through the design space of interesting geometric patterns. The numeric ... to
demonstrate the rules of geometry in architectural design. This also sets.
Exploring Design Shapes with Geometry
Golnaz Mohammadi
A thesis submitted in partial fulfillment of the Requirements for the degree of
Master of Science in Architecture
University of Washington 2004
Program Authorized to Offer Degree: Architecture
University of Washington Graduate School
This is to certify that I have examined this copy of a master’s thesis by
Golnaz Mohammadi
and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made.
Committee Members:_____________________________ Ellen Yi-Luen Do
_____________________________ Mark D. Gross
Date:
______________________________
In presenting this thesis in partial fulfillment of the requirements for a Master’s degree at the University of Washington, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying of this thesis is allowable only for scholarly purposes, consistent with “fair use” as prescribed in the U.S. Copyright Law. Any other reproduction for any purposes or by any means shall not be allowed without my written permission.
Signature:____________________________
Date:_____________________________
University of Washington Abstract Exploring Design Shapes with Geometry
Golnaz Mohammadi
Chair of the Supervisory Committee: Professor Ellen Yi-Luen Do Department of Architecture
Abstract
This thesis presents a collection of geometric shapes and patterns produced with simple programming codes. The codes employ algorithms that define shapes through mathematical equations. Altering or modifying the numeric values and basic variables in the algorithm generate a variety of geometric forms. To enable designers/architects to understand and explore geometric shape generation easily, a catalog of shapes with their numeric values and generating algorithm was complied. This catalog provides an easy navigator through the design space of interesting geometric patterns.
The numeric
values provided in the catalog for each shape may be used as a reference for producing similar shapes. The architects and designers have been using geometric shapes or patterns that may be valuable in their architectural design process. This thesis gives a brief review of various architectural examples from different eras and cultures to demonstrate the rules of geometry in architectural design. This also sets the background and rationale for exploring geometry algorithmically. The programming codes are saved and archived in the Processing environment for generating geometric shapes at a later time. The generated shapes can be saved as image files such as “tiff” or “jpeg” files, or saved as animated java applets for later manipulation.
Mathematical equations are
used in the codes to produce interesting geometric shapes. Finally, a collection of generated shapes was used for design of a library. The generated shapes were used as the pattern for floor plan layouts. The generated shapes and patterns were also used for interior building design, such as tiling pattern window details.
Acknowledgements
I am grateful to several people for their support and advice throughout my tenure as a student at the University of Washington.
The successful
completion of this study would have not been possible without the direction, guidance and persistent support of professor Ellen Yi-Luen Do. Ellen’s deep attention to detail and generous knowledge has impacted my beliefs and principles with regards to Design Computing and Architecture. I would also like to thank professor Mark Gross for his insights. Additionally, I would like to thank my wonderful colleagues in the Design Machine Group: Karen Hanson, for proofreading and administrative support; Ken Camarata, Markus Eng, and Yeonjoo Oh for their continued support and encouragement.
I would also like to thank my parents, Maryam Shoraka and Manouchehr Mohammadi for their support and encouragement throughout this journey. Finally, I am deeply greatful to my dear loving husband, Babak Ziraknejad, for his dedication and support.
Table of Contents
List of Figures....................................................................................................... ii Chapter 1 - Introduction............................................ Error! Bookmark not defined. Chapter 2 - Use of Geometry in architecture ............ Error! Bookmark not defined. 2.1 Tessellation Geometry ............................Error! Bookmark not defined. 2.2 Fractal Geometry ....................................Error! Bookmark not defined. 2.3 Complex curves, surfaces.......................Error! Bookmark not defined. 2.4 Mathematical Geometry and ArchitectureError! Bookmark not defined. 2.5 Thesis Question ......................................Error! Bookmark not defined. 2.6 Goals of the Thesis .................................Error! Bookmark not defined. 2.7 The Structure of the document................Error! Bookmark not defined. Chapter 3 - Background and Concepts .................... Error! Bookmark not defined. 3.1 Fractals ...................................................Error! Bookmark not defined. 3.2 Curves.....................................................Error! Bookmark not defined. 3.3 Supershapes...........................................Error! Bookmark not defined. 3.4 Spirograph...............................................Error! Bookmark not defined. 3.5 Spiral.......................................................Error! Bookmark not defined. 3.6 Surfaces..................................................Error! Bookmark not defined. Chapter 4 - Related Work......................................... Error! Bookmark not defined. 4-1 ArchiDNA ................................................Error! Bookmark not defined. 4-2 Design By Numbers ................................Error! Bookmark not defined. 4-3 FormWriter ..............................................Error! Bookmark not defined. 4-4 ArtiE-Fract: ArtiE-Fract: Interactive Evolution of FractalsError! Bookmark not defined 4-5 Objectile software....................................Error! Bookmark not defined. 4-6 Processing ..............................................Error! Bookmark not defined. Chapter 5 – Generating Geometric Shapes AlgorithmicallyError! Bookmark not defined. 5-1 Processing System .................................Error! Bookmark not defined. 5-2 Format of Generated Shapes..................Error! Bookmark not defined. 5-3 Examples of L-system Applets ................Error! Bookmark not defined. 5-4 Component of the L-system Applets .......Error! Bookmark not defined. 5-5 Curve Codes ...........................................Error! Bookmark not defined. 5-6 Mathematical Curves ..............................Error! Bookmark not defined. 5-7 Spirograph...............................................Error! Bookmark not defined. 5-8 Epicycloid ................................................Error! Bookmark not defined. 5-9 Epitrochoid ..............................................Error! Bookmark not defined. 5-10 Hypocycloid.............................................Error! Bookmark not defined. 5-11 Supershape.............................................Error! Bookmark not defined. 5-12 Surface Codes ........................................Error! Bookmark not defined. Chapter 6 - Building Design with Code Generated ShapesError! Bookmark not defined. 6-1 Parametric Design...................................Error! Bookmark not defined. 6-2 Drawing Methods ....................................Error! Bookmark not defined. Chapter 7 - Conclusions and Future Work ............... Error! Bookmark not defined.
i
List of Figures Figure 2-1: Ceiling Of The Tomb Of Hafiz In Persia, Illustrating The Repetition Of The Star Shape (Dome Interior), (Http://Islamicart.Com/Main/Architecture/Geometry.Html)Error! Bookmark not defined. Figure 2-2: Facade Of Du Monde Arabe Interpreting Arabic Latticework, (Www.Greatbuildings.Com)...................................Error! Bookmark not defined. Figure 2-3: Example Of Arab Latticework, Raymond Tennant, 2003Error! Bookmark not defin Figure 2-4: The Institute Of Du Monde Arabe, Left: Exterior View, Right: Close Up Of The Window Screen, (Www.Greatbuildings.Com)Error! Bookmark not defined. Figure 2-5: A) Circle Repetition In Small Portion Of Rose Window B) Center Part Of The Same Portion Is Magnified To Show The Circle Repetition C) Rose Window At Chartres Cathedral (Lorenz, 2003)Error! Bookmark not defined. Figure 2-6: Mandelbrot Set, (Elert, G., 2003) .......Error! Bookmark not defined. Figure 2-7: Mandelbrot Set Magnified With Different Scales Showing Self Similarity Between Different Scales (Elert, G., 2003)Error! Bookmark not defined. Figure 2-8: Using Fractal Repetition In House 11-A, And House X, (Www.Greatbuildings.Com)...................................Error! Bookmark not defined. Figure 2-9: L Shape Production............................Error! Bookmark not defined. Figure 2-10: Left: Plan Of Sculpture Galleries In St. Gallen Kunst Museum, Right: Galleries Section, (Glform.Com) .................Error! Bookmark not defined. Figure 2-11: Left: Concept Model Of St. Gallen Kunst Museum. Right: Tri Of Maclaurin Curve (Http://Glform.Com/ST_GALLEN.Pdf)Error! Bookmark not defined. Figure 2-12: Two Variation Of Embryological House (Glform.Com)Error! Bookmark not define Figure 2-13: Computer Models Of Embryological House, Glform.Com, (Lynn, 1999) .....................................................................Error! Bookmark not defined. Figure 2-14: Non Standard Architects’ Exhibition Area Defined By Mathematical Curves, (Www.Designboom.Com/Contemporary/Nonstandard.Html)Error! Bookmark not defined. Figure 2-15: Left, A Door Detail Used In Islamic Architecture, Right, A Wall Pattern Design Used In Islamic Architecture .........Error! Bookmark not defined. Figure 2-16: Lily Van Der Stokker, Decorative Patterns Was Used For Building Facade Design, (Www.Vividvormgeving.Nl/Pagina/Stokker.Htm)Error! Bookmark not Figure 2-17: Space Design For La Mia Casa Fair By Karim Rashid. Left: Super Blob 3D Model. Center: Photo Of Interior Space, Right: Photo Of Outside Space, (Karimrashid.Com) .......................Error! Bookmark not defined. Figure 3-1: Castle Del Monte, Apulia (Italy) Has Similar Shape With Mandelbrot Set. (Http://Math.Unipa.It/~Grim/Jsalaworkshop.PDF)Error! Bookmark not defined Figure 3-2: First Four Generations Of Dragon Curve, And Dragon Curve, (Http://Www.Math.Okstate.Edu/Mathdept/Dynamics/Lecnotes/Node17.Html )Error! Bookmark Figure 3-3: Koch Curve Production (First Generation), F+F - -F+F, (Http://Www.Math.Okstate.Edu/Mathdept/Dynamics/Lecnotes/Node18.HtmlError! Bookmark n Figure 3-4: Koch Curve Generations....................Error! Bookmark not defined. Figure 3-5: Julia Sets For Different C Values, (Http://Www.Eddaardvark.Co.Uk/Python_Patterns/Mandj.Html)Error! Bookmark not defined. Figure 3-6: Parabola Graph,.................................Error! Bookmark not defined. Figure 3-7: Hyperbola Graph,...............................Error! Bookmark not defined.
ii
Figure 3-8: Spiral Graph, (Www.Xahlee.Org/Specialplanecurves_Dir/Equiangularspiral_Dir/Equiangular spiral.Html) ............................................................Error! Bookmark not defined. Figure 3-10: Range Of Super Shapes Generated By Varying Parameters In The Super Formula, (Http://Www.Sciencenews.Org/Articles/20030503/Mathtrek.Asp )Error! Bookmark not defined Figure 3-11: Supershapes Compared With Nature (Seashells Flowers Shell Fish Tree Trunk), (Http://Www.Geniaal.Be/ ).........Error! Bookmark not defined. Figure 3-12: Traditional Spirograph ......................Error! Bookmark not defined. Figure 3-13: A Curve Drawn With Spirograph .......Error! Bookmark not defined. Figure 3-14: Church Of The Redeemer With Spiral Tower And Staircase,Error! Bookmark not Figure 3-15: Left: Double Spiral Staircase By Leonardo Da Vinci. Right: Model Of Double Spiral Staircase, (Http://Www.Ams.Org/Notices/200303/FeaColding-Web.Pdf) ..................................................Error! Bookmark not defined. Figure 3-16: Guggenheim Museum A Giant Spiral RampError! Bookmark not defined. Figure 3-17: Seashell Compared With Spiral Shape, (Http://Www.Spirasolaris.Ca/Sbb4d2c.Html) .........Error! Bookmark not defined. Figure 3-18: Left: 2-Dimentional Cartesian Coordinate System, Right: 3Dimentional Cartesian Coordinate System............Error! Bookmark not defined. Figure 3-19 ............................................................Error! Bookmark not defined. Figure 3-20: Mobius Band, (Http://En.Wikipedia.Org)Error! Bookmark not defined. Figure 3-21: Mobius Structure Made Of Cubes, (Http://Members.Tripod.Com)Error! Bookmark Figure 3-22: Headquarters For China Central Television (CCTV) In Beijing, China, Scheduled For Completion In 2008, (Http://Www.Arcspace.Com/Architects/Koolhaas/Chinese_Television/ )Error! Bookmark not de Figure 3-23: Mobius Strip Painting, M.C. Escher's, (Http://Www.Mcescher.Com).................................Error! Bookmark not defined. Figure 3-24: Mobius Building By Peter Eisenman .Error! Bookmark not defined. Figure 3-25: Mobius House Spatial Circulation Diagram, Based On Mobius Band (Van Berkel, 1998.) ......................................Error! Bookmark not defined. Figure 3-26: Left: Mobius House Exterior, Right: Mobius House Interior. (Van Berkel, 1998.) ........................................................Error! Bookmark not defined. Figure 4-1: Eisenman-Like Drawing Generated With Archidna, (Kwon, 2003)Error! Bookmark n Figure 4-2: Axonometric View Of Biocentrum Building, (Eisenman, 1996)Error! Bookmark not Figure 4-3: Applying The Applier-Shape G (Ribbon) To The Base-Shape C (Pentagon) (B) Applying The Applier-Shape G To The Base-Shape A (Arch), (Kwon, 2003) .........................................................Error! Bookmark not defined. Figure 4-4: DBN Programming And Display Environment, (Maeda, 2001)Error! Bookmark not Figure 4-5: Various Shapes Produced With DBN .Error! Bookmark not defined. Figure 4-6: Formwriter Programming And Display Environment, (Gross, 2002) .....................................................................Error! Bookmark not defined. Figure 4-7: 3d Models Of Islamic Architecture Generated With Form Writer. (Gross, 2002) ........................................................Error! Bookmark not defined. Figure 4-8: Artie-Fract Window Area (Chapuis, 2000)Error! Bookmark not defined. Figure 4-9: Shapes Generated With Artie-Fract (Chapuis, 2000.)Error! Bookmark not defined
iii
Figure 4-10: Various Objects Manufactured Using Objectile A) Sculpture, B) Door, C) Surface Sculpture, D) Door, E) Surface Sculpture, F) Table (Www.Archilab.Org )..............................................Error! Bookmark not defined. Figure 5-1: System Architecture Diagram.............Error! Bookmark not defined. Figure 5-2: The Processing Environment And ComponentsError! Bookmark not defined. Figure 5-3: Processing Environment Buttons .......Error! Bookmark not defined. Figure 5-4: Cartesian Coordinates System In The Processing Environment, (Proce55ing.Net) ...................................................Error! Bookmark not defined. Figure 5-5: Rectangle Transformation, Rotation ProcessError! Bookmark not defined. Figure 5-6: Geometric Shape Code Archiving System Within The Processing Environment ..........................................................Error! Bookmark not defined. Figure 5-7: Geometric Shape Generated By Using L-SystemsError! Bookmark not defined. Figure 5-8: Example Of Shapes Generated Using The Same Code. By Changing Variables In The Code...........................Error! Bookmark not defined. Figure 5-9: A) Detla 45 Iteration 4 B) Detla 60 Iteration 4 C) Detla 30 Iteration 5 ..............................................................Error! Bookmark not defined. Figure 5-10: Second Iteration Of The Same Statement And Production Rule With Different Rotation Angle ................................Error! Bookmark not defined. Figure 5-11: Fourth Iteration Of The Shapes Shown In Figure 20Error! Bookmark not defined Figure 5-12: Example Of L-System Tree Using The Push And Pop Commands ............................................................Error! Bookmark not defined. Figure 5-13: Astroid Curve ...................................Error! Bookmark not defined. Figure 5-14: Right Strophoid Curve Made Using The Processing CodesError! Bookmark not d Figure 5-15: The Different Values For Variables A And B Will Result Various Shapes As Shown In The Above Figure................Error! Bookmark not defined. Figure 5-16: The Different Values For Variables A And B Will Result Various Folium With 1, 2, Or 3 Leaves ...............................Error! Bookmark not defined. Figure 5-17: Different Curves Formula And The Related Curve ShapesError! Bookmark not d Figure 5-18: Spirograph Shape ............................Error! Bookmark not defined. Figure 5-19: Variety Of Spirograph Shapes...........Error! Bookmark not defined. Figure 5-20: Variety Of Epicycloid Shapes ............Error! Bookmark not defined. Figure 5-21: Variety Of Epitrochoid Shapes ..........Error! Bookmark not defined. Figure 5-22: Variety Of Hypocycloid Shapes.........Error! Bookmark not defined. Figure 5-23: Varity Of Supershapes. Left: Where M=2, N1=100, N2=10, & N3=10....................................................................Error! Bookmark not defined. Figure 5-24: Variety Of Super Shapes...................Error! Bookmark not defined. Figure 5-25: Variety Of Super Shapes...................Error! Bookmark not defined. Figure 5-26: Variety Of Super Shapes..................Error! Bookmark not defined. Figure 5-27: Super Shapes Source Author Built With ProcessingError! Bookmark not defined Figure 5-28: Astrodal Elipsod ...............................Error! Bookmark not defined. Figure 5-29: Surfaces And The Parameters That Are Used To Generate Surface ..................................................................Error! Bookmark not defined. Figure 5-30: Surfaces And The Parameters Used To Generate ThemError! Bookmark not def Figure 6-1: Variety Of Super Shapes That Was Used For The DesignError! Bookmark not de Figure 6-2: Top View Of Building..........................Error! Bookmark not defined. Figure 6-3: 3d Extrusion Process .........................Error! Bookmark not defined.
iv
Figure 6-4: Building Model....................................Error! Bookmark not defined. Figure 6-5: Different Building Models Developed By Various Compositions Of Supershapes. ........................................................Error! Bookmark not defined. Figure 6-6: Library Floor Plans ..............................Error! Bookmark not defined. Figure 6-7: 3d Model Of Library Spatial CirculationError! Bookmark not defined. Figure 6-8: Floor Pattern Design ...........................Error! Bookmark not defined. Figure 6-9 Building Interiors ..................................Error! Bookmark not defined.
v
27 ornamental building patterns. The Church of the Redeemer in Copenhagen has an outside spiral staircase, designed by Danish architecture Laurids de Thurah (1696-1752), illustrated in Figure 3-13.
Figure 3-13: Church of the Redeemer with spiral tower and staircase, (http://turkey-travel.planetware.com/photos/DK/DKKOB33.HTM)
Leonardo da Vinci used a double spiral staircase in the castle Château de Chambord, in France (1519-1539) shown in Figure 3-14.
Figure 3-14: Left: double spiral staircase by Leonardo da Vinci. Right: model of double spiral staircase, (http://www.ams.org/notices/200303/fea-coldingweb.pdf) The Guggenheim Museum in New York City by Frank Lloyd Wright is literally a giant spiral ramp illustrated in Figure 3-15.
28
Figure 3-15: Guggenheim Museum a giant spiral ramp, http://www.greatbuildings.com/buildings/Guggenheim_Museum.html
Spiral may be seen in natural forms such as seashell growth pattern shown in Figure 3-16.
Figure 3-16: seashell compared with spiral shape, (http://www.spirasolaris.ca/sbb4d2c.html)
29
3.6
Surfaces
Arrays of Lines or curves based on their point coordination in the space (three dimensional environments) compose surfaces. The most common two dimensional systems is the Cartesian system invented by René Descartes (From Wikipedia, the free encyclopedia.)
In Cartesian coordinate system
(Figure 3-17, Left) points are defined by their distances along the side of a horizontal path as (x) axis, and vertical path as (y) axis, from a reference point. The reference for this point is (0, 0). Cartesian coordinates also may be used as three (or more) dimensional systems.
Three-dimensional system is a
useful system for defining surfaces. In the three-dimensional system (Figure 3-17, Right) a (z) axis is added to coordinate system, where (z) axis is the height location of a two dimensional point from original point. The reference for this point is (0,0,0.)
Figure 3-17: Left: 2-dimentional Cartesian coordinate system, Right: 3dimentional cartesian coordinate system
30
Mobiuse, Blobe, Piriform, and 3d super shape are good examples of mathematical surfaces (illustrated in Figure 3-18.) Let’s now look at Mobius surface, and it’s uses in architectural design.
Mobiuse
Blobe
Piriform
3d super
Figure 3-18
A model of the Mobius band is created by joining the ends of a long narrow piece of paper after giving it a half twist, 180 degrees (Figure 3-19).
Figure 3-19: Mobius Band, (http://en.wikipedia.org) In order to walk along a face of a Mobius band, it is expected to have more than one loop around the ring before coming back to the starting point.
31 “Other" faces of the prismatic structure will be visited in the procedure. Such a structure may be built with a proper gathering of cubes (Figure 3-20.)
Figure 3-20: Mobius structure made of cubes, (http://members.tripod.com)
The Office for Metropolitan Architecture (OMA) used Mobius concept design for the China Central Television in Beijing to be completed in 2008, shown in Figure 3-21.
Figure 3-21: Headquarters for China Central Television (CCTV) in Beijing, China, Scheduled for completion in 2008, (http://www.arcspace.com/architects/koolhaas/chinese_television/ )
M.C. Escher used Mobius band in his art work. Figure 3-22 left shows a wood engraved Mobius strip was in 1961. In 1963 he used Mobius in another
32 wood engraving. In this work, nine red ants crawling after each other along both sides of a Mobius surface (Figure 3-22 Right).
Figure 3-22: Mobius strip painting, M.C. Escher's, (http://www.mcescher.com) Avant-garde architects like Peter Eisenman and Ben Van Berkel use Mobius for architectural design. Peter Eisenman was one of the first architects to use Mobius form to design a building in his “Max Reinhardt Haus” building. (Thulaseedas, Krawczyk, 2003) (Figure 3-23). He used Mobius strip as a volume with varied width and sliced it into many pieces and used it as an overall volume of building.
Figure 3-23: Mobius building by Peter Eisenman
33 Ben van Berkel used Mobius concept as a spatial circulation for the design of Mobius house (Ben van Berkel, 1998). Figure 3-24 shows his diagram of spatial circulation for Mobius house concept design.
Figure 3-24: Mobius house spatial circulation Diagram, based on Mobius band (van Berkel, 1998.) Eight twisted belts are working as Mobius flat surfaces that shaping the Mobius house design.
These flat Mobius surfaces are treated as walls,
ceilings, and floors that are interlinking inside spaces, (Figure 3-25, Right) with surrounding exterior, (Figure 3-25, Left) creating a spatial twist through out the house.
Figure 3-25: Left: Mobius house exterior, Right: Mobius house interior. (van Berkel, 1998.)
34
In this chapter we have looked at numerous examples of using mathematical geometry in architectural design. Fractals, mathematical curves and surfaces were introduced and their use in architecture was explored. Next I look at related work of using programming codes to generate geometric shapes and patterns.
35
Chapter 4 - Related Work This chapter examines various methods, and computing tools for generating shapes, or architectural design. This chapter reviews various computational approaches to generate shapes and patterns for design.
4-1
ArchiDNA
ArchiDNA (Kwon, 2003) uses a set of rules to define a certain design style. Common elements of a design arrangement can be described as a set of rules that is specific to a certain design style. The rules are translated into Java code to generate 2D or 3D images. ArchiDNA enables designers to use a computer to algorithmically generate shapes and forms. Kwon’s approach in ArchiDNA allows us to define a certain design style of shape generation resembling that of Peter Eisenman. (Figure 4-1)
Figure 4-1: Eisenman-like drawing generated with ArchiDNA, (Kwon, 2003)
36 ArchiDNA is inspired by Eisenman’s design for the Biocentrum building (Figure 4-2), (Kwon, 2003). One example of ArchiDNA is to use eight shapes and a series of rules to define a design style.
Figure 4-2: Axonometric view of Biocentrum building, (Eisenman, 1997) Eight shapes (2 copies each of the four DNA shape elements) were examined as design components. The shape G (ribbon) was then applied to the baseshape C (pentagon). The ArchiDNA system copies and attaches the appliershape G to every edge of the base-shape C. Figure 4-3 demonstrates the process of applying G shapes to C shapes
Figure 4-3: Applying the applier-shape G (ribbon) to the base-shape C (pentagon) (b) Applying the applier-shape G to the base-shape A (arch), (Kwon, 2003)
37 4-2 Design By Numbers John Maeda proposed DBN, “design by numbers” (Maeda, 2001), for teaching computation to artists and designers. DBN is both a programming environment and also a programming language. The language is called DBN which stands for "design by numbers".
Figure 4-4 shows the DBN
programming (right) and the display environment (Left). The programs written in the right half of DBN environment will be displayed in the left half of the environment.
Figure 4-4: DBN programming and display environment, (Maeda, 2001) DBN applies text commands as programming language to draw visual elements like points and lines. Artists and designers have used DBN to draw visual elements. Such drawings are shown in Figure 4-5.
Figure 4-5: Various shapes produced with DBN
38 4-3
FormWriter
FormWriter (Gross, 2002) is a user-friendly programming language designed especially for architects to explore generating 3D forms using algorithms. FormWriter offers a simple syntax combined with a development environment, with easy access to three-dimensional libraries of geometric shapes.
Figure 4-6 shows the FormWriter working environment. The left side of the FormWriter environment is a 3d image display window with browsing controls. The right side is an editor window for writing programming code.
Figure 4-6: FormWriter programming and display environment, (Gross, 2002)
39 Formwriter was used in a class of Islamic architecture to study traditional building and to generate shapes such as 3-D images of mosques, and star rib dome. Examples of such 3-demonsional forms generated in Formwriter are shown in Figure 4-7.
Figure 4-7: 3d models of Islamic architecture generated with form writer. (Gross, 2002)
40 4-4
ArtiE-Fract: ArtiE-Fract: Interactive Evolution of Fractals
ArtiE-Fract (Chapuis, 2000) is user-friendly software for generating fractal images by using algorithm interactively. Interaction happens by manipulating images via a specialized window.
Geometric, colorimetric and structural
modifications are available via this window (Figure 4-8.)
Figure 4-8: ArtiE-Fract window area (Chapuis, 2000)
41
The distortion tool in the ArtiE-Fract environment generates shapes that look like abstract paintings. Figure 4-9 shows variety of fractal shapes generated with ArtiE-Fract and then distorted in the ArtiE-Fract environment.
Figure 4-9: shapes generated with ArtiE-Fract (Chapuis, 2000.)
42 4-5
Objectile software
Objectile (Cache, 2003) calculates complex curves and surfaces to generate shapes.
An unlimited number of shape variations may be produced and
presented.
Objectile has developed a machining writing module that
manufactures a variety of objects using numerically-controlled machines (Figure 4-10). Objectile merges engineering, mathematics, technology, and philosophy to manufacture curved and variable forms such as sculpture, design, furniture, building components, and architecture. Objectile does this by calculating the designs of curved and variable shapes and manufacturing them using numerically-controlled machines.
A series of objects such as
doors, tables, and sculpture are calculated and manufactured by Objectile are shown in Figure 4-10. A
D
B
E
C
F
Figure 4-10: Various objects manufactured using Objectile A) sculpture, B) door, C) surface sculpture, D) door, E) surface sculpture, F) Table (www.archilab.org )
43
4-6
Processing
Ben Fry and Casey Reas initiated processing (Ben, Casey 2004). Processing is a programming language and environment for computational design. Processing environment includes a sketchbook for writing programming text that supports 2D and 3D graphics. The Processing project is made for a new audience of artist, designer, or programmer. Processing introduces programming in the context of electronic art.
The purpose is to open
electronic art concepts to a programming audience.
The goal of the
Processing concept is to create a text programming language specifically for producing interactive images; the language facilitates sophisticated visual and responsive composition. I am using Processing to manipulate programming codes and to generate geometric shapes. The following chapter will further explain the Processing environment.
44
Chapter 5 – Generating Geometric Shapes Algorithmically In this thesis Processing was used as a programming language and environment to produce 2D and 3D images. Possibilities of generating geometric shapes using mathematical algorithms were explored via Processing programming environment. This chapter also introduces the Processing working environment in order to explain how Processing is used for shape generation. Additionally, it also explains the code archiving system. The code may be stored in an environment to be used later for exploring shape generation by manipulating archived codes. It also describes a variety of geometric shapes and their applets including fractals, curves, and surfaces produced in the Processing environment. This chapter also presents a catalog of shape variety, and patterns produced using the algorithm. These shapes were also used at later time to design a library building, shown in chapter Six. A system architecture diagram and the process of shape generation using programming code are showed in Figure 5-1. Code archiving system and building design development of generated shapes as part of the system architecture is also described.
45
Figure 5-1: System Architecture Diagram 5-1
Processing System
Processing System consists of a menu, a text editor, a message area, a text area, buttons, and a display window (Figure 5-2.) Programming codes are written in the text editor area. The image generated by the code can be viewed in the display window by clicking the ‘play’ button.
Figure 5-2: The Processing environment and components
46
Function and description of the buttons in the Processing Environments are listed below: The Play button: pressing this button executes the programming code. Stop executing the program Creates a New File; Processing calls them sketches. They can also be called applets, programs, or interactive pieces. Opens a pre-existing sketch; the programming code that is written in the text editor area are called sketch. This text may be saved as an sketch in a folder called sketch folder. By pressing open button a menu will pop up and you can access to the sketch collection, which is saved in the special Processing sketch folder. You can also select example sketches written by others to modify and execute the codes.
Saves the current sketch into the Processing sketches folder. Exports the current sketch as Java applet into the Processing sketches folder To execute code; the user utilizes the play button, the stop button will terminate the execution. The new button creates a new file to write code. The open button will open a pre-existing sketch, while the save button saves
47 the sketch in a Processing sketch folder. Finally, the export button will export the current sketch as Java applet into the same folder. The Play Button The Stop Button The New File Button The Open Button The Save Button The Export Button Figure 5-3: Processing environment buttons
48 The Display Window utilizes the Cartesian Coordinate system with the origin in the top-left corner (0,0) (Figure 5-4). The screen is a array of pixels - each individually addressable by a unique (X, Y) coordinate. Adding to the value of Y moves the point downward, while adding to the value of X moves the point to the right side.
(Note: this is opposite to normal Cartesian Coordinate
System.)
Figure 5-4: Cartesian coordinates system in the Processing Environment, (proce55ing.net) Let’s now look at few lines of codes in Processing: size(200, 200); background(255); Stroke(0); translate(100,80); rotate(30); rect(30, 20, 50, 50); The codes above include display size and background, of the display window, components of stroke, translate, rotate, and rect object. It first generates a black rectangle (Figure 5-5, Left.) The display window area is set by “size” at 200 x 200 units. The “Background” function sets the background color of the Processing window. “Colors” are set by values between 0 and 255. The
49 value 255 represents white and 0 represents black. The numbers between this range representing a gray between black and white. The background in this example is white since the value is set at 255. The “Stroke” sets the pen color so stroke(0) sets the pen color to black. “Rect” specifies a rectangle – a four-sided shape with every angle at ninety degrees.
The first two
parameters of rect (30, 20, 50, and 50) set the location, the third sets the width, and the fourth sets the height. “Translate” specifies the amount to move objects in space. The x parameter specifies left/right movement; the y parameter specifies up/down movement.
For example translate (100, 80)
means moving the rectangle from original position (30, 20) to a new coordination point (130, 100).
The rectangle will move from coordination
point (30, 20) 100 units to the right and 80 units upward. The rectangle width and height are 50; “Rotate” rotates an object clockwise, as specified by the angle parameter.
Rotate (30) means rotate the rectangle for 30 degrees
clockwise. Objects are always rotated around their position relative to the origin. The result will be executed in the display window area by pressing the play button.
50
(100, 80)
(30)
(30, 20, 50, 50)
Figure 5-5: Rectangle transformation, rotation process A variety of mathematical equations useful for drawing geometric shapes was converted into Processing codes to generate shape and patterns.
The
programming codes are stored and archived in the processing sketch folder for future use to produce a variety of complex geometric shapes, such as Mandelbrot sets at a later time.
Codes saved and archived in three
categories of fractals, curves, and surfaces.
51 Figure 5-6 illustrates how users can access the archiving system in the processing environment. From the menu bar the user selects “file” and then “open” to select from the 3 primary methods such as fractals, curves, surfaces, or tessellations. Selection of a particular shapes types (e.g.: L-System) can be selected after a method (Fractals) is chosen.
Figure 5-6: Geometric Shape Code Archiving System within the processing environment
5-2
Format of Generated Shapes
The generated shapes in Processing environment may be saved as image files such as “.tiff” or exported as Java applets. A Java Applet is a short program written in Java that is attached to a web page and is executed by a web browser. Applets might be images or animations.
52 5-3
Examples of L-system Applets
L-System concept was utilized in the Processing environment to generate varieties of shapes. In this section, I will show some examples of my applets and also briefly describe the systems and how a designer may modify the codes to generate a wide variety of shapes. 5-4
Component of the L-system Applets
This section outlines and explains the L-System Processing applets. Examples of L-Systems and Turtle Graphics Systems are introduced in this section.
Figure 5-7: Geometric shape generated by using L-systems Line 1 String state = "F-F-F-F"; Line 2 String [] axioms = {"F","f","-","+"}; Line 3 String [] productions = {"F+F-F+F-F+F","f","-","+"}; Line 4 Int iterations = 4; the number of iterations (generations) Line 5 Float unit=4; the unit length Line 6 Float delta=30; the rotation angle
The shape in Figure 5-7 was produced with the use of basic L-system interpreter. For example, the alphabets are defined as (F, f, +,-). In real life this translates to:
53 F: draws a line of unit in current (right) direction f: move forward one unit in current direction, without drawing the line +: turns left one angle unit -: turns right one angle unit Code translation: In the first line of the codes listed in the previous page, the program starts by drawing a single line (F) in right direction and then turns right (-) and draws the next line. Line 2 defines the alphabets that are used within these sets of codes. Line 3 takes the specified rules and applies it to the shape that was generated by Line 1 and the alphabets in Line 2.
Line 4 constitutes the
number of times that Line 1 is reproduced based on production rules in Line 4. Line 5 specifies the length of the unit E. Line 6 defines the degree measure of the angle of rotation; which is the angle at turning point. “String” describes a sequence of alphabets used for identifying alphabets used in the program. “Int” declares a variable data type for integers that are numbers without a decimal point. “Float” declares a variable number with decimal point. The shape generated using these codes are shown in Figure 5-7. The Processing Environment does not have a built-in turtle system. One can use a simple algorithm to simulate turtle movements. simulating Turtle movement is newX=x+unit*cos(radians(zt)); newY=y+unit*sin(radians(zt)); line(x,y,newX,newY);
The algorithm for
54 In turtle algorithm, “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” is rotation angle, the turtles, starting point line is horizontal toward right. This 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”. One can generate shapes by changing the production rules, rotation angle, unit size, and number of iterations. All of the shapes shown in Figure 5-8 are generated using L-system algorithm. Modifying the statement in line 1, productions in line 3, number of iterations in line 4, unit length in line 5, or rotation angle in line 6 would result in generating variety of shapes illustrated in Figure 5-8
Figure 5-8: Example of shapes generated using the same code. By changing variables in the code
Figure 5-9 (a) shows patterns generated using the same code, but with the rotation angle Delta set to 45 and the iteration set at 4. In Figure-5-9 (b), Delta is set at 60, and the iteration is set at 4. In Figure 5-9 (c) the rotation
55 angle is set at 30 and iteration number is set at 5. These examples show that, small parametric changes in the same code can produce very different results.
Figure 5-9: a) Detla 45 Iteration 4 b) Detla 60 Iteration 4
c) Detla 30 Iteration 5
Figure 5-10 illustrates a Varity of shapes generated using L-system code in the Processing environment. All of the shapes were iterated 2 times with the same statement and production rules. The change of rotation angle resulted in the different shapes shown in Figure 5-10. The fourth iteration of the same shapes produced the shapes shown in Figure 5-11.
Figure 5-10: second iteration of the same statement and production rule with different rotation angle
Figure 5-11: Fourth iteration of the shapes shown in Figure 20
56 Here is another set of applets developed using the “turtle” L-system interpreter, but implemented via push/pop turtle logic to facilitate dynamic angle changing. The “push” function saves the current coordinate system to the stack and the “pop” restores the prior coordinate system. For example, a stack of pictures created by "pushing" pictures one at a time onto the top of the stack and in which pictures are retrieved one at a time by taking the top picture off the pile. Pop retrieves the last "pushed" orientation. Push and Pop are used in conjunction with the other transformation methods (e.g., move, rotate, scale) and may be embedded in the programming codes to control the range of the transformations. Alphabets for this system are as follows: F: draw a line of unit in current direction f: move forward one unit in current direction without drawing it +: turn left one angle unit -: turn right one angle unit [: push state ]: pop state
Figure 5-12: Example of L-System tree using the Push and Pop commands
57 We can use push/pop turtle L-systems to generate animated 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 (Figure 5-12.) 5-5
Curve Codes
A series of mathematical equation for generating curves are used to draw variety of different shapes in the Processing environment. The shapes were generated based on two different methods. In the first method the x and the y (point positions in the Cartesian coordinate system) are set to curve parameters. The shapes were generated by drawing variables (x,y) points in the Cartesian coordinate system. For example, for drawing the Astroid curve shown in Figure 5-13, “X” is set to a ! cos (t)^ 3 and ”y” is set to
a ! sin(t )^3 Where “a” is a variable value that increases the Astroid shape size, and t is an angle between -2∏ and + 2∏ (360 degrees). x=a*pow(cos(t),3); y=a*pow(sin(t),3); point(x,y)
Figure 5-13: Astroid curve
58 The second method for generating curves is based on radius (R) X is set to
R ! cos(t ) and y is set to R ! sin(t ) . For example, to draw the Right Strophoid (Figure 5-14), R was set to
a ! cos ( 2 ! t) and x and y (point coordination) are cos (t)
calculated based on “r”. “A” is a variable value that increases the size of the curve while “t” is an angle between -2∏ and +2∏. (Note: 2∏=360 )
a ! cos ( 2 ! t) cos (t) x = r ! cos(t ) y = r ! sin(t ) po int (x, y )
r=
Figure 5-14: Right Strophoid curve made using the processing codes 5-6
Mathematical Curves
Some more curve shapes were generated using the same methods. Figures 5-15, 5-16 and 5-17 show the curves and the mathematical equations that are used for generating curves. Different variation of Rhodonea curves is illustrated in Figure 5-15. The value of “a” and “b” cause curve variations. Increasing the value of “b” increases the number of loops and increasing “a” value increases the curve size.
59
Figure 5-15: The different values for variables a and b will result various shapes as shown in the above Figure Folium curve and its variations are illustrated in Figure 5-16.
The folium
expression stands for leaf-shaped curve. Three forms of the folium may be generated by altering” a” and ”b” values, the simple folium which is a folium with one leaf, the double folium is a folium shape with 2 leaves and the trifolium includes 3 leaves. Folium includes 3 leaves while “a” and “b” is equal. Increasing the “a” value enlarges 2 of the 3 leaves and reduces the third leaf size. The third leaf may reduce in size until it disappears and double folium is generated at this point. Increasing “b “value enlarges 1 of the 3 leaves and reduces the other two leaf sizes. The leave may be reduced in size until they disappear. A simple Folium is generated at this point.
Figure 5-16: The different values for variables a and b will result various folium with 1, 2, or 3 leaves
60
Figure 5-17: Different curves formula and the related curve shapes As shown in Figure 5-17, changing the value of “a” will not result in drastic changes in the shape but rather will increase or decrease the size of the curve.
61 5-7
Spirograph
Spirograph is a curve determined by three different variables “R”, “r”, and “O“, while t is an angle between -2∏ and +2∏. The curve consists of two circles; one circle is rolling around a fixed point inside the other fixed circle. The parametric equations used for drawing Spirograph shapes are as follows:
X = ( R + r ) ! cos(t ) " (r + O) ! cos(
X = ( R + r ) ! sin(t ) " (r + O) ! sin(
(R + r) ) ! t) r
(R + r) ) ! t) r
Here R is the radius of the fixed outer circle, r is the radius of the rotating inner circle, and O is the offset of the edge of the rotating circle. Figure 5-18 shows a variety of shapes generated by different values of variables R, r, O.
Figure 5-18: Spirograph shape
62 The variety of shapes generated using Spirograph equation may become very complex. Some of the shapes generated by different values of “R”, “r” and “o” are illustrated in Figure 5-19.
Some more shapes and patterns may be
generated by trying different values of “R”, “r” and “o”.
Figure 5-19: Variety of Spirograph shapes
63 5-8
Epicycloid
Epicycloid is another shape with two variables “a” and “b”. Again “t” is an angle between -2∏ and +2∏. Different values of “a” and “b” produced the shapes shown in Figure 5-20. Epicycloid is defined by:
a X = (a + b) ! cos(t ) " b ! cos(( + 1) ! t ) b a y = (a + b) ! sin(t ) " b ! sin(( + 1) ! t ) b
Using Epicycloid equation can generate shapes that are very complex. Some of the shapes generated by different values of “a” and “b” are illustrated in Figure 5-20.
Some more shapes and patterns may be generated by
manipulating the values of “a” and “b”.
Figure 5-20: Variety of Epicycloid shapes
64 5-9
Epitrochoid
Epitrochoid is a shape with three variables “a”, ”b” and “c”, while t is an angle between -2∏ and +2∏. Different values of a, b and c resulted variety of Epitrochoid shapes shown in Figure 4-20. The Epitrochoid parameters that were used to generate shapes are: a X = (a + b) ! cos(t ) " c ! cos(( + 1) ! t ) b a y = (a + b) ! sin(t ) " c ! sin(( + 1) ! t ) b
Using Epitrochoid equation can generate complicated shapes. Some of the shapes generated by different values of “a” and “b” and “c” are illustrated in Figure 5-21. More shapes and patterns may be generated by trying different values of “a”, “b” and “c”.
Figure 5-21: Variety of Epitrochoid shapes
65 5-10 Hypocycloid Hypocycloid is another curve shape with three variables “a”, “b” and “c”, while t is an angle between -2∏ and +2∏. Different values of a, b and c generated the Hypocycloid shapes shown in Figure 5-22. The Hypocycloid parameters that were used to generate these shapes are: a x = (a " b) ! cos(t ) " b ! cos(( " 1) ! t ) b a y = (a " b) ! sin(t ) " b ! sin(( " 1) ! t ) b
Shapes produced by means of Hypocycloid equation are complicated. Some of the shapes generated by different values of “a” and “b” are illustrated in Figure 5-22.
Some more shapes and patterns may be generated by
changing the values of “a” and “b”.
Figure 5-22: Variety of Hypocycloid shapes
66 5-11 Supershape Supershape formula was used to generate a wide variety of shapes, ranging from circles and rectangles to organic forms similar to those found in nature, such as flowers and stars. By selecting appropriate values of n1, n2, and n3, one can generate various circles, ellipses, rectangles, and other symmetric forms. The Super Formula equation that was used to generate the shapes in Figures 5-23 through 5-25 is:
1 n1 1 m = " cos( " ! ) r a 4
n3
1 m + " sin( " ! ) b 4
n3
Figure 5-23: Varity of Supershapes. Left: where m=2, n1=100, n2=10, & n3=10. Right: where m=10, n1=100, n2=10, & n3=10 Varying the parameters in the Supershape formula produced the shapes shown in Figure 5-23. There are endless possibilities to the variety of shapes that may be generated using Super formula. Super formula is capable of generating a very wide range of shapes from very geometric forms like rectangle and circle to more organic forms that look like stars and flowers.
67 Predictions of the features of generated shapes are due to complexity of the formula.
Altering the “m”, “n1”,” n2”,” n3” values in the formula would
generate very different shapes. Here I provide a catalog of various super shapes and the related values that generated specific shapes to show the trends of shape.
By looking at these examples, it appears that feature
variations increase the value of “m” will produce shapes with more number of edges.
Figure 5-24: Variety of super shapes
68
Figure 5-25: Variety of super shapes
69 The super shapes shown in Figure 5-26 resemble organic forms and flowers:
Figure 5-26: variety of super shapes
70
As the value of “m” increases, the shapes become increasingly complex. Prediction of the finished product become more difficult with very high values of “m”. One might consider that the resulting shapes appeared to be beyond human imaginations (Figure 5-27.)
Figure 5-27: Super Shapes source author built with Processing
71
5-12 Surface Codes Series of mathematical equations useful for generating surfaces were used to draw 3-dimentional shapes in the Processing environment. Variables x, y, and z (the points positions in the 3-dimentional Cartesian coordinate system) are set as surface parameters. Surfaces are generated by drawing an array of points (x,y,z) in the Cartesian coordinate system. For example, to draw the Astrodal Elipsod (Figure 5-28), the following code is used
x = (a ! cos(u ) ! cos(v))^3 y = (b ! sin(u ) ! cos(v))^3 z = (c ! sin(v))^3 Variables “a”, “b” and “c” are values that will cause change in the shape of the form, while “v” and “u” are angles between -2∏ and +2∏. The generated surface using Astrodal Elipsod equation looks like a trumpet shown in Figure 5-28.
Figure 5-28: Astrodal Elipsod
72 More surfaces such as Mobius, Quatric surface, Spherical Nephroid, that were generated with the same approach are shown in Figures 5-29. The parametric equations used for generating each surface are also shown in the Figure 5-29. Quatric surface might look like a sphere where variable “a”, "b” and “c” are set to an equal value. It looks like a bowl with a stand where “a”, “b” and “c” are not set to equal values. Increasing or decreasing “a” or “b” would stretch bowl radius horizontally or vertically.
Figure 5-29: Surfaces and the parameters that are used to generate surface
73 Lamiscate, Astrodal Elipsod, Cone and Eliptic cylinder are surface examples illustrated in Figure 5-30 different values of “a”, “b” or “c” will stretch the surfaces.
Figure 5-30: Surfaces and the parameters used to generate them
74
Chapter 6 - Building Design with Code Generated Shapes The diversity and complexity of generated shapes in chapter 5 became an inspiration for an architectural building design. Here I explore an architectural design of a library building by using the code generated shapes as architectural design elements.
6-1
Parametric Design
Generating the curves in the previous chapters inspired me to apply these techniques to the design of buildings. A series of shapes generated with Super Formula was chosen and used to develop a variety of building designs. One of these designs was further developed to design a library building. Figure 6-1 shows the Supershapes that were used for the design.
Figure 6-1: variety of super shapes that was used for the design
75
Figure 6-2 shows the floor plan view (top) of 3 different buildings using the Supershapes. The shapes used for each design were similar yet different and consequently the design created 3 different, yet similar buildings.
Figure 6-2: Top view of building
76
6-2
Drawing Methods
Various 2-D Supershapes were generated for the building design.
By
changing the parametric values in the Supershape program in the Processing programming environment, different shapes were generated. Some of the shapes were chosen and further explored for the design development process. The generated shapes were exported into AutoCAD environment. The shapes were traced over in the AutoCAD environment. The shapes were then imported to 3D Studio Max environment and then extruded with different heights to form 3-dimensional shapes. (Figure 6-3)
Figure 6-3: 3d extrusion process
77 Extruded shapes were then combined or subtracted from each other through various stages of design process to shape a building form model. Figure 6-4 shows two different views of a building model developed with this process
Figure 6-4: Building model Some more building models were developed by combining different composition of a variety of Supershapes, these models are shown in Figure 6-5.
Figure 6-5: Different building models developed by various compositions of Supershapes.
78 The building models were further developed for a library building design. The floor plans for this library were drawn in Auto CAD. The floor plan contains spaces such as Entrance, Reception Area, Library, and etc. Different numbers labeled in the floor plans is demonstrated as following. Figure 6-6 shows the first and second floor plan for a library.
First floor plan
Second floor plan 1-Entrance 2-Reception Area 3-Library 4-Sitting areas 5-Reading Areas 6-Restrooms 7-coffee Shop
Figure 6-6: Library floor plans
A 3-D model of interior circulations was developed in AutoCAD 3-D environment showing partition walls and stairs as shown in Figure 6-7.
79
Figure 6-7: 3d model of library spatial circulation In this library design Supershapes were used for 2-D and 3-D form generations.
The extruded 3-D shapes formed the building models and
interior spaces. The 2-D Supershapes were used as patterns for the floor plans and to further define and decorate interior spaces. Figure 6-8 shows the patterns used for the floor plan pattern design.
Figure 6-8: floor pattern design
80
The 2-D transformation of Supershapes into 3-D created interesting 3dimensional spaces.
Figure 6-9 shows the building dome created by
transforming 2-D Supershapes into 3-D architectural spaces.
Figure 6-9 Building interiors
81
Chapter 7 - Conclusions and Future Work In this thesis various architectural examples from different eras and regions was examined. The use of geometric shapes and methods for architectural design were explored. Mathematical parameters of geometric shapes such as fractals, curves, and surfaces were further explored and investigated. An extensive collection of shapes using these methods were generated. A collection of the generated shapes then were used to develop an architectural design of a library building. Generating geometric shapes by using mathematical equation enables the designers to explore possibilities of creating a wide range of shapes and patterns.
This method offers a new approach to draw and explore the
understanding of geometry and the generation of geometric shapes. These generated geometric shapes range from simple shapes like rectangles and circles, to more complex shapes like Islamic lattice patterns as well as organic forms similar to the shapes and forms in the nature like flowers and stars. Shapes generated using parametric methods have a wide range of variations and often yielded surprising outcome. Although, such process may seem cumbersome and time consuming, but in return the wide range of shapes generated were quite interesting. The drawing production process of complex shapes was faster and easier than using traditional methods of drafting. This method of using programming codes to generate shapes provided a vast library of interesting shapes that can be useful and interesting for architectural design. This thesis was a fun journey of exploration for me.
82
The parametric methods used for generating shapes are useful for early design stages. For future work, I would like to develop a parametric design system that supports the designer throughout the entire design process. The shapes generated parametrically will be automatically transformed to 3-D shapes and rendered in the same environment. 3D shapes and architectural floor plans that can be modified and redefined in the same environment would be a useful tool.
83
References Barnsley, M: Fractals Everywhere, 1988. Boston, MA, Academic Press Cache, B.: Objectile, 1999, Archilab, Chapuis, J., and Lutton, E.: Arti-Fract: 2001, Interactive Evolution of Fractals, 4th International Conference on Generative Art, Milan, Italy, pp: 2-12 Cook, T.: 1979, The Curves of Life, Dover Publications, Inc., New York. Curve, Britannica Concise Encyclopedia: 2004, Encyclopedia Britannica Premium Service, Escher, M.: Escher the Graphic work, 1989, Honenzollernring, Germany. Eisenman, P.: 1987, House of Cards, Oxford University Press, Oxford. Eisenman, P.: 1997, Chora L Works, The Monacelli Press, New York. Eisenman, P.: 1999, Peter Eisenman Diagram Diaries, Universe Publishing, New York, pp. 27 – 43. Fry, B. and Reas, C.: Processing, www.proce55ing.net, MIT Media Lab Gross, M.D.: 2001, FormWriter: A Little Programming Language for Generating Three-Dimensional Form Algorithmically. Computer Aided Architectural Design Futures 2001, Kluwer Academic Publishers. Gielis, J.: 2003, A Generic Geometric Transformation That Unifies a Wide Range Of Natural and Abstract Shapes, American Journal of Botany 90(3), pp: 333–338. Ibrahim, M. and Krawczyk, R.: 2001, Generating Fractals Based on Spatial Organizations, College of Architecture, Illinois Institute of Technology, Chicago, IL., pp: 3-9 Kwon, D.: 2003, ArchiDNA: A Generative System for Shape Configuration, Department of Architecture, Master thesis University of Washington, Seattle, WA. Lindenmayer A.: 1968, Mathematical models for cellular interaction in development, Journal of Theoretical Biology, Volume 18, pp: 280-315 Lindenmayer, A.: 1990, The Algorithmic Beauty of Plants Springer-Verlag, New York Lorenz, W: 2003, Fractals and Fractal Architecture, Lynn, G.: 1998, Animate Form, Princeton University Press, Princeton, NJ.
84 Lynn, G.: 2003, Architecture for an Embryologic Housing, Birkhauser-Publishers for Architecture, Basel, Switzerland. Mandelbrot, B.: 1983, The Fractal Geometry of Nature, Freeman, New York. Maeda, J.: 1999, Design By Number, The MIT Press paper edition. Cambridge, MA. Nouvel, J.: 1990, Institut Du Monde Arabe, Chronicle Books, Paris, France. Rashid, K.: 2003, International Design Yearbook 2003, Laurence King Publishing, London, England. Papert, S.: 1980, Mindstorms, children, computers and powerful ideas, Basic Books, New York. Peterson, I.: 2003, A Geometric Superformula, Science News Online, Week of May 3, 2003; Vol. 163, No. 18, pp: 333-338 Processing, http://proce55ing.net/ Ostwald, J.: 2001, Fractal Architecture: Late Twentieth Century Connections between Architecture and Fractal Geometry, Nexus Network Journal, vol. 3, no. 1, pp: 437-453, (Winter 2001), http://www.nexusjournal.com/Ostwald-Fractal.html O'Connor, J. and Robertson, E.: 2000, Famous Curve Index, Tennant, R.: 2003 Islamic Constructions: The Geometry Needed by Craftsmen, International Joint Conference of ISAMA, the International Society of the Arts, Mathematics, and Architecture, pp. 459-463 Tobias, H.: 2003, Disks That Are Double Spiral Staircases, AMS, Volume 50, Number 3, March 2003, pp. 327-339 Thornburg, D.: 1986. Beyond Turtle Graphics, Addison-Wesley Publishing, Boston, Massachusetts. Thulaseedas, J, and Krawczyk, R.: 2003. Möbius concepts in architecture. ISAMA/Bridges, 2003 Conference: Mathematical Connections in Art, Music, and Science. July 23-25, 2005. Granada, Spain, pp. 1-8
