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Morphing: a guide to mathematical transformations for architects and designers PDF

233 Pages·2015·11.435 MB·English
by  ChomaJoseph
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Morphing A Guide to Mathematical Transformations for Architects and Designers Joseph Choma To Ting-Ting Morphing A Guide to Mathematical Transformations for Architects and Designers Joseph Choma CCoonntteennttss 006 Introduction 014 Transformations 074 Combining 122 Combining Transformations Shapes 126 A Mound 018 Shaping 078 Cutting and Spiralling 128 A Meandering Mound 022 Translating 082 Scaling and Spiralling 130 A Leaning Mound 026 Cutting 086 Modulating and Spiralling 132 A Steeper Mound 030 Rotating 090 Spiralling and Ascending 134 A Creased Mound 034 Reflecting 094 Texturing and Spiralling 136 A Creased and 038 Scaling 098 Bending and Spiralling Pinched Mound 042 Modulating 102 Spiralling and Bending 138 A Wedge 046 Ascending 106 Pinching and Spiralling 140 A Ridge and Trench 050 Spiralling 110 Flattening and Spiralling 142 Two Ridges 054 Texturing 114 Spiralling and Flattening 144 Another Ridge and Trench 058 Bending 118 Spiralling and Thickening 146 A Valley 062 Pinching 148 Moguls 066 Flattening 070 Thickening 004 150 Analyzing 196 Developable 220 Assumptions Surfaces 154 Japan Pavilion 200 Plane to Cylinder 228 Bibliography – Shigeru Ban Architects 202 Cylinder with Ascending 160 UK Pavilion 231 Acknowledgments – Heatherwick Studio 204 Cylinder with Texturing and Ascending 166 Mur Island – Acconci Studio 206 Cylinder with Flattening, Texturing and Ascending 172 Son-O-House – NOX/Lars Spuybroek 208 Cylinder with Modulating, Flattening and Texturing 178 Ark Nova – Arata Isozaki and 210 Plane to Cone Anish Kapoor 212 Cone with Spiralling 184 Looptecture F – Endo Shuhei 214 Cone with Spiralling Architect Institute and Texturing 190 Mercedes-Benz Museum 216 Cone with Spiralling, – UNStudio Texturing and Modulating 218 Cone with Spiralling, Texturing and More Modulating 005 Introduction ‘The way in which a problem is decomposed imposes fundamental constraints on the way in which people attempt to solve that problem.’ (Rodney Brooks, 1999) 006 A shape can be defined as anything with If a shape maintains its topological a geometric boundary. Yet, when continuity, it can be defined by a single describing a shape with mathematics, equation. A break in continuity, such as precision is crucial. a sharp edge, requires another equation to define the other ‘part’. Dictionaries define words, but these words do not necessarily define our Breaking continuity and having sharp understanding of the world in which we edges can create aesthetic effects perceive and create. The word ‘cube’ is that allow curves to be expressed in defined as a shape whose boundary is a more objectified manner. Breaking composed of six congruent square faces. the continuity of a shape can also Imagine cutting six square pieces of sometimes facilitate the fabrication paper and gluing the edges together. of particular geometries out of flat The cube, in this case, is created by six sheet material. However, the scope square planes. In mathematics, these of this guide has been constrained to planes are considered discrete elements. the definition of shapes with a single Because each plane in the paper cube parametric equation. meets the others at a sharp edge, technically they are not connected, but A parametric equation is one way of are separate parts, each defined by a defining values of coordinates (x, y, z) unique parametric equation. for shapes with parameters (u, v, w). All of the mathematical equations in Now, imagine a ball of clay. Roll it this guide are presented as parametric around on the tabletop to make it into equations. Think of x, y and z as a sphere. To flatten it into a cube, the dimensions in the Cartesian coordinate ball can be simply compressed in system – like a three-dimensional grid. multiple directions (rather than forming Think of u, v and w as a range of values six planar sides that are joined together, or parameters rather than a single as above). Gradually, the sphere could integer. A single integer would be like a transform into a six-sided shape. single point along a line, while the range Most physical cubes in the world have (u, v or w) would define the end points a certain degree of rounded corners. of that line and then draw all the points If we accept this definition for a cube, between them. then a cube could be defined with one parametric equation. The framework that is used becomes critical to deciding how to go about defining a shape mathematically. 007 Introduction Why trigonometry? Tools inherently constrain the way The designer is no longer designing individuals design; however, designers within a black box, but rather within are often unaware of their tools’ a transparent box. influences and biases. Digital tools in particular are becoming increasingly Because mathematical models can be complex and filled with hierarchical based on a global Cartesian coordinate symbolic heuristics, creating a black box system, the designer can constantly in which designers do not understand redefine a shape by redefining its what is ‘under the hood’ of the tools parametric equations, avoiding the they ‘drive’. Many contemporary digital chore of telling the computer how to tools use a fixed symbolic interface, like redraw it every time. For example, a a visual dictionary. When a designer smooth continuous surface can be wants to create a sphere, he or she confined to the boundary of a cube with clicks on the sphere icon and draws a a single transformation. Designers can radius. The resulting sphere, defined by a begin to think and manipulate in a less single symbol, can only be manipulated linear fashion and constantly redefine as a whole. Like a ball of clay, the sphere the ‘world’ that they create and perceive. can be stretched, twisted and pulled. The Cartesian coordinate system becomes a blank canvas at which any If, however, the shape had been defined type of paint can be thrown! by a parametric equation, it would be defined by a rule-based logic that This pedagogical guide embraces the encompasses both the whole sphere thought that all shapes could potentially and its parts. When the designer be described by the trigonometric manipulates the shape’s trigonometry functions of sine and cosine. But the (or ‘DNA’), it becomes clear that he utility of this guide does not depend on or she has a new range of geometric whether that idea is true or not: this freedom that could not have been guide does not invent a new field of imagined in the other framework. mathematics, but develops a cognitive Since the designer can manipulate the narrative within the existing discipline, smallest building blocks of the shapes, emphasizing the interconnected and understanding how each function plastic nature of shapes. Within this influences a particular transformation guide, sine and cosine are the only becomes straightforward. functions that are examined. 008 A word of warning: trigonometry may Functions (in mathematics) associate an seem like the tool of a designer’s input (independent variable) with an dreams, but just because every shape output (dependent variable). For could be potentially described by example if x→y, x would be considered trigonometric functions does not mean the independent variable while y would that it is necessarily easy to make every be considered the dependent variable. shape. It is important to remember that The function would be what makes x all tools have biases, even mathematics. map to y. In trigonometry, sine and For instance, in order to make a cube cosine are functions. Note that with a single parametric equation one trigonometric functions are a type of must first make a sphere; therefore, periodic functions, whose values repeat shapes that are initially round become in regular periods or intervals. easier to produce. A pedagogical guide is neither purely a A tool is a device that augments an technical reference nor a theoretical text, individual’s ability to perform a but rather the teaching of an inquiry particular task. through a series of instructional frameworks. In this context, heuristics refers to the strategies used to solve problems within software. Think of heuristics as the rules that govern how a machine ‘thinks’ and calculates solutions. 009

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