A Pedagogical Guide into Trigonometric Transformations

A Pedagogical Guide into Trigonometric Transformations

by

Joseph Choma

Bachelor of ArchitectureRensselaer Polytechnic Institute, 2009

SUBMITTED TO THE DEPARTMENT OF ARCHITECTURE IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE IN ARCHITECTURAL STUDIESAT THE

MASSACHSETTS INSTITUTE OF TECHNOLOGY

June 2011

© 2011 Joseph Choma. All rights reservedThe author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies

of this thesis document in whole or in part in any medium now known or hereafter created.

Signature of Author: ….……………………………………………………………………………Department of Architecture

20 May 2011

Certi�ed by: ….……………………………………………………………………………………George Stiny

Professor of Design and Computation Thesis Advisor

Accepted by:….……………………………………………………………………………………Takehiko Nagakura

Associate Professor of Design and ComputationChairman, Department Committee on Graduate Students

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George StinyProfessor of Design and Computation

Thesis Advisor

Nader TehraniProfessor and Head of Department of Architecture

Thesis Reader

Edith K. AckermannHonorary Professor of Developmental Psychology at the University of Aix-Marseille

Thesis Reader

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ABSTRACT

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A tool is a device that augments an individual’s ability to perform a particular task. The more speci�city a tool has, the narrower its instrumentality. Tools inherently constrain the way individuals design; how-ever, designers are often unaware of their in�uence and bias. Digital tools are becom-ing increasingly complex and �lled with hier-archical symbolic heuristics, creating a black box in which designers do not understand what is “under the hood” of the tools they drive. And yet designers are becoming fasci-nated with engineering mentalities: optimi-zation and automation. Simply, it gives a solution. But, this is not design! Designers need to work outside of a �xed atmosphere!

The future of digital instruments is not more complex heuristics, but rather the contrary. It is imperative to go back to the most basic building blocks of these “engines:” math-ematics. Within mathematics, functions can be embedded inside other functions at

anytime, giving designers endless freedom to alter the computational hierarchy. By “playing” with parametric equations and tacit engagement with the algorithm, one can begin to learn explicitly how discrete opera-tions transform shapes in a particular way.

This guide embraces the thought that all shapes could potentially be described by the trigonometric functions of sine and cosine. These functions became the only �xed constraint to instrumentalize. Through the recursive “play” and learn process, a new morphological classi�cation of topological transformations emerged, leading to the development of this guide, the �rst peda-gogical guide into trigonometric transforma-tions. This guide does not invent new realms within the �eld of mathematics, but devel-ops a new cognitive narrative within it, emphasizing the interconnected and plastic nature of shapes.

ABSTRACT

A Pedagogical Guide into Trigonometric Transformations

by

Joseph Choma

SUBMITTED TO THE DEPARTMENT OF ARCHITECTURE ON MAY 20, 2011, IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE IN ARCHITECTURAL STUDIESAT THE

MASSACHSETTS INSTITUTE OF TECHNOLOGY

Thesis Advisor: George StinyTitle: Professor of Design and Computation

Abstract

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During the last two years at the Massachu-setts Institute of Technology I have encoun-tered numerous unusual conversations with creative talents, which have left a lasting impression on me. I would like to especially thank a few of those individuals.

Professor George Stiny is an individual who I feel honored to have studied under while at MIT. When I �rst heard George speak I found myself curiously fascinated with his ideas about embedding. From that moment on, each time he spoke it seems like I have itera-tively embedded more emerging ideas. The fundamental philosophical ideas of his shape grammars has in�uenced and inspired the motivator of this thesis’ inquiry. His work places the bar in�nitely high which I plan to continue to reach for.

Professor Dennis Shelden was the �rst indi-vidual who interacted with me on my research in the area of trigonometric trans-formations. As the work began to reach high levels of complexity, Dennis encouraged me to step back and understand the basic prin-ciples of the mathematics �rst. This led me to focus on a pedagogical guide which revealed the inner workings of the most fundamental trigonometric transformations.

Professor Terry Knight’s class taught me how to develop an inquiry and identify contribu-tions within design research. The clarity of this thesis would not have been possible without Terry’s structured exercises.

Professor Nader Tehrani was the �rst profes-sor I had the privilege to act as a teaching assistant for at MIT. Since that �rst experi-ence, teaching has become an increasingly larger portion of my life. I would also like to thank Nader for his straight forward critiques that often got to the underlying stakes, always reminding me that I am an architect!

ACKNOWLEDGEMENTS

While attending MIT, I often referred back to my experiences working for Vito Acconci. The think tank design atmosphere has trans-formed who I am and who I want to become.

If only I could count the number of times I used the words instrumentation and instru-mentality! I must thank my previous profes-sor at RPI, Ted Krueger. While assisting Ted on his sensory research, he gave me my �rst exposure to the world of design research. A �eld I �nd myself rede�ning each day.

I would also like to thank my friends in the Design and Computation Group who I believe are some of the most uniquely inter-esting individuals I have ever met. The atmo-sphere and energy they added to my educa-tion will never be forgotten.

Lastly, I would like to thank my family, espe-cially my Father.

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TABLE OF CONTENTS

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TABLE OF CONTENTS

Advisor and Readers……………………………………………………………………………3Abstract……………………………………………………………………………………………5Acknowledgements………………………………………………………………………………7

Morphing: Pedagogical Guide…………………………………………………………………11Introduction……………………………………………………………………………………13

Shaping.…………………………………………………………………………………………27Cutting…………………………………………………………………………………………31Scaling…………………………………………………………………………………………35Modulating………………………………………………………………………………………39Ascending………………………………………………………………………………………43Spiraling…………………………………………………………………………………………47Texturing…………………………………………………………………………………………51Bending…………………………………………………………………………………………55Pinching…………………………………………………………………………………………59Flattening………………………………………………………………………………………63Thickening………………………………………………………………………………………67Containing………………………………………………………………………………………71

Teaching…………………………………………………………………………………………75Syllabus…………………………………………………………………………………………79Philosophy………………………………………………………………………………………83

Conclusion………………………………………………………………………………………85Physical Models…………………………………………………………………………………87Bibliography……………………………………………………………………………………89

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MORPHINGa pedagogical guide into

trigonometric transformations Jose

ph C

hom

a

MORPHING

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INTRODUCTION

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“The way in which a problem is decomposed imposes fundamental constraints on the way in which people attempt to solve that prob-lem” (Brooks 1999).

In the 1800’s Joseph Fourier attempted to prove that every shape could be described by trigonometric functions. Fourier’s proof was close to perfection, except it ran into di�culty when trying to solve for square waves. For those less familiar with square waves, it is similar to that of a sine curve only it is composed of zero curvature such that it moves horizontally, then perpendicularly vertically upward, then horizontally to the right, and then vertically downward until it moves horizontally again. Within trigonom-etry, a sine curve can easily become a square curve with slightly rounded corners. This can be done by placing the initial function inside recursions of sine. The more sine functions embedded inside, the more �attened the geometry. However, it is impossible to com-pletely �atten a curve to go around a corner without the slightest rounding because the curve needs to maintain its property of conti-nuity. If the curve had an edge or kink, it would break its continuity and become mul-tiple discrete lines. Therefore, a square wave could be considered a series of discrete lines rather than a single curve. If you imagine an architect using a drafting ruler to draw this, the architect would not draw it with one continuous line. The architect would �rst draw horizontal lines, and then would rotate the ruler to draw the vertical components. If a square wave is simply a series of straight horizontal and vertical lines in space that happen to share a common point, each of those straight lines and their locations can easily be de�ned using trigonometry. Any-thing can be described using trigonometry, it may just simply take more than one equa-tion to de�ne it. Any break in continuity will require an additional equation. When

designing, it may be necessary to use more than one equation to de�ne the entirety of your design.

“Act always so as to increase the number of choices” (Foerster 2002).

This research does not attempt to make an ultimate proof, but rather attempts to �nd the rules and logics behind trigonometry’s transformations that would allow anyone to manipulate the algorithm in an instrumental manner rather a purely deterministic one.

“In dealing with the world rationally, we hold it constant, by means of categories formed in the past. Through intuition, on the other hand, we grasp the world as a whole, in �ux” (Langer 1989).

Spheres, cylinders, and cubes are a small handful of plutonic shapes which can be described by a single word (Fig. 1). However, most shapes cannot be found in the diction-ary. When a designer designs, would a designer prefer to design with the relatively few shapes found in the dictionary or all the possible shapes in the world? This pedagogi-cal guide challenges the linguistically driven �xed world which we live by exposing an alternative plastic world de�ned by trigo-nometry. A mathematical world where all shapes can be described under one system-atic language. Any shape can transform into another, even a cylinder can e�ortlessly transform into a sphere! It is not about static instances; it is about morphing series (Fig. 2). Why make a new shape from scratch when you can morph one shape into another? Designing is action, not a noun. Designers should not use tools to simply record preconceived de�nitions, but should instru-mentalize tools as mechanisms to generate ideas!

INTRODUCTION

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INTRODUCTION

14

Figure 1. A cylinder and a sphere are considered discrete shapes in linguistics.

Figure 2. A cylinder and a sphere are considered plastic shapes that can transform into one another in mathematics.

“For the artist communication with nature remains the most essential condition. The artist is human; himself nature; part of nature within natural space” (Klee 1944).

Everything is part of another part. De�ning “worlds” is one of the primary problems with computational models today. As soon as a designer "de�nes a world" all the computer can do is see the “world” they have symboli-cally de�ned, even though the designer sees more than the computational model of that “world.” If you take an additional step back, we are part of the “world” which we perceive and create. Therefore "world" can never be de�ned permanently because "it" is constantly transforming, as is us and our perception of "it." This is especially true once a designer sees, and embeds beyond what they see.

Many contemporary digital tools use a �xed symbolic interface, similar to that of the dictionary. When a designer wants to create

a sphere, the designer clicks on the icon and draws a radius. If a shape is de�ned by a single symbol, all the designer can do is manipulating the matter of that shape. Like a ball of clay, the sphere can be stretched, twisted, pulled and cut (Fig. 3). If the sphere was de�ned by a parametric equation, it becomes de�ned by a ruled based logic that contains parts smaller than it. When the designer manipulates the shape’s “DNA” or trigonometry, it becomes clear that there is a new range of geometric freedom that could not have been imagined in the other “world” (Fig. 4). Since the designer literally manipu-lates the smallest morphemes themselves, understanding how each function in�uences a particular transformation becomes obvi-ous. The designer is no longer designing within a black box, but rather within a trans-parent box.

Figure 3. A sphere deformed in a typical parametric environment, where it de�nes itself.

Figure 4. A sphere transformed in mathematical environment, where it is de�ned by trigonometry.

“We have dealt so far, and for the most part we shall continue to deal, with our coordi-nate method as a means of comparing one known structure with another. But it is obvi-ous, as I have said, that it may also be employed for drawing hypothetical struc-tures, on the assumption that they have

INTRODUCTION

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INTRODUCTION

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varied from a known form in some de�nite way” (Thompson 1942).

Although the two previous morphologies may seem like a series of arbitrary shapes, in actuality, the sphere transformed by trigo-nometry actually analyzes an existing build-ing geometry. The fourth shape in the �ve shape morphing series is the global form of Acconci Studio’s “Mur Island” in Graz, Austria (Fig. 4). Not only is the mathematics used to analyze the existing building form, but it is used to go beyond the initial vision of the architect, by manipulating the building form’s “DNA” an additional step. Mathemat-ics can be used as an analytical tool, but it can also be used as a generative tool. This pedagogical guide primarily focuses on its application as generative one.

“Shape is one of the essential characteristics of objects grasped by the eyes. It refers to the spatial aspects of things, excepting loca-tion and orientation. That is, shape does not tell us where an object is and whether it lies upside down or right side up. It concerns, �rst of all, the boundaries of masses” (Arnheim 1954).

In addition to the linguistically de�ned inter-faces in contemporary digital tools, most software also uses a linear hierarchical struc-ture, building each rule or logic o� of one another. Such linear structures can become extremely speci�c and �xed to the point where they are only able to compute one task. For example, if you imagine a paramet-ric model that stretches a sphere according to a line, all the algorithm can do is stretch it and distort that closed form. If you wanted to taper the shape rather than stretch it, the structure of the algorithm would have to change. If you wanted an even more extreme change to the transformation, like transform the closed surface of the sphere

INTRODUCTION

into an open surface, you would have to rede�ne how the initial input shape is described. This is quite di�erent from a purely mathematical model. In the math-ematical model, all modes of transforma-tions and �exibility are under one systematic language. The designer never has to rede-�ne the initial structure of the system; the designer has to simply alter within that system. Since mathematical models are based on a global Cartesian coordinate system, the designer can at anytime alter the computational hierarchy and embedded as they like. For example within this guide a transformation called “containing” alters the boundary condition of any shape by placing it within the boundary of another. Suddenly a smooth continuous surface can be con�ned to the boundary of a cube, with one transformation (Fig. 5). Its complete hierar-chy has been altered. Designers can begin to think and manipulate in a less linear fashion and constantly rede�ne the “world” within they create and perceive. Like throwing paint onto a blank canvas, the Cartesian coordinate system becomes that canvas!

Figure 5. A smooth shape placed inside the boundary of a cube.

Figure 6. Altering the thickness of a shape placed inside the boundary of a cube.

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INTRODUCTION

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Trigonometry may seem like the tool of a designer’s dreams, but just because poten-tially every shape could be described by trigonometric functions doesn’t mean that it is necessarily easy to make every shape. It is important to remember that all tools have biases, even mathematics. For instance, in order to make a cube one must �rst make a sphere; therefore initially round shapes become easier to produce.

“The �rst spatial intuitions of the child are, in fact, topological rather than projective or consistent with Euclidean metric geometry. Up to the age four, for example, squares, rect-angles, circles, ellipses, etc., are all repre-sented by a closed curve without straight lines or angles. Topologically, squares and circles are the same �gure. Crosses, arcs, etc., are all represented by an open curve. At this same age, however, children can produce quite accurate copies of a closed �gure with a little circle inside. The topological relation of the inside circle to the enclosing one, or even the relation between a closed �gure and a circle on its boundary is represented by children who are quite incapable of copying a square correctly” (Piaget 1969).

A pedagogical guide is not purely an instruc-tional guide nor is it purely a philosophical text, but rather through a series of instruc-tions a philosophical idea is taught. This guide structures itself around ideas of topol-ogy while revealing the trigonometry behind each transformation. The guide begins the creative process in a section called shaping. In this �rst section, several basic shapes transform into one another, introducing the four basic demonstration shapes for the rest of the guide: a sine curve, a circle, a cylinder and a sphere. These demonstration shapes become the constant which portray how particular shapes transform under eleven fundamental types of transformations. The

eleven transformations covered in this guide include: cutting, scaling, modulating, ascending, spiraling, texturing, bending, pinching, �attening, thickening, and containing. These eleven transformations become the most fundamental design operations when instrumentalizing trigo-nometry.

“The forces coming from within transform the point into a line, can be very diverse. The variation in lines depends upon the number of these forces and upon their combinations” (Kandinsky 1947).

By combining these eleven transformations, one can create possibly any shape imagin-able. There are an in�nite number of hierar-chical ways these transformations could combine to develop more complex forms of transformations. Hierarchy is extremely important in the combining process. Since functions can be embedded inside other functions at anytime, it is imperative that the sequence of operations becomes carefully narrated. For example, if you imagine �attening a cylinder and then bending it, the resultant would be a bent square tube. If the process was reversed, where the cylinder was �rst bent and then �attened, then the entire bent cylinder would be �attened according to the outer boundary of a cube. The entire bent shape would �atten globally rather than �atten locally and then bend globally. One can also achieve di�erent results by transforming portions of paramet-ric equations and not others. For instance, perhaps a designer wants a bent surface to meet a �attened surface, rather than have a bent �attened surface. Anything is possible!

“The outline of the common pattern is set by the fact that every experience is the result of interaction between live creature and some aspect of the world in which he lives. A man

INTRODUCTION

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INTRODUCTION

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does something; he lifts, let us say, a stone. In consequence he undergoes, su�ers, some-thing; the weight, strain, texture of the surface of the thing lifted. The properties thus undergone determine further doing” (Dewey 1934).

By manipulating the smallest pieces that could de�ne any geometry, sine and cosine, it is potentially possible to create any shape! However, there may be times when sine and cosine seems like too small of pieces to manipulate. Within mathematics, it is also possible to de�ne larger pieces to calculate with. For example, if you look at the para-metric equation of a sphere, its x, y and z coordinates are each de�ned by sine and cosine and u and v. If the sphere’s x, y, and z coordinates become de�ned as input variables, like S(x), S(y), and S(z), then they can be used as pieces to calculate with (Fig. 7). The dimensions de�ning the shape could become de�ned by parts of dimensions of other shapes. This tends to allow for a more global type of manipulation, more similar to deforming a ball of clay. Again, anything is possible!

Figure 7. A shape transforming based on calculations with dimensions de�ned by a cylinder and sphere.

“When someone re�ects-in-action, he becomes a researcher in the practice context. He is not dependent on the catego-ries of established theory and technique, but constructs a new theory of the unique case. His inquiry is not limited to a deliberation about means which depends on a prior

agreement about ends. He does not keep means and ends separate, but de�nes them interactively as he frames a problematic situation. He does not separate thinking from doing, ratiocinating his way to a deci-sion which he must later convert to action. Because his experimenting is a kind of action, implementation is built into his inquiry. Thus re�ection-in-action can proceed, even in situations of uncertainty or uniqueness, because it is not bound by the dichotomies of Technical Rationality” (Schön 1983).

Although this guide demonstrates discrete strategies to control trigonometric transfor-mations, arbitrary play is also encouraged! After all, designing is supposed to be fun! By playing with the algorithm it is also possible that the initial lack of control could help gen-erate unpredictable shapes that may guide the design process in a new direction. Try going back and forth between tacit experi-mentation and explicit learning. It is not simply about memorizing speci�c rules, but rather also about developing sensitivity and intuitive understanding of the medium. Like any medium, it can only be truly learned by doing. As you are doing, don’t forget to look back at what had been done and what is being done. Thinking is also part doing!

“In conversation, participants �nd them-selves discussing topics that they’d never thought of, when they began, and they may �nd radical, new ideas in and through conversation” (Glanville 2008).

In the end a machine will always be able to record and calculate better than man can ever, but a machine will never be able to wander as well as we do. It is imperative to think of these mechanisms not purely as deterministic. Even within mathematical models which get to the most fundamental

INTRODUCTION

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building blocks of the computational engines there is the simple logic of an input and an output. This logic alone is not design! It will calculate for us, but we need to inter-vene, and constantly transform the algo-rithm or “world” at hand, according to criteria which we identify during the process. We are the feedback loop, and without us, it’s just automation.

“Evidently, there is scarcely anything that one can say about a ‘single sensation’ by itself, but we can often say much more when we can make comparisons” (Minsky 1985).

One of the unique teaching techniques within this pedagogical guide is the morph-ing documentation process. In order for a designer to understand a transformation, they have to see what its starting shape and resultant. By documenting that transforma-tion iteratively over a couple more instances, it becomes clear how that mechanism can be controlled. For example, in order to demon-strate the steepness one can control with the pinching transformation, it is important to show multiple transformations gradually steepening the pinch. Morphology in this context is like an evolutionary series that reveals a particular type of pattern of trans-formation. This is di�erent than simply four discrete shapes documented with their alge-bras. Morphologies are iterative. For example, imagine starting with a sphere, which then gets pinched, then �attened, and then bent. In a morphing documentation process, the starting sphere would be docu-mented �rst, then the sphere pinched, then the sphere pinched and �attened, and then the sphere pinched �attened and bent. It would not simply be a sphere, a sphere pinched, a sphere �attened, and a sphere bent. The morphology needs to transition from its neighbors and blend, with each step building o� of the previous. Morphing is not

simply about thinking iteratively; it also is about documenting the iterative process. Documenting is part of doing, thinking and analyzing.

“The destructive analysis of a comprehensive entity can be counteracted in many cases by explicitly stating the relation between its particulars. Where such explicit integration is feasible, it goes far beyond the range of tacit integration. Take the case of a machine. One can learn to use it skillfully, without knowing exactly how it works. But the engineer’s understanding of its construction and operation goes much deeper” (Polanyi 1966).

This pedagogical guide into trigonometric transformations is structured around mor-phologies. Each page explicitly documents a transformation in process, openly revealing the algebras causing each step to happen. In order to evaluate your own design iterations it may be helpful to adopt this documenta-tion process as a means to re�ect on what had been done. Design is not just play, but rather rigorous play. In order for it to be rigorous you need to be able to understand the causes and e�ects, and evaluate the results according to criteria you have person-ally identi�ed. For example, if you identify pinches within the design as signi�cant attri-butes of the design, because of the way in which it yields directionality, how do you judge now many or how large of pinches to use in your design? It becomes critical to document the extremes. Looking at pinches that are few and enormous gestures, and small masses of pinches that read as a texture and atmosphere. Saying you like pinches is no longer enough in the process, further criteria needs to evolve throughout the process. Thinking and re�ecting is as important as playing and seeing.

INTRODUCTION

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“One eye sees, the other feels” (Klee 1964).

There is more to seeing than just looking. When we see, our eyes moves back and forth, up and down, constantly, scanning what is in front of us. We may tilt our head or move our body as we engage. We are often unaware of our bodily actions, as we are thinking simul-taneously; we superimpose our mental vision on top of what is physically there. Like a mental tracing paper over reality. There is always more there than can be seen and what is there is always changing. Our cogni-tive embedding becomes an augmented reality.

“First, the embedding relation—what you see is there if you can trace it out, no matter what has gone on before. Second, the transformations—what you see is like given examples of what to look for, maybe things that were noticed in the past and used. And together—embedding and transformations interact as rules are tried to calculate with shapes” (Stiny 2006).

Never let anything be completely �xed, even something like a curve! A curve may be a curve but a curve could also become a tube or space. Anything you imagine could be embedded inside another shape. It is up to the designer to see beyond the initial gener-ated mathematical representations and push it! Although the algebras can get quite com-plex, it is possible to embed spaces on top of other spaces using mathematics (Fig. 8). In the end remember that trigonometry has its biases and constraints. Some operations may be more easily completed in other mediums and with other tools. Designers should not limit themselves to design solely within only one medium. It is imperative that designers understand that every tool has a bias, and that with every constraint lays a speci�c type of design opportunity. Play,

INTRODUCTION

look and enjoy this guide into trigonometric transformations and feel free to use the instrument of mathematics as you see best �t.

Figure 8. A space embedded on top of another space, where each is transformed independent of one another.

The feelings of something being pulled apart, or bending a shape to twist, are trans-formations that also relate to bodily actions. Even without looking at the parametric equation which de�nes the trigonometric transformations within this pedagogical guide, this guide is still relevant to designers. This guide emphasizes that design is about transformations, designers acting on matter, seeing the resultant, and generating ideas. Even if you don’t look at the math, look at the transformations! I challenge you to trans-form whatever is in front of you, and see how your perception transforms. That is design!

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SHAPINGintroducing the basic shapes

SHAPING

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x = cos(u)y = sin(u)z = 0

for allu є [0, 2π]

x = cos(u)y = sin(u)z = v

for allu є [0, 2π]v є [0, 1/3π]


for allu є [0, 2π]v є [0, 2/3π]


for allu є [0, 2π]v є [0, π]

x = uy = 0z = 0


x = uy = cos(u)z = 0


x = uy = sin(u)z = 0




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x = (sin(v))(cos(u))y = sin(u)z = v


x = (sin(v))(cos(u))y = (sin(v))(sin(u))z = v


x = (sin(v))(cos(u))y = (sin(v))(sin(u))z = cos(v)




x = cos(u)y = sin(u)z = u


x = v(cos(u))y = v(sin(u))z = u


x = v(cos(u))y = v(sin(u))z = v


SHAPING

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CUTTINGchanging the period of the shapes

CUTTING

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for allu є [0, 3/2π]


for allu є [0, π]








for allu є [0, π]



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for allu є [0, 3/2π]v є [0, π]


for allu є [0, π]v є [0, π]











CUTTING

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for allu є [0, π]



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SCALING

SCALINGchanging the amplitude of the shapes

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x = cos(u)y = sin(u)/2z = 0


x = cos(u)/2y = sin(u)z = 0


x = cos(u)/2y = sin(u)/2z = 0




x = uy = sin(u)/2z = 0


x = u/2y = sin(u)z = 0


x = u/2y = sin(u)/2z = 0


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SCALING



x = (sin(v))(cos(u))y = (sin(v))(sin(u))/2z = cos(v)


x = (sin(v))(cos(u))/2y = (sin(v))(sin(u))/2z = cos(v)


x = (sin(v))(cos(u))/2y = (sin(v))(sin(u))/2z = cos(v)/2




x = cos(u)y = sin(u)/2z = v


x = cos(u)/2y = sin(u)z = v


x = cos(u)/2y = sin(u)/2z = v


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MODULATINGchanging the frequency of the shapes

MODULATING

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x = cos(u)y = sin(2(u))z = 0


x = cos(u)y = sin(3(u))z = 0


x = cos(u)y = sin(4(u))z = 0




x = uy = sin(2(u))z = 0


x = uy = sin(3(u))z = 0


x = uy = sin(4(u))z = 0


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x = (sin(v))(cos(u))y = (sin(v))(sin(2(u)))z = cos(v)








x = cos(u)y = sin(2(u))z = v






MODULATING

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ASCENDING

ASCENDINGincreasing the trajectory of the shapes

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x = cos(u)y = u+sin(u)z = 0


x = u+cos(u)y = sin(u)z = 0


x = u+cos(u)y = u+sin(u)z = 0




x = uy = u+sin(u)z = 0


x = u+uy = sin(u)z = 0


x = u+uy = u+sin(u)z = 0


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x = (sin(v))(cos(u))y = u+(sin(v))(sin(u))z = cos(v)


x = u+(sin(v))(cos(u))y = (sin(v))(sin(u))z = cos(v)


x = u+(sin(v))(cos(u))y = u+(sin(v))(sin(u))z = cos(v)




x = cos(u)y = u+sin(u)z = v


x = u+cos(u)y = sin(u)z = v


x = u+cos(u)y = u+sin(u)z = v


ASCENDING

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SPIRALING

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SPIRALING increasing the radius of the shapes

SPIRALING

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x = cos(u)y = u(sin(u))z = 0


x = u(cos(u))y = sin(u)z = 0


x = u(cos(u))y = u(sin(u))z = 0




x = uy = u(sin(u))z = 0


x = uy = cos(u)z = 0


x = uy = u(cos(u))z = 0


SPIRALING

48



x = (sin(v))(cos(u))y = u((sin(v))(sin(u)))z = cos(v)


x = u((sin(v))(cos(u)))y = (sin(v))(sin(u))z = cos(v)


x = u((sin(v))(cos(u)))y = u((sin(v))(sin(u)))z = cos(v)




x = cos(u)y = u(sin(u))z = v


x = u(cos(u))y = sin(u)z = v


x = u(cos(u))y = u(sin(u))z = v


SPIRALING

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TEXTURING

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TEXTURINGcombining shapes of di�erent frequencies

TEXTURING

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x = cos(2(u))/2+cos(u)y = sin(2(u))/2+sin(u)z = 0








x = uy = sin(2(u))/2+sin(u)z = 0


x = uy = sin(3(u))/3+sin(u)z = 0


x = uy = sin(4(u))/4+sin(u)z = 0


TEXTURING

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x = cos(2(u))/2+(sin(v))(cos(u))y = sin(2(u))/2+(sin(v))(sin(u))z = cos(v)








x = cos(2(u))/2+cos(u)y = sin(2(u))/2+sin(u)z = v






TEXTURING

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BENDING

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BENDINGde�ecting the shapes globally

BENDING

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x = cos(u)y = sin(v)+sin(u)z = 0


x = cos(v)+cos(u)y = sin(u)z = 0


x = cos(v)+cos(u)y = sin(v)+sin(u)z = 0




x = uy = sin(v)+sin(u)z = 0


x = cos(v)+uy = sin(u)z = 0


x = cos(v)+uy = sin(v)+sin(u)z = 0


BENDING

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x = (sin(v))(cos(u))y = sin(v)+(sin(v))(sin(u))z = cos(v)


x = cos(v)+(sin(v))(cos(u))y = (sin(v))(sin(u))z = cos(v)


x = cos(v)+(sin(v))(cos(u))y = sin(v)+(sin(v))(sin(u))z = cos(v)




x = cos(u)y = sin(v)+sin(u)z = v


x = cos(v)+cos(u)y = sin(u)z = v


x = cos(v)+cos(u)y = sin(v)+sin(u)z = v


BENDING

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PINCHINGsteepening the apex of the shapes

PINCHING

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x = cos(u)3

y = sin(u)3

z = 0


x = cos(u)5

y = sin(u)5

z = 0


x = cos(u)7

y = sin(u)7

z = 0




x = uy = sin(u)3

z = 0


x = uy = sin(u)5

z = 0


x = uy = sin(u)7

z = 0


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x = (sin(v))(cos(u)3)y = (sin(v))(sin(u)3)z = cos(v)








x = cos(u)3

y = sin(u)3

z = v


x = cos(u)5

y = sin(u)5

z = v


x = cos(u)7

y = sin(u)7

z = v


PINCHING

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x = cos(u)3

y = sin(u)3

z = 0


x = cos(u)5

y = sin(u)5

z = 0


x = cos(u)7

y = sin(u)7

z = 0


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FLATTENING

FLATTENINGdecreasing the apex of the shapes

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x = sin(cos(u))y = sin(sin(u))z = 0


x = sin(sin(cos(u)))y = sin(sin(sin(u)))z = 0


x = sin(sin(sin(cos(u))))y = sin(sin(sin(sin(u))))z = 0




x = uy = sin(sin(u))z = 0


x = uy = sin(sin(sin(u)))z = 0


x = uy = sin(sin(sin(sin(u))))z = 0


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FLATTENING



x = sin((sin(v))(cos(u)))y = sin((sin(v))(sin(u)))z = sin(cos(v))


x = sin(sin((sin(v))(cos(u))))y = sin(sin((sin(v))(sin(u))))z = sin(sin(cos(v)))


x = sin(sin(sin((sin(v))(cos(u)))))y = sin(sin(sin((sin(v))(sin(u)))))z = sin(sin(sin(cos(v))))




x = sin(cos(u))y = sin(sin(u))z = v


x = sin(sin(cos(u)))y = sin(sin(sin(u)))z = v


x = sin(sin(sin(cos(u))))y = sin(sin(sin(sin(u))))z = v


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THICKENINGintroducing another dimension to the shapes

THICKENING

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xo = 0yo = 0zo = 0

xw = xo+cos(u)yw = yo+sin(u)zw = zo+0

for allu є [0, 2π]w є [0, 1/3π]

xo = 0yo = 0zo = 0


for allu є [0, 2π]w є [0, 2/3π]

xo = 0yo = 0zo = 0


for allu є [0, 2π]w є [0, π]



xo = 0yo = 0zo = 0

xw = xo+uyw = yo+sin(u)zw = zo+0

for allu є [0, 2π]w є [0, 1/3π]

xo = 0yo = 0zo = 0


for allu є [0, 2π]w є [0, 2/3π]

xo = 0yo = 0zo = 0



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xo = 0yo = 0zo = 0

xw = xo+(sin(v))(cos(u))yw = yo+(sin(v))(sin(u))zw = zo+cos(v)

for allu є [0, 2π]v є [0, π]w є [0, 1/3π]

xo = 0yo = 0zo = 0



xo = 0yo = 0zo = 0


for allu є [0, 2π]v є [0, π]w є [0, π]



xo = 0yo = 0zo = 0

xw = xo+cos(u)yw = yo+sin(u)zw = zo+v


xo = 0yo = 0zo = 0



xo = 0yo = 0zo = 0



THICKENING

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70

CONTAININGplacing a shape inside the boundary of another

CONTAINING

71



xo = uyo = sin(u)zo = 0

xw = sin(xo)yw = sin(yo)zw = zo

for allu є [0, 2π]w є [0, 1/3π]



for allu є [0, 2π]w є [0, 2/3π]







xw = xoyw = sin(yo)zw = zo

for allu є [0, 2π]w є [0, 1/3π]



for allu є [0, 2π]w є [0, 2/3π]




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xo = (sin(v))(cos(u))yo = (sin(v))(sin(u))zo = cos(v)

xw = sin(xo)yw = sin(yo)zw = sin(zo)










xo = cos(u)yo = sin(u)zo = v









CONTAINING

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Student Work: Mallory Demty’s morphing series completed with chaoscope.

Student Work: Zachary Stanesa’s morphing series completed with cage edit.

Student Work: Neil Piatt’s morphing series completed with trigonometric transformations.

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It is impossible to test the potential success of the pedagogical guide into trigonometric transformations without having architecture students attempting to instrumentalize the material. From August 2010 until May 2011, Joseph Choma taught two consecutive advanced workshops on instrumentalizing mathematics as an Adjunct Professor at the Boston Architectural College. At the surface, the course appears to be primarily rooted in trigonometric transformations, however its primary objective is on generating ideas by utilizing tools in a less predictable manner. The course combines technical skill-based tutorials with theoretical lectures and exem-plary readings.

It begins by teaching students to document transformations with morphologies, making students’ intuitive tacit process more explicit. The course emphasizes the internal biases of di�erent digital tools by having students experience utilizing very predictable digital mechanisms: squishing, stretching, deform-ing, in comparison to the much less predict-able: chaos theory. Later, the students manipulate algebra in parametric equations where they learn to understand the math-ematics based on their intuitive dialogue with the tool. Their initial lack of complete control with the medium drives their curios-ity to generate unpremeditated ideas. Even-tually the students learn how to explicitly control such transformations, allowing their shapes to adapt to programmatic and contextual constraints. Shapes become looked at in a less �xed symbolic manner as students understand how to transform a cylinder into a sphere and a sphere into a cube. All shapes in this context are plastic and interconnected. Students also become greater aware of the medium’s underlying bias. For instance, in order to make a cube using trigonometry, one must �rst make a sphere. Therefore, shapes that are more

round become initially easy to construct in that medium. Finally, students are taught to translate spatial intentions into physical prototypes and lastly into rigorous architec-tural consequences. However, the emphasis in the course is not on the �nal architecture outcome, but rather the process. The class is seen as a continuous pedagogical exercise that will transform the way each student thinks about instrumentalizing tools. The philosophical idea of using a tool as an idea generator is a fundamental concept that goes well beyond purely instrumentalizing mathematics. In the end, everything has the potential to be a design mechanism!

It is important to note that the primary objective of the workshop di�ered slightly from that of the pedagogical guide. In the pedagogical guide the fundamental trigono-metric transformations are taught to the designer in the beginning as means to get them to start thinking about design through transformations, combinatorics, and embed-ding. In the workshop, there is an emphasis on playful tacit experimentation, which through that experimentation leads to a new explicit understanding of the mathematical medium. Therefore, the “secrets” behind the most fundamental transformations are not revealed until later in the course. In the beginning of the course the students have to try to �nd what transformations are signi�-cant to them. By not giving away the “secrets” the students are forced to constantly look and engage the algebra. Forcing them to understand a foreign medium such as mathematics in a tacit process is an excellent way to teach students to instrumentalize tools in a generative manner.

TEACHING

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Student Work: Mark Roger’s transformation taxonomy and perspective rendering.

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COURSE DESCRIPTION

This workshop is intended for anyone with a curiosity to understand the link between shapes and the mathematics behind them. The workshop will demonstrate how algo-rithms can be manipulated in order to control formal distortions. Although this workshop will teach discrete strategies to control algebraic topology, arbitrary play is encouraged! After all, designing is intended to be fun! Donald Schön describes a tacit knowledge interaction called re�ection in action, where a designer is in an intuitive responsive dialogue with the tool or medium (Schön 1983). By playing and experimenting with algorithms, it is possible that the lack of control will help generate unpredictable forms that may help guide a design process in a new direction. By going back and forth between explicit learning and tacit experi-mentation, designers can begin to develop a convincing design that can conform to �xed practical constraints. With that spirit in mind, play and look, learn and control.

This workshop will be split into three general phases: introductory, cognitive and architec-tural. During the introductory phase, students will learn to become more aware of the technical and philosophical biases that are embedded within tools. Students will be exposed to tools that yield di�erent levels of predictability. In the cognitive phase, students will learn how to control and gener-ate shapes using parametric equations. Stu-dents will be taught to document, design and think through morphologies, emphasiz-ing the plastic and interconnected nature of shapes. At the conclusion of this phase, students will be asked to de�ne a unique manipulation technique for instrumentaliz-ing mathematics. Lastly, in the architectural phase, students will apply their discovered technique to design a dwelling. At the conclusion of this phase students will be asked to design details and draw a section

perspective of their dwelling design, along with other deliverables. In this workshop, form is never considered static, but rather a description of an intention in a particular phase.

NOTE: No prior background in mathematics or programming is necessary to take this workshop. A basic understanding of Rhinoceros 3D Software will be bene�cial.

Student Work: Poleak Na’s boundary transformation and physical model of a distorted helicoid.

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Student Work: Matthew Morse’s spatial analysis of de�ned boundary conditions.

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SYLLABUS

WORKSHOP SCHEDULE: STRUCTURED BUT NOT PREDICTABLE

INTRODUCTORY PHASE

01 – 26 – 11 LECTURE: Introduction to Instrumentality and Instrumentation PLAY: Di�erent Mediums, Di�erent Results LECTURE: Introduction to Morphologies / Chaos Theory / “Less predictable” TUTORIAL: Transforming Strange Attractors just “because...” ASSIGNMENT 01: Document Attractor Morphology / Reading on Cognition

02 – 02 – 11 PIN UP: What we see? What we do? Why? / Reviewing Assignment 01 LECTURE: Topology and Transformations TUTORIAL: Manual “Controlled” Digital Deformations of a Sphere and Geodesic PLAY: Squishing, Stretching, Twisting, Bending ASSIGNMENT 02: Document Deformation Morphology / Reading on Mathematics

COGNITIVE PHASE

02 – 09 – 11 PIN UP: Why? Why not? / Reviewing Assignment 02 LECTURE: “Explicit Learning” / Lesson in Mathematics 01 TUTORIAL: Drawing a Line / Manipulating Algebra without understanding the Math ASSIGNMENT 03: Document Line Morphology / Reading on Computation

02 – 16 – 11 PIN UP: Complex verse Complicated? / Reviewing Assignment 03 LECTURE: “Explicit Learning” / Lesson in Mathematics 02 TUTORIAL: Drawing Lines in Series / Variations with Algebra ASSIGNMENT 04: Document Pattern Morphology / Reading on Perception

02 – 23– 11 PIN UP: Qualifying Elegance? Why? / Reviewing Assignment 04 LECTURE: “Explicit Learning” / Lesson in Mathematics 03 TUTORIAL: De�ning Geometries with Lines / Tacit and Explicit Experimentation ASSIGNMENT 05: Document Form Morphology / Reading on Cognition

03 – 02 – 11 PIN UP: What was “Less predictable... or Controlled?” / Reviewing Assignment 05 LECTURE: Introduction to Design Computation and Calculation PLAY: Identifying Constraints and Ambitions ASSIGNMENT 06: Document Form Taxonomy / Perspective of Object

03 – 09 – 11 PIN UP: / Reviewing Assignment 06 LECTURE: Learning Morphologies / Pedagogical Books PLAY: Developing a “Cognitive Narrative” / De�ning a Unique Manipulation Technique ASSIGNMENT 07: Document Thinking Process / Reading on Pedagogy

03 – 16 – 11 MIDTERM REVIEW: with Guest Critics / Presenting Assignments 01 through 07

03 – 23 – 11 Spring Break! ARCHITECTURAL PHASE

03 – 30 – 11 LECTURE: Mathematics and Architecture PLAY: Transforming “Geometric to Volumetric to Spatial” ASSIGNMENT 08: Document Shape to Space Morphology / Perspective inside Shape

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Student Work: Neil Piatt’s rendering series and physical model variations.

80

SYLLABUS

04 – 06 – 11 PIN UP: Volumetric or Spatial? / Reviewing Assignment 08 PLAY: Identifying “Spatial Consequences” ASSIGNMENT 09: Document Spatial Morphology / Perspective inside Space 04 – 13 – 11 PIN UP: What scale? / Reviewing Assignment 09 TUTORIAL: Triangulations / Aggregates / Modules / Tiling PLAY: “Rede�ne the Formal Description” TUTORIAL: Unfolding and Unrolling Surfaces / Cut Files for Physical Models ASSIGNMENT 10: Transform Continuous Form into Form De�ned by Modules

04 – 20 – 10 PIN UP: How could it be built? / Reviewing Assignment 10 TUTORIAL: “Drawing Geometries with Architectural Conventions” PLAY: De�ne Hierarchy and Spatial Emphasis ASSIGNMENT 11: Drawing a Section Perspective / Physical Model of Section

04 – 27 – 11 PIN UP: What are the Perceptual Consequences? / Reviewing Assignment 11 LECTURE: Architectural Details / Drawing as a Design Instrument TUTORIAL: Architectural Rendering PLAY: “Designing and Drawing Details” ASSIGNMENT 12: Redrawing the Section Perspective with Details

05 – 04 – 11 PIN UP: / Reviewing Assignments 01 through 12 PLAY: Final Drawing Revisions / Finalize Physical Model

05 – 11 – 11 FINAL REVIEW: with Guest Critics / Presenting Assignments 01 through 12

NOTE: “PLAY” is referring to in class work sessions with individual desk critiques.

Student Work: Naomi Sherman’s architectural section.

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Student Work: Zachary Stanesa’s interweaving wall module.

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PHILOSOPHY

On the �rst day of class Joseph starts with this simple but powerful exercise that teaches the fundamentals of instrumenta-tion and instrumentality (Krueger 1998). Perhaps the most basic motivators of his teaching philosophy can be best expressed by this exercise.

“We are about to begin a design exercise. The instructions will be short and explicit, and after they are given you cannot ask any questions. There will be �ve exercises, each lasting �ve minutes in length.” The students waited, looked at each other and then back at me.

“Using pen and paper, draw a chair.” The students began to draw instantly and �nished early.

“Using pen and paper, design a chair.” The students all stopped, paused and thought. Eventually, they each began to draw.

“Using pen and paper, design a surface which one can sit on.” The students again began to draw, this time more ambiguously. Each only drawing one iteration.

“Using only paper, design a surface which one can sit on.” Finally the students stopped drawing and began to fold the piece of paper, manipulating its material properties. In the previous two instructions the students were not told to draw, yet they still reverted to the common preconceptions that one must draw when given pen and paper. Again, only one design iteration resulted from each student. After another �ve min-utes the last instructions were given.

“Using only paper, design a surface which one can sit on, and make �ve iterations in �ve minutes.” The students instantly began to manipulate the material as quickly as

possible, resulting in less predictable, more curious consequences.

There is a major fundamental di�erence between using a tool as a recording device of preconceived ideas and using a tool as a mechanism to generate ideas. Designers need to learn to instrumentalize any medium in front of them as a design opportunity. They need to be able to look beyond their �xed symbolic preconceptions and literally “play” with the medium at hand. Rigorous “play” should be a designer’s expertise!

Students at the BAC using pen and paper to design a chair.

Students at the BAC using paper to design a surface which one can sit on.

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Figure 10. A textured geodesic dome with respect to its �attened edge unfolding.

Figure 9. A geodesic dome with respect to its �attened edge unfolding.

84

CONCLUSION

A pedagogical guide into trigonometric transformations is a guide that contributes beyond the speci�city of the speci�c math-ematical functions portrayed within it. The guide is �rst and foremost about a radical new way of learning and teaching design. Design is not about recording but it’s about transforming. Transformations within the guide are taught through morphing series, which record each phase of the iterative process. Learning through morphologies has proven to be a graspable, intuitive way of understanding. It is very clear that this is unlike any other math book ever published. This straight forward structure for teaching could in the future yield other pedagogical books. For example, beyond looking at the algebras that de�ne shapes, one could look at the shape’s �attened edge unfolding. When transforming shapes designers don’t often have an intuitive understanding of how that shape would �atten. It is still unproven as to whether or not every convex polyhedron can have a �attened edge unfolding. This future area of research would be less invested in proving such a question and would be more interested in conveying with morphologies more generalized rules of how these shapes’ edge unfolding transform. For example a dome can unfold similar to that of orange peels, parting vertically down-ward (Fig. 9). If texturing was applied to that dome, suddenly the shape would have to unfold in a more spiral like fashion in order to avoid overlaps in the unfolding (Fig. 10). This conveys to the designer that spiral unfolding techniques are more versatile, but take up a larger area of �attened unfolding.

Beyond developing more cognitive narra-tives, the more speci�c research of trigono-metric transformations has numerous excit-ing future areas of study as well. Within the algorithms there are multiple areas to prog-ress. Embedding spaces on top of other

spaces seems like a potentially signi�cant ontological view of world making. Further developing and simplifying the complex algorithms used to undergo such transfor-mations could be further investigated. The pedagogical guide also ends with a category called “containing.” This section and the “thickening” section suggest a wide range of other types of transformations that could occur when a designer begins to de�ne more than one boundary condition. Lastly, there is also the question of translation. Although every designer can design di�erently with trigonometry according to their own bias, the number of designers currently using trigonometry as a design tool is few. Design-ers have the opportunity to push experien-tial consequences in directions that could not have been imagined without instrumen-talizing this medium. A new age of math-ematical design is at the horizon!

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PHYSICAL MODELS

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BIBLIOGRAPHY

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Schön, Donald A. The Re�ective Practitioner: How Professionals Think in Action. New York: Basic, 1983. Print.

Stiny, George. Shape: Talking about Seeing and Doing. Cambridge, MA: MIT, 2006. Print.

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Documents

A Pedagogical Guide into Trigonometric Transformations