Morphing: a guide to mathematical transformations for architects and designers

232 Pages·2015·11.03 MB·English

by  Joseph Choma

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
Contents
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 Refecting 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
– Heatherwick Studio 231 Acknowledgments
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 defned as anything with If a shape maintains its topological
a geometric boundary. Yet, when continuity, it can be defned 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 defne the other ‘part’.
Dictionaries defne words, but these
words do not necessarily defne 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
defned as a shape whose boundary is a more objectifed 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 fat
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 defnition 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 defned by a defning 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 fatten 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 defne the end points
a certain degree of rounded corners. of that line and then draw all the points
If we accept this defnition for a cube, between them.
then a cube could be defned with one
parametric equation. The framework
that is used becomes critical to
deciding how to go about defning
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.
infuences and biases. Digital tools in
particular are becoming increasingly Because mathematical models can be
complex and flled 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 redefne a shape by redefning 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 fxed 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 confned to the boundary of a cube with
clicks on the sphere icon and draws a a single transformation. Designers can
radius. The resulting sphere, defned by a begin to think and manipulate in a less
single symbol, can only be manipulated linear fashion and constantly redefne
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 defned type of paint can be thrown!
by a parametric equation, it would be
defned 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 feld 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
infuences 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 frst 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
Introduction
A point of view

{ (u,v) | 0 ≤ u ≥ 2π, 0 ≤ v ≥ 2π } As we circle around an object, our be diffcult to comprehend in their
perceptual understanding of it is entirety because of this singular view.
x = sin(u)10(cos(v))
transformed. We move in dialogue
y = sin(v)+cos(u)10(cos(v))
z = cos(v)+10(sin(v))sin(u) with the observed object, as in a dance In the series above, there are four
performance. Similarly, the pages within distinct shapes. On paper, they look
this book document shapes according different; however, they are defned by
to particular, chosen perspectives. Most the same mathematical formula. As an
of the shapes within this guide are three individual moves around the object
dimensional, but their representation is anti-clockwise, the shape begins to
constrained to the two-dimensional reveal an illusion. Initially, the shape
plane of the page. Some shapes may appears to be rectangles overlapping
010