A Pedagogical Guide into Trigonometric Transformations
by
Joseph Choma
Bachelor of Architecture
Rensselaer Polytechnic Institute, 2009
SUBMITTED TO THE DEPARTMENT OF ARCHITECTURE IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN ARCHITECTURAL STUDIES
AT THE
MASSACHSETTS INSTITUTE OF TECHNOLOGY
June 2011
© 2011 Joseph Choma. All rights reserved
The 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
Certified by: ….……………………………………………………………………………………
George Stiny
Professor of Design and Computation
Thesis Advisor
Accepted by:….……………………………………………………………………………………
Takehiko Nagakura
Associate Professor of Design and Computation
Chairman, Department Committee on Graduate Students
2
George Stiny
Professor of Design and Computation
Thesis Advisor
Nader Tehrani
Professor and Head of Department of Architecture
Thesis Reader
Edith K. Ackermann
Honorary Professor of Developmental Psychology at the University of Aix-Marseille
Thesis Reader
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4
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 STUDIES
AT THE
MASSACHSETTS INSTITUTE OF TECHNOLOGY
Abstract
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 operations transform shapes in a particular way.
A tool is a device that augments an
individual’s ability to perform a particular
task. The more specificity a tool has, the
narrower its instrumentality. Tools inherently
constrain the way individuals design; however, designers are often unaware of their
influence and bias. Digital tools are becoming increasingly complex and filled with hierarchical 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 fascinated with engineering mentalities: optimization and automation. Simply, it gives a
solution. But, this is not design! Designers
need to work outside of a fixed atmosphere!
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 fixed
constraint to instrumentalize. Through the
recursive “play” and learn process, a new
morphological classification of topological
transformations emerged, leading to the
development of this guide, the first pedagogical guide into trigonometric transformations. This guide does not invent new realms
within the field of mathematics, but develops a new cognitive narrative within it,
emphasizing the interconnected and plastic
nature of shapes.
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:” mathematics. Within mathematics, functions can
be embedded inside other functions at
Thesis Advisor: George Stiny
Title: Professor of Design and Computation
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ACKNOWLEDGEMENTS
During the last two years at the Massachusetts Institute of Technology I have encountered 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.
While attending MIT, I often referred back to
my experiences working for Vito Acconci.
The think tank design atmosphere has transformed who I am and who I want to become.
If only I could count the number of times I
used the words instrumentation and instrumentality! I must thank my previous professor at RPI, Ted Krueger. While assisting Ted
on his sensory research, he gave me my first
exposure to the world of design research. A
field I find myself redefining each day.
Professor George Stiny is an individual who I
feel honored to have studied under while at
MIT. When I first 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 iteratively embedded more emerging ideas. The
fundamental philosophical ideas of his
shape grammars has influenced and inspired
the motivator of this thesis’ inquiry. His work
places the bar infinitely high which I plan to
continue to reach for.
I would also like to thank my friends in the
Design and Computation Group who I
believe are some of the most uniquely interesting individuals I have ever met. The atmosphere and energy they added to my education will never be forgotten.
Professor Dennis Shelden was the first individual who interacted with me on my
research in the area of trigonometric transformations. As the work began to reach high
levels of complexity, Dennis encouraged me
to step back and understand the basic principles of the mathematics first. This led me to
focus on a pedagogical guide which
revealed the inner workings of the most
fundamental trigonometric transformations.
Lastly, I would like to thank my family, especially my Father.
Professor Terry Knight’s class taught me how
to develop an inquiry and identify contributions within design research. The clarity of
this thesis would not have been possible
without Terry’s structured exercises.
Professor Nader Tehrani was the first professor I had the privilege to act as a teaching
assistant for at MIT. Since that first experience, 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!
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TABLE OF CONTENTS
Advisor and Readers……………………………………………………………………………3
Abstract……………………………………………………………………………………………5
Acknowledgements………………………………………………………………………………7
Morphing: Pedagogical Guide…………………………………………………………………11
Introduction……………………………………………………………………………………13
Shaping.…………………………………………………………………………………………27
Cutting…………………………………………………………………………………………31
Scaling…………………………………………………………………………………………35
Modulating………………………………………………………………………………………39
Ascending………………………………………………………………………………………43
Spiraling…………………………………………………………………………………………47
Texturing…………………………………………………………………………………………51
Bending…………………………………………………………………………………………55
Pinching…………………………………………………………………………………………59
Flattening………………………………………………………………………………………63
Thickening………………………………………………………………………………………67
Containing………………………………………………………………………………………71
Teaching…………………………………………………………………………………………75
Syllabus…………………………………………………………………………………………79
Philosophy………………………………………………………………………………………83
Conclusion………………………………………………………………………………………85
Physical Models…………………………………………………………………………………87
Bibliography……………………………………………………………………………………89
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MORPHING
a pedagogical guide into
trigonometric transformations
11
Joseph Choma
MORPHING
12
INTRODUCTION
“The way in which a problem is decomposed
imposes fundamental constraints on the way
in which people attempt to solve that problem” (Brooks 1999).
designing, it may be necessary to use more
than one equation to define the entirety of
your design.
“Act always so as to increase the number of
choices” (Foerster 2002).
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
difficulty 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 trigonometry, 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 flattened the
geometry. However, it is impossible to completely flatten a curve to go around a corner
without the slightest rounding because the
curve needs to maintain its property of continuity. If the curve had an edge or kink, it
would break its continuity and become multiple 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 first
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 defined using trigonometry. Anything can be described using trigonometry,
it may just simply take more than one equation to define it. Any break in continuity will
require an additional equation. When
This research does not attempt to make an
ultimate proof, but rather attempts to find
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 flux”
(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 dictionary. 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 pedagogical guide challenges the linguistically driven
fixed world which we live by exposing an
alternative plastic world defined by trigonometry. A mathematical world where all
shapes can be described under one systematic language. Any shape can transform into
another, even a cylinder can effortlessly
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 definitions, but should instrumentalize tools as mechanisms to generate
ideas!
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14
INTRODUCTION
a sphere, the designer clicks on the icon and
draws a radius. If a shape is defined 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 defined by a parametric equation, it
becomes defined 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 manipulates the smallest morphemes themselves,
understanding how each function influences
a particular transformation becomes obvious. The designer is no longer designing
within a black box, but rather within a transparent box.
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. Defining
“worlds” is one of the primary problems with
computational models today. As soon as a
designer "defines a world" all the computer
can do is see the “world” they have symbolically defined, 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
defined 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.
Figure 3. A sphere deformed in a typical parametric
environment, where it defines itself.
Figure 4. A sphere transformed in mathematical
environment, where it is defined by trigonometry.
“We have dealt so far, and for the most part
we shall continue to deal, with our coordinate method as a means of comparing one
known structure with another. But it is obvious, as I have said, that it may also be
employed for drawing hypothetical structures, on the assumption that they have
Many contemporary digital tools use a fixed
symbolic interface, similar to that of the
dictionary. When a designer wants to create
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16
INTRODUCTION
varied from a known form in some definite
way” (Thompson 1942).
into an open surface, you would have to
redefine how the initial input shape is
described. This is quite different from a
purely mathematical model. In the mathematical model, all modes of transformations and flexibility are under one systematic
language. The designer never has to redefine 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
confined to the boundary of a cube, with one
transformation (Fig. 5). Its complete hierarchy has been altered. Designers can begin to
think and manipulate in a less linear fashion
and constantly redefine the “world” within
they create and perceive. Like throwing
paint onto a blank canvas, the Cartesian
coordinate system becomes that canvas!
Although the two previous morphologies
may seem like a series of arbitrary shapes, in
actuality, the sphere transformed by trigonometry actually analyzes an existing building geometry. The fourth shape in the five
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. Mathematics 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 location 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,
first of all, the boundaries of masses”
(Arnheim 1954).
In addition to the linguistically defined interfaces in contemporary digital tools, most
software also uses a linear hierarchical structure, building each rule or logic off of one
another. Such linear structures can become
extremely specific and fixed to the point
where they are only able to compute one
task. For example, if you imagine a parametric 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
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
Trigonometry may seem like the tool of a
designer’s dreams, but just because potentially 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 first make a
sphere; therefore initially round shapes
become easier to produce.
eleven transformations covered in this guide
include: cutting, scaling, modulating,
ascending, spiraling, texturing, bending,
pinching, flattening, thickening, and
containing. These eleven transformations
become the most fundamental design
operations when instrumentalizing trigonometry.
“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).
“The first 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, rectangles, circles, ellipses, etc., are all represented by a closed curve without straight
lines or angles. Topologically, squares and
circles are the same figure. Crosses, arcs, etc.,
are all represented by an open curve. At this
same age, however, children can produce
quite accurate copies of a closed figure with
a little circle inside. The topological relation
of the inside circle to the enclosing one, or
even the relation between a closed figure
and a circle on its boundary is represented by
children who are quite incapable of copying
a square correctly” (Piaget 1969).
By combining these eleven transformations,
one can create possibly any shape imaginable. There are an infinite number of hierarchical 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
flattening a cylinder and then bending it, the
resultant would be a bent square tube. If the
process was reversed, where the cylinder
was first bent and then flattened, then the
entire bent cylinder would be flattened
according to the outer boundary of a cube.
The entire bent shape would flatten globally
rather than flatten locally and then bend
globally. One can also achieve different
results by transforming portions of parametric equations and not others. For instance,
perhaps a designer wants a bent surface to
meet a flattened surface, rather than have a
bent flattened surface. Anything is possible!
A pedagogical guide is not purely an instructional guide nor is it purely a philosophical
text, but rather through a series of instructions a philosophical idea is taught. This
guide structures itself around ideas of topology while revealing the trigonometry behind
each transformation. The guide begins the
creative process in a section called shaping.
In this first 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
“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
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20
INTRODUCTION
does something; he lifts, let us say, a stone. In
consequence he undergoes, suffers, something; the weight, strain, texture of the
surface of the thing lifted. The properties
thus undergone determine further doing”
(Dewey 1934).
agreement about ends. He does not keep
means and ends separate, but defines them
interactively as he frames a problematic
situation. He does not separate thinking
from doing, ratiocinating his way to a decision which he must later convert to action.
Because his experimenting is a kind of
action, implementation is built into his
inquiry.
Thus reflection-in-action can
proceed, even in situations of uncertainty or
uniqueness, because it is not bound by the
dichotomies of Technical Rationality” (Schön
1983).
By manipulating the smallest pieces that
could define 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 define larger pieces to calculate
with. For example, if you look at the parametric equation of a sphere, its x, y and z
coordinates are each defined by sine and
cosine and u and v. If the sphere’s x, y, and z
coordinates become defined 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 defining the shape could
become defined 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!
Although this guide demonstrates discrete
strategies to control trigonometric transformations, 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 generate unpredictable shapes that may guide
the design process in a new direction. Try
going back and forth between tacit experimentation and explicit learning. It is not
simply about memorizing specific 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 find themselves discussing topics that they’d never
thought of, when they began, and they may
find radical, new ideas in and through
conversation” (Glanville 2008).
Figure 7. A shape transforming based on calculations
with dimensions defined by a cylinder and sphere.
“When someone reflects-in-action, he
becomes a researcher in the practice
context. He is not dependent on the categories 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
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
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INTRODUCTION
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 intervene, and constantly transform the algorithm 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.
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).
“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 morphing documentation process. In order for a
designer to understand a transformation,
they have to see what its starting shape and
resultant. By documenting that transformation iteratively over a couple more instances,
it becomes clear how that mechanism can be
controlled. For example, in order to demonstrate 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 transformation. This is different than simply four
discrete shapes documented with their algebras. Morphologies are iterative. For
example, imagine starting with a sphere,
which then gets pinched, then flattened, and
then bent. In a morphing documentation
process, the starting sphere would be documented first, then the sphere pinched, then
the sphere pinched and flattened, and then
the sphere pinched flattened and bent. It
would not simply be a sphere, a sphere
pinched, a sphere flattened, and a sphere
bent. The morphology needs to transition
from its neighbors and blend, with each step
building off of the previous. Morphing is not
This pedagogical guide into trigonometric
transformations is structured around morphologies. 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 documentation process as a means to reflect 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 effects, and evaluate the
results according to criteria you have personally identified. For example, if you identify
pinches within the design as significant attributes 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 reflecting is as
important as playing and seeing.
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INTRODUCTION
“One eye sees, the other feels” (Klee 1964).
look and enjoy this guide into trigonometric
transformations and feel free to use the
instrument of mathematics as you see best
fit.
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 simultaneously; 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 cognitive embedding becomes an augmented
reality.
Figure 8. A space embedded on top of another space,
where each is transformed independent of one
another.
“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).
The feelings of something being pulled
apart, or bending a shape to twist, are transformations that also relate to bodily actions.
Even without looking at the parametric
equation which defines 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 transform whatever is in front of you, and see how
your perception transforms. That is design!
Never let anything be completely fixed, 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 generated mathematical representations and push
it! Although the algebras can get quite complex, 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 specific type of design opportunity. Play,
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26
SHAPING
SHAPING
introducing the basic shapes
27
x= u
y= 0
z= 0
x= u
y = cos(u)
z= 0
x= u
y = sin(u)
z= 0
x = cos(u)
y = sin(u)
z= 0
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
x = cos(u)
y = sin(u)
z= 0
x = cos(u)
y = sin(u)
z= v
x = cos(u)
y = sin(u)
z= v
x = cos(u)
y = sin(u)
z= v
for all
u є [0, 2π]
for all
u є [0, 2π]
v є [0, 1/3π]
for all
u є [0, 2π]
v є [0, 2/3π]
for all
u є [0, 2π]
v є [0, π]
28
SHAPING
x = cos(u)
y = sin(u)
z= 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
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
x = cos(u)
y = sin(u)
z= v
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)
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
29
30
CUTTING
CUTTING
changing the period of the shapes
31
x= u
y = sin(u)
z= 0
x= u
y = sin(u)
z= 0
x= u
y = sin(u)
z= 0
x= u
y = sin(u)
z= 0
for all
u є [0, 2π]
for all
u є [0, 3/2π]
for all
u є [0, π]
for all
u є [0, 1/2π]
x = cos(u)
y = sin(u)
z= 0
x = cos(u)
y = sin(u)
z= 0
x = cos(u)
y = sin(u)
z= 0
x = cos(u)
y = sin(u)
z= 0
for all
u є [0, 2π]
for all
u є [0, 3/2π]
for all
u є [0, π]
for all
u є [0, 1/2π]
32
CUTTING
x = cos(u)
y = sin(u)
z= v
x = cos(u)
y = sin(u)
z= v
x = cos(u)
y = sin(u)
z= v
x = cos(u)
y = sin(u)
z= v
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 3/2π]
v є [0, π]
for all
u є [0, π]
v є [0, π]
for all
u є [0, 1/2π]
v є [0, π]
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))
z = cos(v)
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))
z = cos(v)
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))
z = cos(v)
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))
z = cos(v)
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 3/2π]
v є [0, π]
for all
u є [0, π]
v є [0, π]
for all
u є [0, 1/2π]
v є [0, π]
33
34
SCALING
SCALING
changing the amplitude of the shapes
35
x= u
y = sin(u)
z= 0
x= u
y = sin(u)/2
z= 0
x = u/2
y = sin(u)
z= 0
x = u/2
y = sin(u)/2
z= 0
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
x = cos(u)
y = sin(u)
z= 0
x = cos(u)
y = sin(u)/2
z= 0
x = cos(u)/2
y = sin(u)
z= 0
x = cos(u)/2
y = sin(u)/2
z= 0
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
36
SCALING
x = cos(u)
y = sin(u)
z= v
x = cos(u)
y = sin(u)/2
z= v
x = cos(u)/2
y = sin(u)
z= v
x = cos(u)/2
y = sin(u)/2
z= v
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))
z = cos(v)
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))/2
z = cos(v)
x = (sin(v))(cos(u))/2
y = (sin(v))(sin(u))/2
z = cos(v)
x = (sin(v))(cos(u))/2
y = (sin(v))(sin(u))/2
z = cos(v)/2
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
37
38
MODULATING
MODULATING
changing the frequency of the shapes
39
x= u
y = sin(u)
z= 0
x= u
y = sin(2(u))
z= 0
x= u
y = sin(3(u))
z= 0
x= u
y = sin(4(u))
z= 0
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
x = cos(u)
y = sin(u)
z= 0
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
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
40
MODULATING
x = cos(u)
y = sin(u)
z= v
x = cos(u)
y = sin(2(u))
z= v
x = cos(u)
y = sin(3(u))
z= v
x = cos(u)
y = sin(4(u))
z= v
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))
z = cos(v)
x = (sin(v))(cos(u))
y = (sin(v))(sin(2(u)))
z = cos(v)
x = (sin(v))(cos(u))
y = (sin(v))(sin(3(u)))
z = cos(v)
x = (sin(v))(cos(u))
y = (sin(v))(sin(4(u)))
z = cos(v)
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
41
42
ASCENDING
ASCENDING
increasing the trajectory of the shapes
43
x= u
y = sin(u)
z= 0
x= u
y = u+sin(u)
z= 0
x = u+u
y = sin(u)
z= 0
x = u+u
y = u+sin(u)
z= 0
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
x = cos(u)
y = sin(u)
z= 0
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
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
44
ASCENDING
x = cos(u)
y = sin(u)
z= 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
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))
z = cos(v)
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)
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
45
46
SPIRALING
SPIRALING
increasing the radius of the shapes
47
x= u
y = sin(u)
z= 0
x= u
y = u(sin(u))
z= 0
x= u
y = cos(u)
z= 0
x= u
y = u(cos(u))
z= 0
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
x = cos(u)
y = sin(u)
z= 0
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
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
48
SPIRALING
x = cos(u)
y = sin(u)
z= 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
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))
z = cos(v)
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)
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
49
50
TEXTURING
TEXTURING
combining shapes of different frequencies
51
x= u
y = sin(u)
z= 0
x= u
y = sin(2(u))/2+sin(u)
z= 0
x= u
y = sin(3(u))/3+sin(u)
z= 0
x= u
y = sin(4(u))/4+sin(u)
z= 0
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
x = cos(u)
y = sin(u)
z= 0
x = cos(2(u))/2+cos(u)
y = sin(2(u))/2+sin(u)
z= 0
x = cos(3(u))/3+cos(u)
y = sin(3(u))/3+sin(u)
z= 0
x = cos(4(u))/4+cos(u)
y = sin(4(u))/4+sin(u)
z= 0
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
52
TEXTURING
x = cos(u)
y = sin(u)
z= v
x = cos(2(u))/2+cos(u)
y = sin(2(u))/2+sin(u)
z= v
x = cos(3(u))/3+cos(u)
y = sin(3(u))/3+sin(u)
z= v
x = cos(4(u))/4+cos(u)
y = sin(4(u))/4+sin(u)
z= v
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))
z = cos(v)
x = cos(2(u))/2+(sin(v))(cos(u))
y = sin(2(u))/2+(sin(v))(sin(u))
z = cos(v)
x = cos(3(u))/3+(sin(v))(cos(u))
y = sin(3(u))/3+(sin(v))(sin(u))
z = cos(v)
x = cos(4(u))/4+(sin(v))(cos(u))
y = sin(4(u))/4+(sin(v))(sin(u))
z = cos(v)
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
53
54
BENDING
BENDING
deflecting the shapes globally
55
x= u
y = sin(u)
z= 0
x= u
y = sin(v)+sin(u)
z= 0
x = cos(v)+u
y = sin(u)
z= 0
x = cos(v)+u
y = sin(v)+sin(u)
z= 0
for all
u є [0, 2π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
x = cos(u)
y = sin(u)
z= 0
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
for all
u є [0, 2π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
56
BENDING
x = cos(u)
y = sin(u)
z= 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
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))
z = cos(v)
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)
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
57
58
PINCHING
PINCHING
steepening the apex of the shapes
59
x= u
y = sin(u)
z= 0
x= u
y = sin(u)3
z= 0
x= u
y = sin(u)5
z= 0
x= u
y = sin(u)7
z= 0
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
x = cos(u)
y = sin(u)
z= 0
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
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
60
PINCHING
x = cos(u)
y = sin(u)
z= 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
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))
z = cos(v)
x = (sin(v))(cos(u)3)
y = (sin(v))(sin(u)3)
z = cos(v)
x = (sin(v))(cos(u)5)
y = (sin(v))(sin(u)5)
z = cos(v)
x = (sin(v))(cos(u)7)
y = (sin(v))(sin(u)7)
z = cos(v)
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
61
62
FLATTENING
FLATTENING
decreasing the apex of the shapes
63
x= u
y = sin(u)
z= 0
x= u
y = sin(sin(u))
z= 0
x= u
y = sin(sin(sin(u)))
z= 0
x= u
y = sin(sin(sin(sin(u))))
z= 0
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
x = cos(u)
y = sin(u)
z= 0
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
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
for all
u є [0, 2π]
64
FLATTENING
x = cos(u)
y = sin(u)
z= 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
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))
z = cos(v)
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))))
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
for all
u є [0, 2π]
v є [0, π]
65
66
THICKENING
THICKENING
introducing another dimension to the shapes
67
x= u
y = sin(u)
z= 0
xo = 0
yo = 0
zo = 0
xo = 0
yo = 0
zo = 0
xo = 0
yo = 0
zo = 0
for all
u є [0, 2π]
xw = xo+u
yw = yo+sin(u)
zw = zo+0
xw = xo+u
yw = yo+sin(u)
zw = zo+0
xw = xo+u
yw = yo+sin(u)
zw = zo+0
for all
u є [0, 2π]
w є [0, 1/3π]
for all
u є [0, 2π]
w є [0, 2/3π]
for all
u є [0, 2π]
w є [0, π]
x = cos(u)
y = sin(u)
z= 0
xo = 0
yo = 0
zo = 0
xo = 0
yo = 0
zo = 0
xo = 0
yo = 0
zo = 0
for all
u є [0, 2π]
xw = xo+cos(u)
yw = yo+sin(u)
zw = zo+0
xw = xo+cos(u)
yw = yo+sin(u)
zw = zo+0
xw = xo+cos(u)
yw = yo+sin(u)
zw = zo+0
for all
u є [0, 2π]
w є [0, 1/3π]
for all
u є [0, 2π]
w є [0, 2/3π]
for all
u є [0, 2π]
w є [0, π]
68
THICKENING
x = cos(u)
y = sin(u)
z= v
xo = 0
yo = 0
zo = 0
xo = 0
yo = 0
zo = 0
xo = 0
yo = 0
zo = 0
for all
u є [0, 2π]
v є [0, π]
xw = xo+cos(u)
yw = yo+sin(u)
zw = zo+v
xw = xo+cos(u)
yw = yo+sin(u)
zw = zo+v
xw = xo+cos(u)
yw = yo+sin(u)
zw = zo+v
for all
u є [0, 2π]
v є [0, π]
w є [0, 1/3π]
for all
u є [0, 2π]
v є [0, π]
w є [0, 2/3π]
for all
u є [0, 2π]
v є [0, π]
w є [0, π]
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))
z = cos(v)
xo = 0
yo = 0
zo = 0
xo = 0
yo = 0
zo = 0
xo = 0
yo = 0
zo = 0
for all
u є [0, 2π]
v є [0, π]
xw = xo+(sin(v))(cos(u))
yw = yo+(sin(v))(sin(u))
zw = zo+cos(v)
xw = xo+(sin(v))(cos(u))
yw = yo+(sin(v))(sin(u))
zw = zo+cos(v)
xw = xo+(sin(v))(cos(u))
yw = yo+(sin(v))(sin(u))
zw = zo+cos(v)
for all
u є [0, 2π]
v є [0, π]
w є [0, 1/3π]
for all
u є [0, 2π]
v є [0, π]
w є [0, 2/3π]
for all
u є [0, 2π]
v є [0, π]
w є [0, π]
69
70
CONTAINING
CONTAINING
placing a shape inside the boundary of another
71
x= u
y = sin(u)
z= 0
xo = u
yo = sin(u)
zo = 0
xo = u
yo = sin(u)
zo = 0
xo = u
yo = sin(u)
zo = 0
for all
u є [0, 2π]
xw = xo
yw = sin(yo)
zw = zo
xw = xo
yw = sin(yo)
zw = zo
xw = xo
yw = sin(yo)
zw = zo
for all
u є [0, 2π]
w є [0, 1/3π]
for all
u є [0, 2π]
w є [0, 2/3π]
for all
u є [0, 2π]
w є [0, π]
x = cos(u)
y = sin(u)
z= 0
xo = u
yo = sin(u)
zo = 0
xo = u
yo = sin(u)
zo = 0
xo = u
yo = sin(u)
zo = 0
for all
u є [0, 2π]
xw = sin(xo)
yw = sin(yo)
zw = zo
xw = sin(xo)
yw = sin(yo)
zw = zo
xw = sin(xo)
yw = sin(yo)
zw = zo
for all
u є [0, 2π]
w є [0, 1/3π]
for all
u є [0, 2π]
w є [0, 2/3π]
for all
u є [0, 2π]
w є [0, π]
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CONTAINING
x = cos(u)
y = sin(u)
z= v
xo = cos(u)
yo = sin(u)
zo = v
xo = cos(u)
yo = sin(u)
zo = v
xo = cos(u)
yo = sin(u)
zo = v
for all
u є [0, 2π]
v є [0, π]
xw = sin(xo)
yw = sin(yo)
zw = zo
xw = sin(xo)
yw = sin(yo)
zw = zo
xw = sin(xo)
yw = sin(yo)
zw = zo
for all
u є [0, 2π]
v є [0, π]
w є [0, 1/3π]
for all
u є [0, 2π]
v є [0, π]
w є [0, 2/3π]
for all
u є [0, 2π]
v є [0, π]
w є [0, π]
x = (sin(v))(cos(u))
y = (sin(v))(sin(u))
z = cos(v)
xo = (sin(v))(cos(u))
yo = (sin(v))(sin(u))
zo = cos(v)
xo = (sin(v))(cos(u))
yo = (sin(v))(sin(u))
zo = cos(v)
xo = (sin(v))(cos(u))
yo = (sin(v))(sin(u))
zo = cos(v)
for all
u є [0, 2π]
v є [0, π]
xw = sin(xo)
yw = sin(yo)
zw = sin(zo)
xw = sin(xo)
yw = sin(yo)
zw = sin(zo)
xw = sin(xo)
yw = sin(yo)
zw = sin(zo)
for all
u є [0, 2π]
v є [0, π]
w є [0, 1/3π]
for all
u є [0, 2π]
v є [0, π]
w є [0, 2/3π]
for all
u є [0, 2π]
v є [0, π]
w є [0, π]
73
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.
74
TEACHING
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 exemplary readings.
round become initially easy to construct in
that medium. Finally, students are taught to
translate spatial intentions into physical
prototypes and lastly into rigorous architectural consequences. However, the emphasis
in the course is not on the final 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 differed slightly
from that of the pedagogical guide. In the
pedagogical guide the fundamental trigonometric transformations are taught to the
designer in the beginning as means to get
them to start thinking about design through
transformations, combinatorics, and embedding. 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 find what transformations are significant 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.
It begins by teaching students to document
transformations with morphologies, making
students’ intuitive tacit process more explicit.
The course emphasizes the internal biases of
different digital tools by having students
experience utilizing very predictable digital
mechanisms: squishing, stretching, deforming, in comparison to the much less predictable: chaos theory. Later, the students
manipulate algebra in parametric equations
where they learn to understand the mathematics based on their intuitive dialogue
with the tool. Their initial lack of complete
control with the medium drives their curiosity to generate unpremeditated ideas. Eventually 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 fixed 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 first make a
sphere. Therefore, shapes that are more
75
Student Work: Mark Roger’s transformation taxonomy and perspective rendering.
76
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 algorithms 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 reflection 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 experimentation, designers can begin to develop a
convincing design that can conform to fixed
practical constraints. With that spirit in mind,
play and look, learn and control.
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 beneficial.
This workshop will be split into three general
phases: introductory, cognitive and architectural.
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 different levels of
predictability.
In the cognitive phase,
students will learn how to control and generate shapes using parametric equations. Students will be taught to document, design
and think through morphologies, emphasizing the plastic and interconnected nature of
shapes. At the conclusion of this phase,
students will be asked to define a unique
manipulation technique for instrumentalizing 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
Student Work: Poleak Na’s boundary transformation
and physical model of a distorted helicoid.
77
Student Work: Matthew Morse’s spatial analysis of defined boundary conditions.
78
SYLLABUS
WORKSHOP SCHEDULE: STRUCTURED BUT NOT PREDICTABLE
INTRODUCTORY PHASE
01 – 26 – 11
LECTURE: Introduction to Instrumentality and Instrumentation
PLAY: Different Mediums, Different 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: Defining 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” / Defining 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
79
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: “Redefine the Formal Description”
TUTORIAL: Unfolding and Unrolling Surfaces / Cut Files for Physical Models
ASSIGNMENT 10: Transform Continuous Form into Form Defined by Modules
04 – 20 – 10
PIN UP: How could it be built? / Reviewing Assignment 10
TUTORIAL: “Drawing Geometries with Architectural Conventions”
PLAY: Define 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.
81
Student Work: Zachary Stanesa’s interweaving wall module.
82
PHILOSOPHY
On the first day of class Joseph starts with
this simple but powerful exercise that
teaches the fundamentals of instrumentation and instrumentality (Krueger 1998).
Perhaps the most basic motivators of his
teaching philosophy can be best expressed
by this exercise.
possible, resulting in less predictable, more
curious consequences.
There is a major fundamental difference
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
fixed symbolic preconceptions and literally
“play” with the medium at hand. Rigorous
“play” should be a designer’s expertise!
“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 five exercises, each
lasting five 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
finished 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.
Students at the BAC using pen and paper to design a
chair.
“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 five minutes the last instructions were given.
“Using only paper, design a surface which
one can sit on, and make five iterations in five
minutes.” The students instantly began to
manipulate the material as quickly as
Students at the BAC using paper to design a surface
which one can sit on.
83
Figure 9. A geodesic dome with respect to its flattened edge unfolding.
Figure 10. A textured geodesic dome with respect to its flattened edge unfolding.
84
CONCLUSION
A pedagogical guide into trigonometric
transformations is a guide that contributes
beyond the specificity of the specific mathematical functions portrayed within it. The
guide is first 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 define shapes, one could look
at the shape’s flattened edge unfolding.
When transforming shapes designers don’t
often have an intuitive understanding of
how that shape would flatten. It is still
unproven as to whether or not every convex
polyhedron can have a flattened 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 downward (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 flattened unfolding.
spaces seems like a potentially significant
ontological view of world making. Further
developing and simplifying the complex
algorithms used to undergo such transformations 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 define more
than one boundary condition. Lastly, there is
also the question of translation. Although
every designer can design differently with
trigonometry according to their own bias,
the number of designers currently using
trigonometry as a design tool is few. Designers have the opportunity to push experiential consequences in directions that could
not have been imagined without instrumentalizing this medium. A new age of mathematical design is at the horizon!
Beyond developing more cognitive narratives, the more specific research of trigonometric transformations has numerous exciting future areas of study as well. Within the
algorithms there are multiple areas to progress. Embedding spaces on top of other
85
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PHYSICAL MODELS
87
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