ project/57/Library-of-Birmingham Metaform. Apartment Building. Cessange, 2011, www.metaform.lu/projects/ residence-building-cessange Sanjay Puri Architect. Aria Resort & Spa. Nasik, 2020, https://sanjaypuriarchitects. com/interior/hospitality/aria-resort-spa-nasik/ ZalewskiArchitectureGroup. DomCube-2-Box. Myslowice, 2015, www.zalewskiag. com/projects/project/dom_cube2box

CARVE Jakob + Macfarlane Architects. The Orange Cube. Lyon, 2011, www.jakobmacfarlane. com/en/project/orange-cube/ Lorcan O’Herlihy Architects. Mariposa1038. Los Angeles, 2017, http://loharchitects. com/work/mariposa1038 Steven Holl Architects. Hunters Point Library. Queens, NY, 2019, www.stevenholl. com/projects/hunters-point-library Takato Tamagami Architectural Design. Sunwell Muse Kitasando. Tokyo, 2008, https://takatotamagami.net/en/works/office/arswh/ Toyo Ito & Associates, Architects. Porta Fira Towers. Barcelona, 2010, www.toyo-ito. co.jp/WWW/Project_Descript/2010-/2010-p_01/2010-p_01_en.html TRU Architects. Simone 0914 Flagship Store. Gangnam-Gu, 2017, www.trugroup. co.kr/simone-0914 Zaha Hadid Architects. Opus. Dubai, 2020, www.zaha-hadid.com/architecture/opus/

NEST Atelier Hitoshi Abe. Ftown Building. Sendai, 2008, https://aha.design/ftown-building/ Lewis Tsurumaki Lewis. Bornhuetter Hall. Wooster, OH, 2004, http://ltlarchitects. com/bornhuetter-hall Ming Architects. Cube House. Singapore, 2017, www.mingarchitects.com/projectsitem/cube-house/ Office of Mcfarlane Biggar Architects + Designers. Telus Garden. Vancouver, 2011, www.officemb.ca/work/telus-garden/ Studio Zhu Pei. Shou County Culture and Art Center. Anhui, 2019, www.studiozhupei. com/en/show/?id=424&page=1&siteid=1 Tabanlioglu Architects. Dogan Media Center. Ankara, 2009, www.tabanlioglu.com/ project/dogan-media-center/ Victoria Acebo & Angel Alonso. National Museum of Science and Technology. Coruña, 2006, www.aceboxalonso.es/museum

TRIM BIG. Via 57 West. New York, NY, 2016, https://big.dk/#projects-w57 Centerbrook. Biomass Heating Facility. Lakeville, CT, https://centerbrook.com/ project/hotchkiss_school_biomass_heating_facility Miller Hull Partnership. Bullitt Center. Seattle, WA, 2013, https://millerhull.com/ project/bullitt-center/ MPH Architects. Thebarton Community Centre.Thebarton, 2013, www.mpharchitects. com.au/portfolio_page/thebarton-community-centre/ Safdie Architects. Exploration Place. Wichita, KS, 2000, www.safdiearchitects.com/ projects/exploration-place

REFERENCED BUILT WORKS

Shigeru Ban Architects. Centre Pompidou-Metz. Metz, 2010, www.shigeru banarchitects.com/works/2010_centre-pompidou-metz/ Toshiko Mori Architect. Thread Artists’ Residences & Cultural Center. Sinthian, 2015, https://tmarch.com/thread

QUAD David Jameson Architect. Manifold House. Arlington, 2018, www.davidjameson architect.com/?project=manifold Degelo Architekten. Freiburg University Library. Freiburg, 2015, www.degelo.net/ projekte/Universitaetsbibliothek-Freiburg-D.php Digsau. 150 Rouse Boulevard. Philadelphia, 2012, www.digsau.com/projects/ 150-rouse Domiuque Perrault Architecture. Fukoku Life Osaka Project. Osaka, 2010, www. perraultarchitecture.com/en/projects/2473-fukoku_life_osaka_project.html Ernst Giselbrecht + Partner. Kiefer Technic Showroom. Bad Gleichenberg, 2007, http://giselbrecht.at/projekte/gewerbe_industriebauten/kiefer/index.html INNOCAD. C&P Corporate Headquarters. Graz, 2017, https://innocad.at/projects/cp/ Pasel Kunzel Architects. V36K0809 House. Leiden, 2009, https://paselkuenzel. com/portfolio/v36k0809/

DIAMOND Diller Scofidio + Renfro. The Broad. Los Angeles, CA, 2015, https://dsrny.com/ project/the-broad Foster + Partners. 30 St Mary Axe. London, 2004, www.fosterandpartners.com/ projects/30-st-mary-axe/ Herzog & de Meuron. Prada Aoyama. Tokyo, 2003, www.herzogdemeuron.com/ index/projects/complete-works/176-200/178-prada-aoyama.html OMA. Seattle Central Library. Seattle, WA, 2004, www.oma.com/projects/seattlecentral-library SHAU. Microlibrary. Semarang, 2020, www.shau.nl/en/project/84 Twelve Architects. The Diamond. Sheffield, 2015, https://twelvearchitects.com/ project/the-diamond/ Woods Bagot. South Australian Health and Medical Research Institute. Adelaide, 2014, www.woodsbagot.com/projects/south-australian-health-and-medicalresearch-institute/

TESSELLATE AHR. Al Bahr Towers. Abu Dhabi, 2012, www.ahr.co.uk/Al-Bahr-Towers Amit Khanna Design Associates. Tri Tessellate. Noida, 2018, www.akda.in/ tri-tessellate-1 Ateliers Jean Nouvel. Doha High Rise Office Tower. Doha, 2012, www.jeannouvel. com/en/projects/doha-9-high-rise-office-tower/ Carey Jones Chapman Tolcher. University of Leeds Multi Story Car Park. Leeds, 2016, www.cjctstudios.com/portfolio/university-of-leeds-mscp/ UN Studio. Lane 189. Shanghai, 2017, www.unstudio.com/en/page/11768/lane-189 Wulf Architekten. P22a Multi Story Car Park. Cologne, 2018, www.wulfarchitekten. com/projekte/detail/show/fassade-messeparkhaus-zoobrueckep22a/ ZAS Architects. Bergeron Center for Engineering Excellence. Toronto, 2015, www. zasa.com/bergeronyork

349

350

REFERENCED BUILT WORKS

ATTRACTOR Behet Bondzio Lin Architekten. Administration Building Textilverband. Munster, 2017, www.2bxl.com/#/tvm/ Ferrer Arquitectos. North Mediterranean Health Center. Almería, 2010, www. ferrerarquitectos.com/project/centro-de-salud-mediterraneo-norte/ IwamotoScott Architecture. City View Garage. Miami, 2015, https://iwamotoscott. com/projects/miami-city-view-garage KAAN Architecten. Crematorium Heimolen. Sint-Niklaas, 2008, https://kaan architecten.com/project/crematorium-heimolen/ Koning Eizenberg Architecture. Children’s Museum of Pittsburgh. Pittsburgh, 2005, www.kearch.com/childrens-museum-of-pittsburgh Lacime Architects. Suzhou Financial Center Exhibition Hall. Suzhou, 2019, www. lacime-sh.com/index/index/Caseinfo.html?id=111 Studio forArchitecture. Lafayette 148. Shantou, 2014, www.studioforarchitecture. net/house-of-doors-1

IRREGULAR C.F. Møller Architects. Faculty of Engineering, University of Southern Denmark. Odense, 2015, www.cfmoller.com/p/The-Technical-Faculty-SDU-i2571.html David Adjaye Associates. Sugar Hill Development. New York, NY, 2014, www. adjaye.com/work/sugar-hill-mixed-use-development Estudi Massip-Bosch Architects. Diagonal ZeroZero Telefonica Tower. Barcelona, 2011, www.emba.cat/?p=313 Morphosis. Perot Museum of Nature and Science. Dallas, 2012, www.morphosis. com/search/125/?s=perot&m=project Nex. Times Eureka Pavilion. London, 2011, www.nex-architecture.com/projects/ times-eureka-pavilion Scott Brownrigg Architects. Pierhead Street Multi-Story Car Park. Cardiff, 2008, www.scottbrownrigg.com/design-research-unit/articles-publications/ the-wave-structure-façade-pierhead-street-mscp-cardiff-bay/ SOM. U.S. Census Bureau Headquarters. Suitland, 2007, www.som.com/projects/ u-s-census-bureau-headquarters

LAYER Behnisch Architekten. Unilever Headquarters. Hamburg, 2009, https://behnisch. com/work/projects/0344 Gerber Architekten. King Fahad National Library. Riyadh, 2015, www.gerber architekten.de/en/project/king-fahad-national-library/ Henning Larsen. University of Southern Denmark – Campus Kolding. Kolding, 2014, https://henninglarsen.com/en/projects/0900-0999/0942-sdu-campus-kolding KieranTimberlake. U.S. Embassy in London. London, 2017, https://kierantimberlake. com/page/embassy-of-the-united-states-of-america Perkins & Will. Gardner Neuroscience Institute. Cincinnati, OH, 2019, https:// perkinswill.com/project/gardner-neuroscience-institute/ Yazdani Studio of Cannon Design. CJ Blossom Park. Suwon-Si, 2016, www. cannondesign.com/our-work/work/cj-corporation-blossom-park/ ZGF. University of Arizona Cancer Center. Phoenix, 2015, www.zgf.com/work/ 148-the-university-of-arizona-the-university-of-arizona-cancer-center-atdignity-health-st-joseph-s-hospital-and-medical-center

INDEX

2D populate 95–101, 174, 303 3D populate 150–153 Absolute Towers 62–63, 347 Aroyo M. 242, 267 array: linear 9, 13, 17, 30; polar

31, 66 Atelier Hitoshi Abe 143, 348 attractor 268–271 Bergeron Center for Engineering

Excellence 252–253, 349 Boolean: difference 9, 13, 17, 29,

122, 151, 162, 311; intersect

29, 152; union 29, 147 Brand W. 14, 35 Carabenchel Housing 38–39, 347 carve 114–115; see also Boolean

difference Centre Pompidou-Metz 166–167,

349 City View Garage 272–273, 350 code 11, 14, 18, 29 conic 124–125 David Adjaye Associates 301, 350 David Jameson Architect 197, 349 diagrid 29, 216, 221, 224–225,

239, 255, 275, 323 diamond 214–215 difference 29, 98–99, 101, 217,

276, 278, 304 Eck D. 238, 267 extrude 9, 12, 16, 30, 36–37, 85, 113, 164 Foreign Office Architects 39, 347 Ftown Building 142–143, 348

Gardner Neuroscience Institute

324–325, 350 Gates Center 88–89, 347 graphs 73–75, 81–83, 259 grid: hex 30, 248–250, 279;

irregular 193, 335; regular 32,

46–47, 193; triangular 32, 47,

248–250; see also diagrid

Herzog & de Meuron 219, 349 intersect 30, 100, 126, 148,

310–311 irregular 293–298 IwamotoScott Architecture 273,

350 Jakob + Macfarlane Architects

117, 348 Joyce D. 238, 267 layer 322–323 loft 9, 12, 16, 31, 57–60, 121, 153,

311 Mack Scogin Merrill Elam

Architects 89, 347 MAD Architects 63, 347 Manifold House 196–197, 349 nest 139–141 Orange Cube, The 116–117, 348 patch 172–175 Perkins & Will 325, 350 Prada Aoyama 218–219, 349 project 183 quad 191–195 region 32, 45–47, 203–204, 313 RhinoScript 11, 35

352

INDEX

scaffold: façade 24–27; mass 20–23; model 18–28, 32 Shigeru Ban Architects 167, 349 stack 85–87 Sugar Hill Development 300–301,

350 tessellate 238–241; see also wallpaper group

topography 175 topology 28, 32, 34 trim 164–165 union 32, 97–98, 101, 217 Voronoi 152, 296 wallpaper group 238, 242–250 ZAS Architects 253, 349