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Geometric Shape Generator (GSG) 1
Geometric Shape Generator:
Abstract
Geometric Shape Generator (GSG) is an end-user-programmable
archiving and interface system for designers that is capable of
manipulating codes to generate recursive shapes.
GSG generates a wide variety of shapes and patterns. Simple or
complex algorithmic formulas and/or transformation rules may be
utilized to alter, modify, replicate, or transform the current
geometric shapes or patterns.
There are endless numbers of design styles, which are created by
limited sets of rules. GSG is a design tool that allows the
designer to choose and apply individual rules and algorithm to
generate geometric shapes or patterns. Additionally, depending on
how the formulas and order of transformation is applied, we may
also alter, modify, replicate, or transform the current chaos
theory based shapes or patterns.
1.1 Introduction
Different design styles from different regions and eras have
used diverse methods of geometry to generate the design. An example
is the Islamic architecture; designers used fractals and
transformation rules to generate various designs.
Figure 1 exemplifies the use of geometry and geometric patterns
in Islamic architecture. The illustrated wall is divided into a
number of panels, each with its own distinctive pattern and logic.
Within each panel the same design logic holds true.
This logic is not exclusively limited to walls or large objects;
the same logic is often applied to generate shapes and patterns
ranging from textiles to metalwork.
Figure 1: Wall pattern of Friday Mosque at Herat in
Afghanistan
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Figure 2 is an example of classic-western design used in a
Chartres Cathedral in the 13th century. The designers used the
concept of fractals in the design of this Gothic style window. The
basic logic used in this shape is the repetition of different scale
circles. The most outer shape is a circle, this circle is reduced
and repeated in the middle part and is surrounded by twelve smaller
circles. In the external part of the window there are again twelve
other circles; which are surrounded by another twelve circles.
Figure 2: Window rose fractal repetition at Chartres
Cathedral
Even avant-garde architects such as Peter Eisenman were
influenced by fractals and the use of fractal concept. Eisenman
exhibited house 11-a in July of 1978, less than a year after the
English language publication of Fractals: Form, Chance and
Dimension by scientist Benoit Mandelbrot.
Eventually, House (11a) became a motif in Eisenman's house
design. He used the concept of fractal scaling - a process that he
describes philosophically as entailing "three destabilizing
concepts: discontinuity, which confronts the metaphysics of
presence; recursively, which confronts origin; and self-similarity,
which confronts representation and the aesthetic object."
House 11a, a composition of Eisenmans signature Ls is a
combination of transformation rules; such as rotation scaling. The
L is actually a square which has been divided into four quarters
and then one quarter square is removed. Eisenman viewed this
resulting L shape as symbolizing an unstable or in-between state;
neither a rectangle nor a square. The three dimensional
variation
Figure 3:Using fractal repetition in house 11-a and House X
respectively
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is a cubic octant removed from a cubic whole, rendering the L in
three dimensions.
Greg Lynn uses self intersecting curves to create volumetric
pockets within continuous surfaces. According to Greg Lynn, blebs
are pockets of space formed when a surface intersects itself,
making a captured space. He uses a class of geometric curves
beginning with the folium of Descartes and including Limacon of
Pascal Maclaurins Trisectrix, TschirnHauss cubic, Cubic curves
Freeths, Nephroid Stronoid and Plateau curves. Refer to figures
5-7.
From the examples above, we can make the assumption that
regardless of time, geographical location, or cultures, designers
utilize geometric rules to create their designs.
1.2 GOALS OF THE THESIS
The goal of this thesis is to generate a wide variety of shapes
and patterns ranging from Classic-Western, Islamic architecture, to
the work of Peter Eisenman and Greg Lynn, as well as shapes and
patterns of nature, using evolutionary algorithms as design
computing tools.
Figure 5: ST. GALLEN KUNST MUSEUM. Three volumes sandwiched
between outdoor ceiling and St. Gallen Kunst
Museum containing sculpture galleries
Figure 6: Tri Of Maclaurin was used to create the volumes in
figure 5
Figure 7: Examples of astroid and limacon of pascal curves used
by Greg Lynn for building design.
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1.3 METHODOLGY
The focus of this thesis is on three different categories using
evolutionary algorithms in the processing environment to generate
shapes to explore fractals, curves and surfaces. The thesis will
also explore the rules that apply to algorithms to regenerate new
shape pattern volumes and spaces. The algorithms used to generate
shape will be organized in a library system for end user
programmers to use them for their own design generation and code
manipulation. The designer can use specific algorithms in a
specific category to generate space volumes or detailed patterns,
which can be used for building detail designs in windows, door
details, or tiling. The thesis is more interested in generating
more complex spaces or volumes like those in Greg Lynn or Frank O
Gehrys work, or even more organic forms using curves or surfaces.
As far as building details, the algorithms will generate fast,
detailed patterns. Drafting such details in conventional CAD
environments, using traditional methods of drafting, will require
time, knowledge of geometry, and creativity. Additionally, the
thesis will add colorimetric properties to the pattern. This will
be useful for coloring and filling the patterns created in the
processing environment
A number of applets have been developed using processing for
this thesis. For making the applets, I have explored l-systems
which is a part of fractals to generate patterns similar to Islamic
architecture details. Also, some very similar to what we can see in
nature; like trees and microscopic details of some living
organs
1.4 THE PARTS OF THE THESIS PROPOSAL
Section 2 discusses the related work and Annotated Bibliography
in order to place GSG in the framework of Computational Design as
an evolutionary shape generative tool that can be used to design
architectural forms or patterns with ubiquitous design style.
Section 3 introduces L-systems and explains how the current
processing applets have used l-system to generate shape and
patterns. Section 4 proposes fractals, curves and surfaces as
shapes with complete mathematical structure that GSG can provide
indirect access to their characteristics and therefore allows shape
manipulation and regeneration. Section 4 also explores different
fractal types as well as curves and surfaces. Section 5 is the
conclusion and future work.
2-RELATED WORK
The thesis will explore related work in two categories: The
history of evolutionary algorithms and its influence in
architecture and
art Shape Generative computing and its rule in design.
2.1-Fractals and Fractal Architecture
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Wolfgang E. Lorenz, Fractals and Fractal Architecture, from URL
http://www.iemar.tuwien.ac.at/modul23/fractals/subpages/10home.html.
Fractals will make you see everything differently. ... You risk
the loss of your childhood vision of clouds, forests, galaxies,
leaves, flowers, rocks, mountains, torrents of water, carpets,
bricks, and much more.
The word fractal was initiated by Benoit Mandelbrot in 1975. He
was describing fractals as asymmetrical curves. Mandelbrot set,
which is one of the most well-known fractal types, was defined for
the first time by Mandelbrot.
Wolfgang believes we should first define how fractals can help
us in a specific field. Then find applications of fractal geometry
in different fields: natural science, medicine, market analysis,
manufacturing, ecology - and architecture? He also believes there
are aspects of fractal applications in fine art, city planning and
architecture; further research should be done.
Until recently, scientists described nature through mathematics
of smooth forms such as lines, curves and planes geometry. The new
science does not try to replicate the rugged quality of nature
through smooth forms, but it deals with the irregularity of the
structure itself - this field of mathematics is expressed in the
language of fractal geometry: The whole is more than its parts. The
fractal new geometric art shows surprising similarity to Grand
Master paintings or Beaux Arts architecture. An evident reason is
that classical visual arts, like fractals, involve many scales of
length and support self-similarity.
Since we can find similarity between nature and architecture
with regard to material and structure, some of the fractal
attributes can be found in architecture. Fractal geometry may help
us investigate the complexity of Middle Age towns, cathedrals and
other man-made objects thru time. It may also help us transfer the
complexity of newly planned cities and buildings - cities and
buildings may then also be reduced to simpler algorithms: the
automatic architecture"
2.2-Generating Fractals Based on Spatial Organizations
Magdy M. Ibrahim and Robert J. Krawczyk, Generating Fractals
Based on Spatial Organizations, Illinois Institute of Technology,
Chicago
The authors classify fractals in two main categories depending
on the technique they are generated from and the mathematical
processes that are used to calculate them.
From perspective of drawing, the categories are: Line or vector
fractals, which are generated from the collection of vector
substitution; like Dragon curves shown in figure 8a &
8b.
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Point fractals are groups of points in a complex plane like the
Mandelbrot set and the Julia set shown in Figure 9
From the perspective of mathematics we can classify fractals
into: IFS iterated function systems like Koch snowflake, cantor
set, Barnesleys
Fern, and the Dragon Curve. Fractal generates from any set of
vectors or any defined curve.
Complex number fractals. Two-dimensional, three-dimensional or
multiple-dimensional. They represent a single case of the IFS that
is using the complex numbers or the hyper complex numbers in a
Cartesian plane to
plot the fractals. The Mandelbrot set and Julia set are
examples. Orbit fractals. Plotting an orbit path in two or
three-dimension generates
the fractal space. Examples include the Bifurcation orbit,
Lorenz Attractors, Rossler Attractors, Henon Attractors, Pickover
Attractors,
Figure 8a: First Four Generations of Dragon Curve
Figure 8b: Dragon Curve
Figure 9 Mandelbrot set
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Gingerbreadman, and Martin Attractors. These are related with
the chaos theory.
2.3-Architecture, Patterns, and Mathematics
Nikos A. Salingaros, Architecture, Patterns, and Mathematics,
University of Texas, Published in the NNJ vol. 1 no. 2 (April
1999)
Salinagarose introduces Mathematics as a science of patterns.
The human mind distinguishes connections and interrelations between
concepts and ideas, and links them together. The human neural
development in response to a persons environment enables him/her to
create patterns. Mathematical theories explain the relations
between patterns that occur inside ordered, logical structures.
Patterns in the mind imitate patterns in nature as well as man-made
patterns, which is probably the reason why humans developed
mathematics in the first place. People generate patterns out of
some fundamental inner need: it externalizes connective structures
generated in the mind thinking process, which explains the
ubiquitousness of visual patterns in the traditional art and
architecture of mankind.
Patterns are also essential to human intellectual development.
Daily activity is planned around natural rhythms. Annual events
become a society's fixed points. Additionally, these can help
individuals understand a periodic natural phenomena like seasons
and their effects. Mathematics itself arose out of the need to
observe patterns in space and time. Repeating gestures become
theater and dance, and are incorporated into myth, ritual, and
religion. The development of voice and music responds to the need
to encapsulate rhythmic patterns and messages. All of these
activities occur as patterns on the human range of time scales.
Self organization in a complex physical and chemical system
generates pattern in space or time.
Patterns manifest the inherent creative aptitude of human beings
for mathematics. Child psychologists necessitate patterns for
developing a childs visual environment.
Salingaro believes patterns are indispensable from architectural
form. The study of hidden patterns in chaotic systems is
Mathematical chaos. The basic aim of mathematics, which is to
discover patterns, has not changed from Newtonian to chaotic
models. Built examples vary from one extreme to another: from the
empty modernist model into random forms. Decoration on contemporary
buildings is either so minimal or hardly noticeable, or it is
deliberately disarrayed and broken, consequently it is jumbled.
Although, Deconstructivist architects avoid organized complexity,
their design concept is still a proper mathematical concept.
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2.4-Towards a generative system for intelligent design
support
John Hamilton Frazer, Ming xi tang, and Sun Jian school of
design, Towards a generative system for intelligent design support,
The Hong Kong polytechnic university
This article approaches evolutionary design as employing
different evolutionary computing techniques in the various stages
of the design process. They find evolutionary design process a
vigorous method since it can be formulated as a wide-ranging
purposed problem solver with a gift similar to human design
aptitude but with an enormity of speed and efficiency.
The role of designers in creative decision-making must be
supported rather than undermined. In this sense, evolutionary
design techniques as general purpose that design support tools
require to be incorporated with knowledge-based design techniques
in order to reveal designers expertise and experience in any
automated generative processes.
2.5-Computational design
Processing Ben Fry and Casey Reas initiated processing. It is
currently developed at the MIT Media Lab, UCLA, Interaction Ivrea,
and by a group of distributed developers across the Net.
The Processing project brings in a new audience to computer
programming who is a mixture of artist/designer/programmer. It
incorporates a programming language, development environment, and
teaching methodology into a unified structure for learning. Its
goal is to introduce programming in the context of electronic art
and to open electronic art concepts to a programming audience. The
Processing concept is to create a text programming language
particularly for making reactive images, the language facilitates
sophisticated visual and responsive composition
DBN John Maeda proposed DBN design by numbers for teaching
computation to artists and designers. DBN is both a programming
environment and language. The environment provides an integrated
space for writing and running programs and the DBN brings in the
basic ideas of computer programming within the context of drawing.
Visual elements such as dots, lines, and fields are combined with
the computational ideas of variables and conditional statements to
generate images.
Form Writer Mark Gross, Form Writer is an easy- programming
language designed especially for architects to explore generating
forms using algorithms. Form Writer offers a
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simple syntax, combined with development environment, and easy
access to three-dimensional libraries.
ArchiDNA ArchiDNA uses a simplified syntax code that can be
translated into Java code to generate 2D or 3D images. ArchiDNA
enables designers to use a computer to algorithmically generate
shapes and forms. Doo Young Kwons approach in ArchiDNA enables you
to define a certain design style of shape generation which looks
like Peter Eisenmans design style
ArtiE-Fract ArtiE-Fract is user friendly software for generating
fractal images using interactive evolutionary algorithms.
ArtiE-Fract generates fractal images based on the iteration
function system. These algorithms have been built up in an easy to
use interface with advance interactive tools. 3-GSG System
3.1 L-Systems
Lyndemayer System (l-systems) introduced in 1968 are capable of
generating complex structures from small data sets. L-systems take
biological knowledge of cell differentiation and growth and
interpret it within a proper mathematical language. The l-system
represents growth by repeated cell subdivision mechanism. The
arrangement arises in l-system is a result of cell subdivision
reiteration, and the product is not symbolic unless we interpret
them as meaningful objects.
An L-system consists of 3 components:
Alphabet a, b Axiom a (also called the initiator) Set of
re-writing rules, (next state functions) b goes to a & a goes
to b
The technique is to apply the rules to each element of axiom,
then write the result below it, then take every element of the
latest row and apply the rules, writing the result below it, this
action then keeps on adding axioms. (Figure 10)
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Figure 10: An example of a simple map L-system
Fig 11 shows a plant being grown using L-systems and turtle
graphics. We can use L-systems principal to generate a variety of
structure from Islamic patterns to plant simulations. This
phenomenon is due to the mutual use of evolutionary algorithms (EA)
and as a result makes l-systems a very dynamic design tool.
Figure 11-plant simulation using l-systems
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3.2-Turtle graphics and L-systems
Turtle graphics were invented by Seymore Papert as a computer
graphic system that works based on predefined statements (position
and orientation) and a small number of rules against that statement
(forward, turn, pen up & down). Turtle geometry was used as a
simple technique to teach geometry to kids using algorithmic
computing. The statement was called the turtle and rules taught the
turtle how to draw.
Turtle graphic was used to build up l-systems by providing a
geometrical interpretation to the l-systems dynamics. The following
symbols can be used to develop an l-system:
F: Draw forward one step in current direction +: Turn right -:
Turn left
The Koch curve L-system as an example to show how such a system
works:
Alphabet = {F,+,-} axiom= F production rules: F F+F- - F+F
If we presume Generation 0 is just a straight horizontal line;
and the rotation angle (r) is 60, the production means replace a
straight line segment F by the following arrangement of four line
vectors as shown in figure-12.
Figure -12: Koch curve production (first generation) Some more
generations of the Koch curve are shown in figure 13.
Figure- 13: Koch curve generations
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The Koch L-system grows fairly quick; each line segment is
replaced by four more at the next generation, the n-th generation
has segments. For instance, generation 8 plotted above has 65536
segments.
3.3 Processing applets
By using the l-system and Turtle graphics, I have developed a
series of applets in the processing environment. In this section I
will show some examples of my applets and also briefly describe
their system and how a designer can interact with my codes to
generate a wide variety of shapes. Figure -14 shows an example of
shapes I created using l-system concepts in the processing
environment
Figure-14. Example of shapes generated with processing
Example of geometric shapes Using a turtle-graphics control
language and start with an initial axiom string, the program
carries out string substitutions to the specified number of times
(the iteration), and plots the resulting shape.
In order to produce the shapes in figure-15, I have used basic
turtle l-system interpreter. My alphabets are defined as (F, f,+,-)
you can see the alphabet description as :
F : draws a line of unit in current direction f : move forward
one unit in current direction, without drawing the line + : turns
left one angle unit - : turns right one angle unit
String state = "F-F-F-F"; String [] axioms = {"F","f","-","+"};
String [] productions = {"F+F-F+F-F+F","f","-","+"}; Int iterations
= 4; the number of iterations (generations) Float unit=4; the
vector unite length Float delta=30; the rotation angle
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The processing environment does not have a built in turtle
system, I used a simple algorithm to define a turtle. An example of
the simple algorithm for simulating turtle movements is:
newX=x+unit*cos(radians(zt)); newY=y+unit*sin(radians(zt));
line(x,y,newX,newY);
X and y are starting coordination points in the Cartesian
coordinate system. NewX and newY are defining the turtle movement
in the coordinate system. Zt defines the turtle turning angle from
left or right. Zt=Zt+Delta which delta is our rotation angle, and
at the turtles starting point our vector line should be horizontal.
Which means zt=0 and the iteration=0, delta is a rotating angle
with a horizontal line and is used to turn the turtle left or right
with the Delta angle. Finally, line command draws a line base on
the line coordination points. I have to mention here that if we
draw another shape instead of a line and apply the same rules to
our shape the same system still works and basically shape
replacement gives the designer the ability to generate a variety of
shapes; needless to mention that applying these simple rules to the
single line can generate a wide range of shapes by itself. The
designer can generate shapes by changing the production statement
or rotation angle unite size or number of iteration. Figure-15 is a
simple example of geometric patterns that a designer can produce
using l-system and Turtle graphics.
Figure 15: Geometric shape generated by using the l-system.
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Here is the programming code that was used to generate
figure-15
String state = "F-F-F-F"; String[] axioms = {"F","f","-","+"};
String[] productions = {"F+F-F+F-F+F","f","-","+"}; int iterations
= 4; float unit=15; float delta=30; int c=0; float x,y,zt; int
index; color dColor; String newState; void setup(){ size(400,400);
background(220,150,100); for(int i=0;i
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Figure-16: Detla 45 Iteration 4 Figure-17: Detla 45 Iteration 4
Figure-18: Detla 30 Iteration 5
Here is another set of applets that have been developed using
the turtle l-system interpreter; but by the implemented via
push/pop turtle logic to facilitate dynamic angle changing. Push()
and pop() requires the understanding of the matrix stack concept.
The push() function saves the current coordinate system to the
stack and the pop() restores the prior coordinate system. Push()
and pop() are used in conjunction with the other transformation
methods and may be embedded to control the scope of the
transformations. Alphabets for this system are:
F : draw a line of unit in current direction f : move forward
one unit in current direction, no draw + : turn left one angle unit
- : turn right one angle unit [ : push state ] : pop state
The code shown below is used to generate a turtle l-system tree
implemented with push and pop. An image of the tree is shown in
figure-19
String state = "F"; String[] axioms = {"F","f","-","+","[","]"};
String[] productions = {"F-[-F+F+F]+[+F-F-F]","f","-","+","[","]"};
int iterations = 4; float unit=80 (the vector unite length) float
delta=150;(rotation angle)
Figure 19-Example of a L-System tree
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We can use push/pop turtle l-systems to generate a wide range of
shapes and patterns by changing the rotation angle, iteration
number, and unit size as well as production statement to create
interesting shapes or manipulate existing shapes. The applets can
work as knowledge based support systems to support the designer to
create interesting shapes quickly with little understanding of
l-systems structure. The artist can interact with the code to
manipulate or generate a range of shapes.
Figure 20: example of shape generator using push and pop turtle
L-System
4.1 Applet extensions
In the next part of the thesis, I would like to further explore
fractals, curves, and surfaces and the related algorithm that
generate a wider range of shape. Then I will explore the
possibility of interaction with the end userprogrammable library
system that covers fractal, curves and surfaces to manipulate and
generate shapes.
4.2 Fractals
Various methods using different algorithms have been discovered
to produce fractals. Ant Automaton, Barnsley Mandelbrot/Julia Set,
Bifurcation, Cellular Automata, Martin Attractors, Circle, Peterson
Variations, Formula, Diffusion Type, Dynamic System, Pick over
Popcorn, Frothy Basins, Gingerbreadman, Halley, Hyper Complex,
Newton, Icon and Icon3d ,Julia Sets ,Inverse Julias, L-Systems (2d,
3d) are some of the most well-known fractal types that I am going
to produce and explore ways to interact with their evolutionary
algorithmic structure to generate patterns and to find easier more
powerful interaction and code manipulation to easily and quickly
generate shapes. Figure 21 and 22 show patterns that have been
produced using different fractal types.
The fractal shapes generated by designers can be used as forms
for architectural floor plan design, or the patterns can be used
for architectural building details, such as tiling windows and
doors.
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Ant type Barnsleyj2 type Basic bifurcation fractal
Cellular Automata Martin Type Circle type
Pickover Popcorn Complex Newton type cmplxmarsjul type Figure
21: Shapes produced with various fractal types
Diffusion type Dynamic type Frothy basin type
Manzpowr type Icon type Henon type
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Halley type Julia type l-system type
Kamtorus type Inverse Julia type Gingerbread type Figure 22:
Shapes produced with various fractal types
4.3-3d Fractals
Manipulating fractals in a 3d environment will provide the
designer a large variety of volume and space, which is useful for
architectural design. 3d L-systems is a good example of such a 3d
environment that I will further explore in this thesis to set a
series of parameters in my programming code that enables the
designer to easily manipulate and generate shapes in 3d environment
or apply rules to the single or complex 3d axiom to generate
architectural spaces.
Figure 23 exemplifies a 3 dimensional structure of the 2D
carpet. The unit cube is divided to slice the width of the cube by
1/3. The central pieces of the resulting cubes located at the
center of the faces are removed from original cube. This process is
repeated perpetually with the resulting cubes. Such a process could
be useful to generate Eisenmans like designs.
Result Figure 23: Example of cube generated using 3d
l-systems
Another area to explore in regard to 3d fractals is to transform
or extrude existing 2d fractals like Mandelbrot or Julia sets along
z axis to generate more organic type of volumes
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Figure 24: 3d Julia set generated by Pov-Ray
4.4-curves
Mathematic algorithms can be used to generate a variety of
curves useful for generating architectural floor plans,
architectural ornamentation as well as generating geometric spaces
for building domes or Greg Lynn and Frank Gehrys organic forms and
spaces. For another section of this thesis I would like to explore
the use of algorithms that generate curves and also transforming
the resulting curve to produce 3d shapes. Table 1 are samples of
the curve types that I will explore and manipulate their
evolutionary algorithm and possibilities of generating forms from
architectural spaces to geometric patterns using these curves in my
future work.
Table 1 Super circle Piecewise cubic Bezier Butterfly curve
Chrysanthemum Cycloid Spline curves Conic sections Archimedes
Spiral Equiangular Spiral Fermats Spiral Hyperbolic Spiral Lituus
Spiral Parabolic spiral
Square Archimedes Spiral Sinusoidal Spiral Coth spiral Tanh
spiral Cornu Spiral Cayleys sextic Cissoid of Dioches Viviani
Epicycloid Concoid of Nicromedes Cissoid of Dioches Viviani
Epicycloid
Lemniscate Deltoid Strophoid Nephroid 2D Bow curve Bow curve
Klein Cycloid Krishna Anklets Mango Leaf Snake Kolam Cardioid
Parabola Spherical Cardioid Astroid Hyperbola Hypocycloid Agnesi
curve
Bicorn curve Glissette Diamond curve Kappa curve Piriform curve
Trisectrix of Maclaurin Tractrix Folium curve Baseball seam Eight
knot Helix Limacon Spherical Nephroid Freeths Nephroid Borromean
rings
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Super shapes
Knots
Spherical Cardioid
Figure 25: examples of various curve families can be produced
using algorithms
4.5-surfaces Surfaces as well as curves have been constantly
used in architectural design. Greg Lynn Blobs are good examples of
using geometric surfaces in architectural design. Voluptuous
undulating surfaces can define our living spaces. Figure 26
illustrates EMBRYOLOGICAL HOUSE design by Greg Lynn and use of
folding surfaces in architecture design. Mathematical algorithms
may be used to generate such surfaces which are useful for
architectural design.
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Figure 26: Voluptuous undulating surfaces in Embryological
House
Table 2 illustrates some of the surface categories that would be
used to generate forms. Surface manipulation of the evolutionary
algorithm will be explored for possible form generation.
Table 2Spherical Harmonics 3D Super Shape Pseudo sphere
Steinmetz solid Bifolia surface Twisted plan Kuen's surface Cross
Hole Twisted Fano Fano planes Slipper surface Tranguloid Trefoi
Chladni plates Nose, Witch hat Tangle, Strophoid surface Chair
surface Fish, Cresent Sea Shells Steiner look-a-like Cross Cap
Steiner surface Maeder's Owl Twisted Triaxia Hunt surface Stiletto
surface Mitre surface Blob, Nodal cubic Boy surface Apple shape
Klein Cycloid
Hearts Mobius strip Pisod triaxial Dini's surface Tear drop
Verrill surface Cassini Ova Glob teardrop Piriform Gerono lemniscat
Piriform Cassini Oval Glob teardrop Jet surface Superellipse
Cymbelloid Bezier surfaces Spline surfaces Triaxial teardrop
Whitney umbrella Lemniscape Tractrix Pseudocatenoi Twisted heart
Torus/Supertoroid Triaxial Tritorus Elliptic Torus Gumdrop Torus
Saddle Torus Bohemian Dome
Twisted pipe Devil surface Swallow P1 atomic orbital Ghost Plane
Folium Bent Horns Tubey Catenoid minimal surface Helicoids minimal
surface Catalan minimal surface Henneburg minimal surface Scherk
minimal surface Bour minimal surface Ennepers minimal surface
Richmond minimal surface Handkerchief, Kidney Monkey saddle Pillow
shape Snail surface Kinky torus Kusner Schmitt McMullen K3 mode
Weird surface Tri-Trumpet Gerhard Miehlich Kampyle of Eudoxus Barth
sextic Cayley surface Tooth
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Golnaz Mohammadi University of Washington Thesis Proposal
February 16, 2004
Geometric Shape Generator (GSG) 22
Figure 27: examples of Blobs and super shapes surfaces that may
be produced using algorithms
4.6-GSG end user programmable library system GSG will be
organized in a library system. Based on the wide variety of shapes
produced, the designer can effortlessly access the categories of
his desired interest
5.1 Conclusions Geometry as part of mathematics has always been
used in architectural design process. We can see the use of
geometry in western and eastern classical architecture to modern
and avant-gardes architecture. Higher end architects have been
trying to imitate from nature to create more organic forms that we
refuse to see any geometric relation in their form. Exploring
evolutionary algorithm will enlighten the geometric structure of
organic forms. The only reason human being refuses to see geometric
relationship of nature is due to their complex structure. Further
and more in-depth attention and exploration is needed to visualize
the complex structures. Computer may be used as a tool to study and
achieve such structures. Since computers can process huge amounts
of information in a very short time frame, computing is not, and
will not block human creative abilities. GSG is simply a tool that
allows the designer to explore more design possibilities. Human
creativity and abilities are beyond our imagination, providing the
right tools to the designer may indeed further human creativity and
abilities.
5.2 Future Works Evolutionary structure of fractals is capable
of creating music as well as shape for further exploration and
expansion of this thesis. I hypothesis, that the possibility of
generating music using the same algorithm has been used to generate
the form to explore the harmony between shapes and music. Currently
GSG system is capable of generating shapes which are useful for
early stage of design. I would like to offer a knowledgebase design
critique to evaluate the resulting shape to support the designer
for the entire design process.
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Golnaz Mohammadi University of Washington Thesis Proposal
February 16, 2004
Geometric Shape Generator (GSG) 23
6. Schedule 10 weeks in Spring of 2004: To research related
work, develop and expand
programming skills
Week 1-3 of Spring of 2004: Completion of implementation project
as well as enhancements
Week 3-7 of Spring of 2004: Writing documents Week 7-10 of
Spring of 2004: Modifying the interface and writing documents
7. Available Resources: Faculty Support: Ellen Yi-Luen Do and
Mark D Gross
8. Space Support MS program in the dept of architecture (Design
Machine Group) will provide the thesis space during Spring
quarter.