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Home , Archimedean solid, Catalan solid, Disphenoid, Geodesic dome, Pentagon, Platonic solid

UNIVERSITY OF CINCINNATI

Date:______February 16, 2005

I, ______,Nicola Laila D’souza hereby submit this work as part of the requirements for the degree of: Master of Architecture

in: the School of Architecture & Interior Design at the College of Design, Architecture, Art & Planning

It is entitled: Natural Forms Through and Structure: Design of the Parachute Pavilion

This work and its defense approved by:

Chair: ______Barry Stedman ______Robert Burnham ______Thomas Bible ______

Natural Forms Through Geometry And Structure : Design of the Parachute Pavilion.

A thesis submitted to the Division Of Research and Advanced Studies of the University of Cincinnati

In partial fulfillment of the requirements for the degree of

Master of Architecture

In the School of Architecture and Interior Design of the College of Design, Architecture, Art and Planning

2005 by Nicola Laila D’souza Bachelor of Architecture, University of Mumbai, 1999

Thesis Committee:

Barry Stedman Robert Burnham Thomas Bible ABSTRACT “Mathematics is only a means for expressing the that govern phenomena” stated Albert Einstein. Similarly, geometry is only a means of defining the laws that govern natural structures, both organic and inorganic. Hence it is a powerful and ubiquitous explanator of form in the natural world.

Nature is abundant with the most profound examples of delicate strength. All natural forms have an intrinsic geometrical order with an inherent flexibility for variable outcomes. This thesis is inspired by the geometrical order that is responsible for beauty, variety, strength, stability, functionality and economy in nature. It hypothesizes that the geometry that has shaped the natural world, can be used at an intrinsic level to create form in the built environment.

The result of this exploration is the Golden Tetrahedral System. It is a triangulated and dimensionally stable geometric system for light weight construction. It is derived from the geometry of the Golden Section. Six tetrahedral blocks, composed from four different lengths and triangular faces, form the basic units of the system. These versatile blocks can be combined to result in planar, spherical, cylindrical and other curved surfaces. With the use of this system, it is also possible to create naturally curved surfaces like those found in a flower petal or a leaf. The modularity and use of repetitive elements enables economic manufacture and construction.

The system is tested out in the design project, ‘The Parachute Pavilion’. It is used to form a space frame structure that derives its form from a collapsing parachute, frozen in time, just before it meets the ground at the base of the tower in whose shadow it stands. The intricate complexity of the geometry in this building is integrated with simplicity of form and construction technique so as to be easily constructable.

3 4 ACKNOWLEDGEMENTS

For their love and support, I would like to thank my family; my parents, Lawrence D’souza and Lynette D’souza and my sisters and brothers-in-law; Sunita D’souza and Paul Crosser, Liezel D’souza and Are Andreson.

For their academic guidance, I would like to thank my thesis committee: Barry Stedman, Robert Burnham and Thomas Bible. I would also like to thank Vincent Sanzalone and Masud Taj and all the faculty at The School of Architecture and Interior Design and Rizvi College of Architecture.

For the professional learning that I have had, I would like to thank the staff from Kamal S. Malik (Bombay), Zimmer Gunsul & Frasca Partnership (Washington DC) and Perkins Eastman (New York)

I would like to thank Steven Albert, Kevin Schreur and Brandon Kelly for their support and help.

For all his support, love and patience I would like to thank John Gallagher.

I would like to thank all the scientists and mathematicians who have tried to explain the wonders in nature.

And of course, the creator of it all.

5 TABLE OF CONTENTS. Abstract 2 Chapter III: The & Fivefold Acknowledgements 4 . Table of Contents 5 Introduction 59 List of Tables 7 History of the Golden Ratio 60 List of Illustrations 8 Mathematical Properties of the Golden Ratio 62 Introduction 17 Observations of the Golden Ratio in Nature 64 Conclusions 66 Chapter I: Plane and Introduction 21 Chapter IV: Penrose Tilings and . Concepts of Space and Geometry 22 Introduction 67 Dimensional Aspects of Space 24 Penrose Tilings 68 Planar Geometry 25 Quasicrystals 70 Curves 26 Surfaces from Geometry 28 Planar Surfaces 72 Two Dimensional 30 Non Planar Surfaces 74 Three Dimensional Geometry Closed Seed Patterns 76 Classification 34 Open Seed Patterns 82 Three Dimensional Tessellations 36 Replacing Planes by a Pattern 84 Polyhedra 38 Joining of the Modules Together 85 Curved Surfaces 43 Use of Quasicrystal Geometry in Art & Dimensional Stability 44 Architecture 87 Dimensionally Stable Configurations in Conclusions 88 Architecture The Octet 46 Chapter V: The Golden Tetrahedral System. Geodesic 48 Introduction 89 Twisted Domes 50 The Golden Tetrahedral System 90 Conclusion. 52 Mathematical Formulae 91 The Geometry of the System 101 Chapter II: Aperiodic Geometry Basic Combinations of the Tetrahedra 102 Introduction 53 Surfaces with Uniform Curvature 103 Projections from Higher Dimensional Space. 54 Surfaces with Non-Uniform Curvature 105 The Infinite Family of Aperiodic Tessellations. 56 Conforming the System to a Predetermined 106 Conclusion: The Fivefold Case 58 Form. Conclusions 107 6 Bibliography 109

Appendix I: Space Frame Systems Introduction 113 Concepts of SpaceFrames 114 Advantages of SpaceFrames 115 Disadvantages of SpaceFrames 116 Types of SpaceFrames 117 Connector Plates Hub Type 118 Spherical Hub Type 119 Extruded Hub Type 120 The N55 Spaceframe (Hub-less Type) 121 The Aurodyne Hyperframe System 123 Bambutec Technology 124 Conclusion. 125

Appendix II: The Competition Brief Introduction 126 Coney Island History. 127 Site History. 129 Site Description 130 Program Requirements 132 Site Plan 133 Site Images 134

7 LIST OF TABLES Chapter I: Plane and Solid Geometry

Table 1.1: Operatons for the Creation of Three Dimensional Objects. Table 1.2: The relation between the values of p and q and the geometry of the {p,q}

Chapter II: Aperiodic Geometry

Table 2.1 : A table showing the infinite family of rhombi projected from ‘n’.

Chapter IV: Penrose Tilings and Quasicrystals.

Table 4.1 The 17 Different Possible Around a Node

Chapter V: The Golden Tetrahedral System.

Table 5.1: Comparision of the lengths Obtained with Different Starting Assumptions Table 5.2 : Ratios of the Different Lengths.

8 LIST OF ILLUSTRATIONS Introduction as Addison Wesley 0.1 Flowers exhibiting Geometry. Royalty Free images from Burton Networks 1.3: Unfolding of Spatial . Retrieved on May 12, 2005 from http:// Keith Critchlow. Order in Space, pg 7 www.burtonnetworks.com/downloads/ photos/flowers/ 1.4.a: Open and Closed Curves. Eric W. Weisstein. “Closed Curve.” Retrieved Chapter I: Plane and Solid Geometry May 12,2005 From MathWorld--A Wolfram 1.1.a: Limit III Web Resource. http:// M. C. Escher, pg 97 mathworld.wolfram.com/ClosedCurve.html Photograph from the Catalogue compiled by the authors, F.H. Bool, J.L. Locher & F. Wierda. 1.4.b: Simple and Non Simple Curves. Eric W. Weisstein. “Simple Curve.” Retrieved 1.1.b: Circle Limit IV From MathWorld--A Wolfram Web Resource. M. C. Escher, pg 98 http://mathworld.wolfram.com/ Photograph from the Catalogue compiled by SimpleCurve.html the authors, F.H. Bool, J.L. Locher & F. Wierda. 1.4.c: Jordan and Non Jordan Curves. 1.2.a: Flat Eric W. Weisstein. “Jordan Curve.” Retrieved http://astro.uwaterloo.ca/~mjhudson/teaching/ May 12, 2005 from MathWorld--A Wolfram sci238/notes/node17.html Web Resource. http://mathworld.wolfram.com/ Copyright©2004 Pearson Education, Publishing JordanCurve.html as Addison Wesley 1.5: Circle. 1.2.b: Spherical Space Eric W. Weisstein. “Circle.” Retrieved May http://astro.uwaterloo.ca/~mjhudson/teaching/ 12, 2005 from MathWorld--A Wolfram Web sci238/notes/node17.html Resource. http:// mathworld.wolfram.com/ Copyright©2004 Pearson Education, Publishing Circle.html as Addison Wesley 1.6: Conics. 1.2.c: Eric W. Weisstein. “Conic Section.” Retrieved http://astro.uwaterloo.ca/~mjhudson/teaching/ May 12, 2005 from MathWorld--A Wolfram sci238/notes/node17.html Web Resource. http://mathworld.wolfram.com/ Copyright©2004 Pearson Education, Publishing ConicSection.html 9 1.7: Ellipse. 1.13: Skew Polygons. Eric W. Weisstein. “Ellipse.” Retrieved May Robert Williams. Natural Structure, pg 34 12, 2005 from MathWorld--A Wolfram Web Resource. http:// mathworld.wolfram.com/ 1.14: Regular Tessellations. Ellipse.html & 1.15: Archimedian Tessellations. Eric W. Weisstein. “Tessellation.” Retrieved 1.8: Parabola. May 12, 2005 from MathWorld--A Wolfram Eric W. Weisstein. “Parabola.” Retrieved May Web Resource. http://mathworld.wolfram.com/ 12, 2005 from MathWorld--A Wolfram Web Tessellation.html Resource. http://mathworld.wolfram.com/ Parabola.html 1.16: Duals of Regular Tessellations. & 1.17: Duals of Archimedian Tessellations. 1.9: Hyperbola. Eric W. Weisstein. “Dual Tessellation.” Retrieved Eric W. Weisstein. “Hyperbola.” Retrieved May 12, 2005 from MathWorld--A Wolfram May 12, 2005 from MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ Web Resource. http://mathworld.wolfram.com/ DualTessellation.html Hyperbola.html

1.18: Demi-Regular or Polymorph Tessellations. 1.10: Polygons. Eric W. Weisstein. “Tessellation.” Retrieved Eric W. Weisstein. “.” Retrieved May May 12, 2005 from MathWorld--A Wolfram 12, 2005 from MathWorld--A Wolfram Web Web Resource. http://mathworld.wolfram.com/ Resource. http://mathworld.wolfram.com/ Tessellation.html Polygon.html

1.19: Rotations. 1.11: Regular Polygons. Eric W. Weisstein. Retrieved May 12, 2005 Eric W. Weisstein. “.” Retrieved From MathWorld--A Wolfram Web Resource. May 12, 2005 from MathWorld--A Wolfram http://mathworld.wolfram.com/ Web Resource. http://mathworld.wolfram.com/ RegularPolygon.html 1.20: Extrusions. Mandara - The World of Uniform Tessellations 1.12: Metamorphpsis of Regular Polygons. Retrieved May 12, 2005 from http:// Keith Critchlow, Order in Space, pg 33 www2u.biglobe.ne.jp/~hsaka/ mandara/index.html 10 1.21:Revolutions. 1.28: Catalan Solids. Eric W. Weisstein. “Torus.” Retrieved May 12, Eric W. Weisstein. “.” Retrieved 2005 from MathWorld--A Wolfram Web Resource. May 12, 2005 from MathWorld--A Wolfram http://mathworld.wolfram.com/Torus.html Web Resource. http://mathworld.wolfram.com/ CatalanSolid.html 1.22:Extrusion and . & 1.23:Spherical, Euclidean & Hyperbolic 1.29: Process Illustrating the of a Tessellations. Mandara - The World of Uniform Tessellations , Space and symmetry, pg 84 Retrieved May 12, 2005 from http:// Photograph by Doug Kendall www2u.biglobe.ne.jp/~hsaka/ mandara/index.html 1.29: Process Illustrating the stellation of a 1.24: Platonic Solids. Shapes, Space and symmetry, Holden, Alan. Shapes, Space and symmetry, pg 1 Photograph by Doug Kendall Photograph by Doug Kendall 1.31: Stellated Dodecahedron. 1.25: Study illustrating why there cannot be more 1.32: Stellated Icosaahedron. than five regular polyhedra. Exhibit, catalogue, and web site made possible Holden, Alan. Shapes, Space and symmetry, pg 1 by a University of Arizona award to the Photograph by Doug Kendall Department of Mathematics for outstanding achievement in undergraduate education, 1.26: Archimedian Solids 1994. Rawles, Bruce. Home Page. http://math.arizona.edu/~models/ Retrieved May 12, 2005 from Stellated_Polyhedra/ http://www.intent.com/sg/ 1.33: Kepler-Poinsot Solids. Retrieved May12, 1.27: Quasi regular polyhedra. 2005 from http://www.math.unb.ca/~barry/ Holden,40,41 Alan. Shapes, Space and symmetry, pg figures/kep.htm

Photograph by Doug Kendall 1.34: Other Stellated Polyhedra. http://home.aanet.com.au/robertw/ .html Copyright © 2001-2005, Robert Webb. 11 1.35: Polyhedral Packing of . equation. . De , Sketch by Compilation of images from : Eric W. . Retrieved May 12, 2005 from Weisstein. Retrieved May 12, 2005 from http://www.georgehart.com/virtual-polyhedra/ MathWorld--A Wolfram Web Resource. http:// figs/leonardo-cubes.jpg mathworld.wolfram.com/

1.36: Polyhedral Packing of Lord Kelvin’s (The following illustrations have the same source tetrakaidecahedron. and are all credited after Figure 1.45) Synergetic Geometry of R. , Edmondson, Amy C. A Fuller Explanation: The 1.42: Dimensionally Stable Platonic Solids. http://www.angelfire.com/mt/marksomers/ 1.43: Other Stable Deltahedra. Fig12.4.html 1.44: Stable Configurations based on Tetrahedra. 1.45: Other Stable Deltahedra. 1.37: Polyhedral Packing of Sphenoid Images by the Author. Created using AutoCad Hendecahedra. software. Inchbald, Guy. Five space-filling polyhedra. The 3-D views of polyhedra were created by 1.46: Alexander Graham Bell and the Octet Truss. WimpPoly and Polydraw software from Fortran & 1.47: The first built Octet Truss. Friends, PO Box 64, Didcot, Oxon OX11 0TH. Retrieved May 12, 2005 from http:// Retrieved May 12, 2005 from www.grunch.//bell.html http://www.queenhill.demon.co.uk/ The photographs are copyrighted by the polyhedra/five_sf/five.htm National Geographic Society.

1.38: Curved Surfaces of Revolution. 1.48: The geometry of the Octet Truss. Eric W. Weisstein. “Surface of Revolution.” Retrieved May 12, 2005 fromhttp:// Retrieved May 12, 2005 from MathWorld--A www.grunch.net/synergetics/ivm.html Wolfram Web Resource. http:// mathworld.wolfram.com/SurfaceofRevolution.html 1.49: Stable with a corrugated profile. 1.39: Curved Surfaces derived by revolving an & 1.50: Stable Configuration with voids. open curve around an axis. Images by the Author. Created using AutoCad & 1.40: Curved Surfaces derived by revolving a software. closed curve around an axis. & 1.41: Curved Surfaces derived from a parametric 12 1.51: Architect Buckminister Fuller and a from http://www.geom.uiuc.edu/~banchoff/ with his patented geodesic system. Flatland/ Photograph from the website of the ‘Buckminister Fuller Institute’ Retrieved May 2.3: A Four-Dimensional object passing through 12, 2005 from http://www.bfi.org/domes/ three-dimensional space takes on the index.htm appearance of various polyhedra as it passes through space. 1.52: Geodesic domes from an . Hesse,in Three Bob. Dimensions. Viewing Four-dimensional Objects Retrieved May 12, 2005 from http:// retrived May 12, 2005 www.arbeit.ro/icosa/icosa.htm from http://www.geom.uiuc.edu/docs/forum/ / 1.53: Geodesic domes from a Cuboctahedron. 2.4: Increasing Dimensions. 1.54: Construction Process of a Twisted Dome Saltsman,the Fourth Eric. Dimension? The Fourth Dimension: What is & 1.55: Twisted Domes from non convex deltahedra retrived May 12, 2005 from http://www.geocities.com/ & 1.56: Family of Twisted Domes derived from a CapeCanaveral/7997/index.html , and and an Icosahedron. 2.5: Octagonal Ammann-Beenker Tiling: Aperiodic Gailiunas, Paul. Twisted Domes, Retrieved May Rhombic Tiling for n=4. 12, 2005 from F. Gähler. Aperiodic colored tilings. Retrived http://web.ukonline.co.uk/polyhedra/Twist.pdf May 12, 2005 from http://www.itap.physik.uni-stuttgart.de/ Chapter II: Aperiodic Geometry ~gaehler/tilings/oct.html 2.1: Pinwheel Aperiodic Tessellation.(by Danzer) F. Gähler. Aperiodic colored tilings. Retrived 2.6: Aperiodic Rhombic Tiling for n=5. May 12, 2005 from F. Gähler. Aperiodic colored tilings. Retrived http://www.itap.physik.uni-stuttgart.de/ May 12, 2005 from ~gaehler/tilings/pin.html http://www.itap.physik.uni-stuttgart.de/ ~gaehler/tilings/pen.html 2.2: A takes on the appearance of ofof differentmany dimensions, sizesas it passes through a plane. 2.7: Dodecagonal Shield Tiling: Aperiodic Modified Abbott, Edwin A. , 1884. Flatland: A romance Rhombic Tiling for n=6. retrived May 12, 2005 F. Gähler. Aperiodic colored tilings. Retrived 13 May 12, 2005 from http://www.itap.physik.uni-stuttgart.de/ 3.4 The Isosceles and the ~gaehler/tilings/sh.html pentagram and from which they can be derived. 2.8: Variations of the Aperiodic Rhombic Tiling for Knott,and the Ron(Dr) Golden . section:Two-dimensional Fascinating Geometry Flat Facts n=6. about Phi. F. Gähler. Aperiodic colored tilings. Retrived May Retrived May 12, 2005 from 12, 2005 from http://www.mcs.surrey.ac.uk/Personal/ (Dodecagonal Wheel Tiling) R.Knott/Fibonacci/phi2DGeomTrig.html http://www.itap.physik.uni-stuttgart.de/~gaehler/tilings/wh.html. (Dodecagonal Socolar Tiling) 3.5 The appearance of fivefold symmetry in http://www.itap.physik.uni-stuttgart.de/~gaehler/tilings/soc.html. flowers. (Dodecagonal Plate-Tiling) Hargittai,535, 539 István. Fivefold symmetry. pg 532, http://www.itap.physik.uni-stuttgart.de/~gaehler/tilings/pl.html . World Scientific, 1992.

2.9: Vector Star Method for the derivation of the 3.6 Phyllotaxis and the Golden . three Rhombi for the n=7 case. Hargittai, István. Fivefold symmetry. pg 18. Hargittai, István. Fivefold symmetry. pg 101. World Scientific, 1992. World Scientific, 1992. 3.7 Other appearances of the Golden angle in plant Chapter III: The Golden Ratio & Fivefold Symmetry. life 3.1 Church with a pentagonal Plan Knott,Nature. Ron(Dr) Fibonacci Numbers and Hargittai, István. Fivefold symmetry. pg 242. Retrived May 12, 2005 from World Scientific, 1992. http://www.mcs.surrey.ac.uk/Personal/ R.Knott/Fibonacci/fibnat.html 3.2 : sacred Proportions of the Human Body by Leonardo Da Vinci 3.8 Fivefold Symmetry and the Golden ratio as Hargittai, István. Fivefold symmetry. pg 21. observed in human and animal life. World Scientific, 1992. The Golden Section in Nature Copyright 1997-2005 GoldenNumber.net 3.3 Le Modulor: A harmonic proportioning system Retrived May 12, 2005 from based on the Golden ratio. http://goldennumber.net/nature.htm. Images Livio, Mario. The Golden Ratio. pg 173. courtesy of Gary B. Meisner. Broadway Books, 2002. 14 Chapter IV: Penrose Tilings and Quasicrystals. 4.6: An Electron Diffraction Pattern With Fivefold 4.1: Aperiodic Rhombic Penrose Tilings. Symmetry Exhibited by Quasicrystals F. Gähler. Aperiodic colored tilings. Retrived Lifshitz,Condensed Ron . Quasicrystals Physics. California- Introduction. May 12, 2005 from Institute of Technology http://www.itap.physik.uni-stuttgart.de/ . Retrived May12 ~gaehler/tilings/pen.html from http://www.cmp.caltech.edu/~lifshitz/ quasicrystals.html 4.2: The Second Set of Penrose Tiles Made up of Six Different Shapes 4.7: Quasicrystal Geometry and Methods of tiles Steve Edwards. and Penrose Spacefilling with Quasicrystals. . Retrived May 12, 2005 from Inchbald, Guy. A 3-D Quasicrystal Structure, http://www2.spsu.edu/math/tile/ http://www.queenhill.demon.co.uk/polyhedra/ aperiodic/penrose/penrose1.htm quasicr/quasicr.htm

4.3: The Two pentagram Isosceles Triangles (The following illustrations have the same source Knott,and the Ron(Dr) Golden Two-dimensional section: Fascinating Geometry Flat Facts and are all credited after Figure 4.23) about Phi. Retrived May 12, 2005 from 4.8: The Original Quasicrystals and their http://www.mcs.surrey.ac.uk/Personal/ Dimensionally Stable Versions R.Knott/Fibonacci/phi2DGeomTrig.html 4.9: Flat Planar Surfaces Formed by Quasicrystals 4.10: Corrugated Planar Surfaces Formed by 4.4: The and Dart Quasicrystals. F. Gähler. Aperiodic colored tilings. Retrived 4.11: Patterned Planar Surfaces Formed by May 12, 2005 from Quasicrystals. http://www.itap.physik.uni-stuttgart.de/ 4.12: Transformation of a 2D Penrose Tiling ~gaehler/tilings/kitedart.html to a Patterned Planar Surface Formed by Quasicrystals 4.5: The Rhombic Penrose Tiling 4.13: An Open Seeded Penrose Tiling and its F. Gähler. Aperiodic colored tilings. Retrived associated Patterned Planar Surface Formed May 12, 2005 from by Quasicrystals. http://www.itap.physik.uni-stuttgart.de/ 4.14: Changing the Scale of the Pattern by ~gaehler/tilings/pen.html Repeating the Symmetry around a node to 15 Force it out of a Plane. System. 4.15: Closed seed pattern with Five and Ten Planes 5.7: Random Curved Surfaces Described by the Meeting at a Node. System. 4.16: Closed Seed Patterns. 5.8: An Anticlastic Surfaces Described by the 4.17: Open Seed Patterns. System. 4.18: Surfaces Derived from Open Seed Patterns. 5.9: Conforming the system to approximate any 4.19: Replacing the planes of closed Seed Pattern Predetermined Form Four and Six with a Pattern. 5.10: Shadow Patterns. 4.20: Replacing the planes of Closed Seed Pattern Images by the Author. Created using AutoCad Four by Two Planes With a Pattern. software. 4.21: Joining the Modules Together. 4.22: Joining modules with a Pattern Together. Appendix 1: Space Frame Systems. 4.23: Joining parts of Modules with a Pattern 6.1: The Unistrut System. Together. Borrego,Skeletal FrameworksJohn. Space andGrid Stressed-SkinStructures: Images by the Author. Created using AutoCad Systems, pg 33, 53 software. . The MIT Press, 1968.

4.24: Art Work By Portraying Higher 6.2: The Nodus Hub. Dimensional Space. Chilton, John. Space Grid Structures, pg 37. Robbin, Tony Images retrieved May 12, 2005 Oxford: Architectural Press, 2000. from http://tonyrobbin.home.att.net/ 6.3: The Mero System. (The following illustrations have the same source Borrego,Skeletal FrameworksJohn. Space andGrid Stressed-SkinStructures: and are all credited after Figure 5.10) Systems, pg 18 . The MIT Press, 1968. Chapter V: The Golden Tetrahedral System. 5.1: The Four Basic Lengths of the System. 6.4: The Triodetic System. 5.2: The Four Basic Triangular Faces of the Borrego,Skeletal FrameworksJohn. Space andGrid Stressed-SkinStructures: System. Systems, pg 88, 89, 91 5.3: The Four Basic Tetrahedra of the System. . The MIT Press, 1968. 5.4: Combinations of The Four Basic Tetrahedra. 5.5: Domes Described by the System. 6.5: The Geometrica System. 5.6: Circles, Cylinders and Vaults Described by the Photographs from the company website

16 retrieved May 12,2005 from http:// 7.9: View of the Site and Parachute Tower from www.geometrica.com/Architectural/Index.html the Beach.

6.6: The N55 Spaceframe. Photographs and images from the competition website & 6.7: The N55 Spaceframe: Assembly and Details. http://www.vanalen.org/competitions/ConeyIsland/ downloads.htm Photographs from the groups website retrieved May 12,2005 from Photography http://www.n55.dk/MANUALS/SPACEFRAME/ Jonathan Cohen-Litant (aerial and site SPACEFRAME123.html photography), David Starr-Tambor (Coney Island and site 6.8: The Aurodyne Hyperframe System. photography) Photographs from the company website Elizabeth Stoel (Coney Island photographs) retrieved May 12,2005 from http:// Brooklyn Public Library-Brooklyn Collection www.aurodyn.com/index.html (Historical photographs)

6.9: A Bambutec Bridge Drawings/Diagrams & 6.10: Bambutec Technology. Eva Vela, Van Alen Institute Photographs from the company website retrieved May 12,2005 from http:// www.bambutec.org/html/bambutec_home_english.html

(The following illustrations have the same source and are all credited after Figure 5.10)

Appendix 2: The Competition Brief. 7.1: Map of New York. 7.2: Map of Coney Island. 7.3: Historic View of the Parachute Jump. 7.4: Site Plan. 7.5: Site View. 7.6: Aerial View of the Site and its Surroundings. 7.7: Site Viewed from the Riegelmann Boardwalk. 7.8: Aerial View of the Site. 17 Introduction

Nature is abundant with the most profound examples of delicate strength. Natural forms have evolved over the centuries to meet a vast range of environmental conditions, yet they all have a geometrical order. This order contains within itself an inherent flexibility to allow variable outcomes, combinations and permutations for adaptability to local conditions, in accordance with Darwins ‘Survival of the Fittest’ theory. Geometric order is responsible for beauty, variety, strength, stability, functionality and economy in natural forms.

Science, is an attempt to understand the beauty and order of nature. Art traditionally, is an attempt to replicate this beauty and order. Architecture, being both a science and an art, is their meeting point. It is therefore hypothesized that architecture should look towards science to create art.

This thesis attempts to do just that in a specific way. It hypothesizes that the geometry that has shaped beauty, functionality and economy in the natural world can be used to do the same in the built environment. Hence, geometrical principles that define natural structure and form, are examined for their potential to define architectural Fig. 0.1 : Flowers structure, form and components. exhibiting geometry. 18 This concept has been used repeatedly in the fields The thesis research looks at geometry with the of art and architecture. For example, the golden ratio intention of defining a limited number of units that was discovered to be the proportion of many natural can function as the ‘bricks’ of the system and that forms, ranging from flowers and leaves, to can be put together in different ways to create a and frogs, and even humans. This ratio has been variety of surfaces. For the design of the ‘bricks’, used extensively in art and architecture, particularly emphasis is on modularity and use of repetitive during the Renaissance. More recently, the noted elements, to achieve economy in construction. architect Le Corbusier has studied the golden ratio and used it to define ‘Le Modulor’, which is a Geometry, therefore becomes the main focus of harmonic proportioning system. Countless other study. In particular, geometry is examined for historic and modern examples of the use of geometry rules that have the potential to result in a variety in architecture exist, ranging from the Pantheon in of possible outcomes. Penrose and other aperiodic Greece to the Taj Mahal in India. tilings (that will be discussed further on in this document) are an example of such geometric rules This thesis does not attempt to design with the use that this thesis investigates. of geometry in a similar manner. It does not start by using geometry as a proportioning system, nor Chapter one begins by introducing some common as an external form giving system. Instead, this terms and in geometry. It describes only thesis hypothesizes that as in the case of natural .1 Begining with planar geometry, structures, the use of geometry should be intrinsic. the different objects that can be drawn a plane, In other words, geometry should be used to define and their arrangements are described. Next three the building blocks or bricks that go together to dimensional objects and tessellations are described, make the overall form. followed by space filling objects. Introduction. 1The familiar three dimensional geometry of our everyday experience. was the first to synthesize all earlier knowledge of

it and hence it is credited with his name : 0 19 The concept of dimensional stability and ways in can tile a plane aperiodically. Of the many sets of which it can be applied to two-and three-dimensional rhombi that produce aperiodic tilings with different tesselations is described next. Finally the geometry symmetries, the case with fivefold symmetry is cited of the octet truss system and geodesic domes by as a special case because of certain properties that Buckminister Fuller are presented as examples in it possesses. These properties are briefly described which concepts of geometry and dimensional stability in this chapter and are elaborated in greater detail in have been used in architecture. the next chapter.

Although it is clear at the end of chapter one, that Chapter three focuses on the aperiodic fivefold the geometry that has been described thus far, can symmetry of penrose tilings. The golden ratio that be used to produce building blocks for architecture, is employed in the geometry of these tilings is the blocks still produce planar surfaces, folded planes described. The chapter looks at some of its history, or uniform curves like domes and cylinders. They do properties and instances where it is observed in the not resemble the more asymmetric curves found in natural world. nature. In chapter four, quasicrystals, whose geometry also Chapter two initiates a study of aperiodic geometry, makes use of the Golden Ratio are explored for in the hope that when applied to architecture at the a potential to use their geometry in architecture. scale of a building block, this geometry will produce The work of Artist Tony Robbin from New York, natural forms. Aperiodic geometry is a geometry in who has experimented with the use of quasicrystal which the rules that describe it, allow for a great geometry in sculpture, and who has proposed its amount of variation. This chapter describes some use in architecture (though no building using this of the work of Haresh Lalvani, who is credited for geometry has been built yet) is also presented. the discovery of an infinite family of rhombi that Introduction. : 0 20 The chapter concludes with the proposition that In the conclusion of chapter five, a method of quasicrystal geometry can be used to design a creating geometric surfaces with the tetrahedral variety of surfaces, including flat planes, planar units, that could potentially be used in architecture surfaces with penrose patterns, folded planes, folded is presented. These geometric surfaces, composed planes with patterns, and surfaces that start to of lines, planes and nodes, can be translated into describe curves. However, it is still not possible to space frame structures with hub, strut and skin use this geometry to design smooth surfaces with construction. The challenge to architecture then lies constantly varying curvature that approximate the in the detailing of a system that can be used to curvature found in natural objects like a flower. construct this geometry.

The study of quasicrystal geometry provides evidence Although, the detailing of such a system is beyond that the golden ratio can produce shapes that tile the scope of this text, a brief study of space three-dimensional space aperiodically. With this fact frame construction is presented in Appendix 1. Here, in mind, the golden ratio has been explored further concepts of space frame design, its advantages in chapter five. The original quasicrystal units are and disadvantages, and the currently available broken down to basic . The geometry of technology is studied. Construction systems with these irregular tetrahedrons has been used to derive steel, aluminum and bamboo are described. the ‘Golden Tetrahedral System’ (patent pending by the author). It has been discovered by the author The final test which would evaluate the applicability that only four different lengths are required to of such a system would be to design a building with it, produce four different triangles that combine to form with consideration of all the other factors that go into the six basic tetrahedra used in the system. These making architecture. For this purpose, the ‘Parachute tetrahedra can be joined in a variety of ways to Pavilion’ Architectural Design Competition has been produce surfaces that approximate natural curves. selected as an appropriate project. Appendix 2, presents the competition brief. It studies the site, Introduction. history of the site, cultural context and program :

requirements of the pavilion. 0 21 Chapter 1: Plane and Solid Geometry

INTRODUCTION A study of the concepts of space and geometry is necessary to define space filling patterns and forms that make up buildings or parts of buildings. For design purposes, it has long been sufficient for an architect to have a good understanding of the laws of geometry that are applicable to the ‘flat’ two and three dimensional Euclidean space of our everyday experience.

This thesis looks at geometry, with the intention of defining a limited number of units, that can function as the ‘bricks’ of a new and innovative system, that can be put together in different ways to create a variety of surfaces.

A review of the basic concepts of geometry paves the way for an understanding of the more complex geometry of chapter two. Familiar concepts of two-and three-dimensional geometry are examined in a few examples where geometry has been used to define ‘building blocks’ that go together to make architecture.

22 CONCEPTS OF SPACE AND GEOMETRY. All study of geometry is based on an idea of space. The idea of space that we are most familiar with, is three-dimensional Euclidean space, with three axes. It was named after the greek mathematician Euclid, who lived in 300 BC. He systematically synthesized earlier knowledge about space and geometry in his book, ‘The Elements’. His model for reasoning was thought to be correct until about the mid 1800’s when science proposed different models of space and hence geometry.

Although these more advanced and intangible ideas of space andFig. 1.1a: Circle Limit III dimensionality exist in the realms of math and science, for the most part, they are not directly relevant to architecture. None the less, these concepts have been introduced in this text, because it is possible to ‘project’ from higher dimensional space onto the two-and-three dimensional space that we design and build in. Architectural practice is familiar with the concept of projecting three-dimensional objects on to a flat two-dimensional plane, for representation, in the form of ‘plans’, ‘elevations’ and ‘sections’. Similarly, projections from curved space, onto a two-dimensional plane can be observed in the art of M.C. Escher. In chapter two of this text, the concept of projecting from higher dimensional space is introduced. Fig. 1.1b: Circle Limit IV Most of the aperiodic tilings discussed there, are projections of periodic arrangements in higher dimensional space. Fig. 1.1: The art of M.C. Escher depicting projections from It is for this reason that the concepts of curved space and higher hyperbolic space onto a two dimensional plane. a) Circle Limit dimensionality have been introduced in this chapter. Plane and Solid Geometry. III, b) Circle Limit IV : 1 23 The other concepts of space that were developed in the early nineteenth century, are as logically consistent as Euclidean geometry. They include: * (Lobachevsky-Bolyai-Gauss geometry) Fig 1.2.a: Flat *Elliptic geometry (Riemannian geometry) Euclidean *Spherical geometry Geometry

Unlike in Euclidean geometry, where the curvature of space is ‘0’ or flat, in these curved concepts of space, space itself may have a constant positive or negative curvature, as a result of which all objects conceived in these spaces are bent according to the curvature of the space. In these systems of geometry and concepts Fig 1.2.b: of space, no straight lines are able to exist. The very Spherical nature of space is bent and curved in different ways. In Geometry hyperbolic space the curvature is negative and in elliptic and spherical space the curvature is positive.

Fig 1.2.c: Hyperbolic

Geometry Plane and Solid Geometry. : 1 24 THE DIMENSIONAL ASPECTS OF SPACE. In Geometry, dimensionality is usually conceived as By transforming the same point out of the two a set of mutually perpendicular space co-ordinates dimensional plane, we can create the simplest of the x,y,z.... Euclidean geometry describes the first three three dimensional forms or polyhedra, which is a dimensions and all constructions in this geometry tetrahedron. An infinite array of regular and irregular can be described within these three dimensions. polyhedra can be created in three dimensional space.1 For convenience we use the notation x , x , x , x , 1 2 3 4 ...... x to denote the different spatial dimensions. Mathematically, it is possible to work with higher n In fig. 1.3, the most simple entity that can be dimensional space, but practically it is difficult generated in the first three spatial dimensions is to imagine them. The fourth dimension can be drawn. visualized in curved as the dimension For n= 0 the simplest entity is a around which the first three dimensions ‘bend’. This point. text does not discuss these concepts, other than the By transforming a point along a single aperiodic projections from higher dimensional space, dimension say ‘the x-axis’ we can that are introduced in chapter two. generate a line. For n= 1, the simplest entity is a line. By transforming the same point along a different dimension say ‘the y-axis’ we can generate the simplest in a two dimensional plane (n= 2) which is a . An infinite number of shapes are possible in two Fig 1.3: Unfolding of Spatial Dimensions. (Keith Critchlow. Order in Space, pg 7)

dimensions. Plane and Solid Geometry.

1 Critchlow, Keith. Order In Space, pg 4. New York, New York: Thames & Hudson, 1987. : 1 25 PLANAR GEOMETRY. Planar geometry consists of objects that are contained LINE: A single dimensional geometric figure formed in a plane. These include points, lines, curves and by a point moving along a fixed direction and the polygons. A study of planar geometry is relevant reverse direction. It is the only geometric object for to architecture because all plans and elevations which n=1. A line extends indefinitely in opposite are based on some regular or irregular geometric directions. A part of a line that has a defined start configuration. Most often, and for obvious reasons, and end point is called a ‘Segment’ and rectangular grids and compositions are used in architecture. CURVE: “A curve (n=2) is the graph of a function1 on a coordinate plane.”1 It can also be defined as Architecture makes use of patterns at different a line that deviates from straightness in a smooth, scales, ranging from that of a texture, to mosaics, continuous fashion, defined by a function. There are to grids, to patterns on an elevation, to city blocks. various types of curves, the simplest of which is a Hence the different objects of planar geometry and circle. Other curves include ellipses, hyperbolas and ways of joining these two dimensional objects to parabolas. form planar patterns are described. More important is the fact that two dimensional patterns can be used POLYGON: A polygon (n=2) is an ordered collection to create three dimensional patterns, by modification of points and segments joining the points. It is or addition of certain parameters. a plane figure bounded by segments and vertices. Most often they are named ‘n-gons’ based on the The different objects of planar geometry include: number of sides and vertices. Triangles, , POINT: It can be defined as a dimensionless , are some of the more well geometric object (n=0) having no properties except known polygons. location.

1A function is a relation that uniquely associates members of one set with members of another set. More formally, a function Plane and Solid Geometry. from A to B is an object f such that every is uniquely associated with an object . . A function is therefore a many : to-one (or sometimes one-to-one) relation. Weisstein, Eric W. “Mathworld: The web’s most extensive mathematics resource.” Retrieved May 12,2005 From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ 1 26 CURVES. A curve is a two dimensional object, i.e. (n=2). It is the graph of a function,1 on a coordinate plane. It can also be defined as a line that deviates from straightness in a smooth, continuous fashion, defined by a function1.

OPEN AND CLOSED CURVES: In a plane, a closed curve is a curve with no Fig. 1.4.a: Open & Closed Curves end points and which completely encloses an . An open curve has end points and does not enclose an area. SIMPLE AND NON SIMPLE CURVES: In a plane, a curve is simple if it does not cross itself. It is non simple if it crosses itself.

JORDAN AND NON JORDAN CURVES: In a plane, a Jordan curve is simple Fig. 1.4.b: Simple & Non Simple Curves and closed. A non Jordan curve is either not simple, or open or both.

FAMILY OF CURVES: “A set of curves whose equations are of the same form but which have different values assigned to one or more parameters in the equations. Families of curves arise, for example, in the solutions to differential equations with a free parameter.” 2 Fig. 1.4.c: Jordan & Non Jordan Curves

CIRCLE: A circle is the set of points in a plane that are equidistant from a given fixed point O. The distance r from the center fixed point to a point on the circle is called the , and the fixed central point O is called the center. Twice the radius is known as the diameter (d=2r ) . The angle a circle 3 subtends from its center is a full angle, equal to 360° or 2π radians. A circle has the maximum possible area for a given , and the minimum Fig. 1.5: Circle possible perimeter for a given area. Plane and Solid Geometry.

1see note on pg 25. : 2 Harris, J. W. and Stocker,p. 649. H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, 1 1998. 27 3∏. is the sum of the of a triangle in Euclidean space which is 180 degrees. CONICS: The conic sections are the non degenerate curves generated by the intersections of a plane with one or two nappes of a cone. For a plane perpendicular to the axis of the cone, a circle is produced. For a plane that is not perpendicular to the axis and that intersects only a single nappe, the curve produced is either an ellipse or a parabola . The curve produced by a plane intersecting both nappes is a hyperbola1

Fig. 1.6: Conics ELLIPSE: An ellipse is a curve that is the locus of all points in the plane the sum of whose distances and from two fixed points and (the foci) separated by a distance of is a given positive constant2 Fig. 1.7: Ellipse

PARABOLA: “A parabola is the set of all points in the plane equidistant from a given line L (the conic section directrix) and a given point F not on the line (the focus). The focal parameter (i.e., the distance between the directrix and focus) is therefore given by , where a is the distance from the to the directrix or focus.” 3

HYPERBOLA: “A hyperbola is a conic section defined as the locus of all points P in the plane the difference of whose Fig. 1.8: Parabola distancesand Their Surfaces and from of Revolution. two fixed §2 points in Geometry (the foci and theand Imagination. ) separated 3 by a distance is a given positive constant k.” Plane and Solid Geometry.

1 Hilbert, D. and Cohn-Vossen, S. The Cylinder, the Cone, the Conic Sections, : p. 8. New

York: Chelsea, pp. 7-11, 1999. 1 2 (Hilbert and Cohn-Vossen 1999, p. 2) 28 3 Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed., p. 45. Boca Raton,Fig. 1.9: FL: HyperbolaCRC Press, 1997. POLYGONS. A polygon is an ordered collection of points and segments joining the points. The points and the segments are known as vertices and sides of the polygon. So it is a plane figure bounded by vertices. Most often they are named ‘n-gons’ based on the number of sides and vertices. A polygon whose vertices all belong to a single plane is called planar. Every polygon separates the plane into its interior (finite) and exterior (infinite) regions. Polygons are often identified with their interior regions.

Fig. 1.10: Polygons A polygon may be concave or convex. It is convex when every vertex angle is less than ∏1. It is concave when at least one vertex angle is more than ∏. It is considered simple when its edges can be continuously deformed into a circle without changing its topological properties.

A polygon is said to be equilateral if its sides are all equal. It’s said to be equiangular if its angles are all equal. A polygon can be equilateral without being equiangular and can be equiangular without being equilateral. Table of properties of regular polygons

A polygon that is both equilateral and equiangular is called regular. A regular n-gon is often denoted as {n}.

The sum of the vertex angles in a polygon is π(p-2)

The vertex angle of a regular polygon is π(p-2)/p and its center angle is Fig. 1.11: Regular Polygons

2∏/p Plane and Solid Geometry.

1 ∏. is the sum of the angles of a triangle in Euclidean space which is 180 degrees. : 1 29 Two kinds of circles can be drawn with points on a regular polygon. An in-circle to the mid point of each A circum-circle passing through the vertices.

SKEW POLYGONS A regular polygon {p} can be decomposed into p equilateral triangles. If the angular relationships of these triangles is altered so that each triangle becomes isosceles with the central angle > 2 ∏/p then this would result in a in which opposite pairs of vertices are pulled out of the original plane. If these triangles were replaced by a single surface with curvature, the surface would have the constant negative curvature found in hyperbolic space. This surface would be Fig. 1.12: Metamorphosis of regular polygons from {3} to {12} considered minimal when it has the least possible required to define the corresponding skew polygon.1 These skew polygons are also called saddle polygons.

Skew polygons can be tiled to describe cylindrical surfaces.

Fig. 1.13: Skew Polygons Plane and Solid Geometry.

1 Williams, Robert. Natural Structure: Toward a Form Language.pg 34. Moorpark California: Eudaemon Press, 1972. : 1 30 CIRCLE PACKINGS AND TESSELLATIONS. CIRCLE PACKING: A circle packing is an REGULAR TESSELLATIONS assembly of circles in which each circle “A regular tessellation in one in which all of the following is in contact with at least three or more conditions are met; circles and each circle is tangent to the 1.All the polygons in the tessellation are congruent. same number of circles. A circle packing 2.All the polygons in the tessellation are regular is stable “ ... if each circle is fixed by 3.All the vertices in the tessellation are congruent, i.e. the other circles; i.e. if on each circle the all the vertices are n-connected and form an n-connected greatest ‘free’ arc, which contains no point network, where n denotes the number of lines meeting at a of contact, is smaller than a semicircle.1 vertex.” 3 If the center of each circle is joined to the 4.Since the tessellation is planar, the angles at each vertex centers of the adjacent circles it results in must be an integer sub multiple of 2π and the sum of all the a tesselation of regular polygons. angles that meet at a vertex must be 2π.

TESSELLATION: A tessellation can be The only tessellations possible that satisfy all of the above defined as a regular tiling of polygons conditions are those with polygons {3}, {4} and {6} and (in two dimensions), polyhedra (three they are the only regular tessellations. They are assigned the dimensions), or (n dimensions). Schläfli symbols {3,6}, {4,4} and {6,3}2 Tessellations can be specified using a Schläfli2 symbol.

2D TESSELATION: A 2D tessellation is an assembly of polygons that cover a plane Fig. 1.14: Regular Tessellations. without overlap and without interstices. Plane and Solid Geometry. 1Fejas Toth, L. 1960/61. “On the Stability of a Circle Packing” Ann Univ. Sci. Budapestinensis, Sect. Math. v. 3-4, pp 63-66. The central angle (lamda) of the free arc is called the liability and the stability (rho) of a circle in the packing is determined by : subtracting (lambda) from ∏ 1 2The symbol {p,q} means p-gons, q at a vertex 31 3Williams, Robert. Natural Structure: Toward a Form Language.pg 35. Moorpark California: Eudaemon Press, 1972. SEMIREGULAR OR ARCHIMEDEAN TESSELLATIONS: “By relaxing the condition of the congruency of polygons, eight additional circle packings would yield tessellations that are called semiregular or archimedean tessellations.” 1 Plane and Solid Geometry. Fig. 1.15: Archimedian Tessellations. 1Williams, Robert. Natural Structure: Toward a Form Language.pg 37. : Moorpark California: Eudaemon Press, 1972. 1 32

DUAL TESSELLATIONS: “The dual tessellations are tessellations that are generated by interconnecting polygon centers with new edges. The edges of the duals are perpendicular to the edges of the original tessellations. The duals of the above (which are also regular tessellations) are: The dual of the {3,6} regular tessellation is a {6,3} regular tessellation. The dual of the {4,4} regular tessellation is another {4,4} tessellation. The dual of the {6,3} regular tessellation is a {3,6} regular tessellation.”1

Fig. 1.16: Duals of Regular Tessellations. Plane and Solid Geometry. Fig. 1.17: Duals of Archemedian Tessellations. 1Williams, Robert. Natural Structure: Toward a Form Language.pg 37. : Moorpark California: Eudaemon Press, 1972. 1 33

PLANE TESSELLATIONS WITH REGULAR POLYGONS OF NON-EQUIVALENT EDGES: These tessellations can be generated by removing specific vertices and edges of the regular and semiregular tessellations.

DEMI-REGULAR OR POLYMORPH TESSELLATIONS: By relaxing the conditions (1) and (3) the number of possible combinations of polygons increases to include fourteen more orderly composites of the three regular and eight semi-regular tessellations called polymorphs or demi- regular tessellations.1

Language.pg 43. Plane and Solid Geometry.

Fig. 1.18: Demi-regular or Polymorph Tessellations. 1Williams, Robert. Natural Structure: Toward a Form : Moorpark California: Eudaemon Press, 1972. 1 34 THREE DIMENSIONAL GEOMETRY CLASSIFICATION AND CATEGORIZATION: There EXTRUSIONS : The operation of exists an infinite number of three dimensional ‘Extrusion’ adds a single dimension geometric objects. One of the methods to study to a 2 dimensional object to create and classify these objects is based on their method simple three dimensional objects. of creation. A three dimensional(3n) object can be Prisms, Cylinders and some curved created from a one(1n) or two(2n) dimensional object surfaces fall into this category of by performing certain operations on the object. The 3D objects. operations may add one or two dimensions to the object. The total of the original dimension of the ROTATIONS : The operation of object + the dimensions added by performing the ‘Rotation’ adds a single dimension operation should add up to 3n or greater for a three to a 2 dimensional object to create a dimensional object three dimensional object. , Some of these operations are listed in table below. Spheroids, Ellipsoids etc. fall into this category of 3D objects.

OPERATION - PERFORMED ON

EXTRUSION(1n) - 2n : curve, polygon

ROTATION(1n) - 2n : curve, polygon

REVOLUTION(2n) - 1n & 2n : lines, curves, polygons

EXTRUSION + ROTATION(2n) - 1n & 2n : lines, curves, polygons

EXTRUSION + ROTATION + BEND/TAPER(2n) - 1n & 2n : lines, curves, polygons

TESSELLATION (0n) or (1n) - 2n : polygon

PARAMETRIC SURFACES : - Defined by two or more curves

Table 1.1: Operatons for the Creation of 3 Dimensional Objects. Fig. 1.19: Rotations Fig. 1.20: Extrusions. Plane and Solid Geometry. : 1 35 REVOLUTIONS : The operation of ‘Revolution’ COMBINATIONS OF THE ABOVE adds additional dimensions to a one or two THREE OPERATIONS dimensional object to create a more complex It is possible to have different three dimensional object. It involves revolving an combinations of the above three object around an axis that is away from its center operations to create a greater variety point. The axis of revolution may or may not be of 3D solids. A still greater variety parallel to the axis of the object. Cylinders, Cones, results if parametric changes are Toroids, Catenoids, Hyperboloids, Paraboloids etc. added into the process. A parametric fall into this category of 3D objects. change can be made by altering any A CYLINDER is formed by revolving a line about parameter of the original object (say an axis that is parallel to the axis of the line and length of side or radius or slant angle) at some distance away from it. as the operation is performed on it. A CONE is formed by revolving a line about an An example is mentioned below: axis that is inclined to the axis of the line and at some distance away from it. EXTRUSION + ROTATION : A one TOROIDS: A surface of revolution obtained by or two dimensional object may be rotating a closed plane curve about an axis rotated and extruded at the same parallel to the plane which does not intersect the time to create a three dimensional curve. The simplest toroid is the torus. object. The simplest example of this type would be a helix. also fall into this category. The rotations maybe a half turn, a full turn or any number of turns around the axis of Fig. 1.21: Revolutions revolution. Plane and Solid Geometry.

Fig. 1.22: Extrusion and Rotation : 1 36 THREE DIMENSIONAL TESSELLATIONS. Three dimensional tessellations can be created from any two dimensional tessellation by modifying the condition at the nodes.

A necessary condition for all two dimensional tessellations is that the sum of the angles at every node, has got to be equal to 2∏ or 360°.

By changing this condition, i.e. if the sum of the angles at every node is not equal to 360°, then the tessellation has moved into the third dimension. Table 1.2: The relation between the values of p and q and the geometry of the tessellation{p,q} Modifying the sum of the angles at every node may be done in two ways: The table above summarizes all the possible 1) Increasing or decreasing the value of the angles regular tessellations in Euclidean, Spherical and 2) Changing the polygon, i.e. substituting a / Hyperbolic Space. for a from the 2D tiling On the following page, a series of planar Euclidean If the sum of the angles at every node is less then tessellation in the middle column, have been 360°, then the tiling is spherical. Spherical tilings are modified to spherical tilings on the left and forms of polyhedra. hyperbolic tilings on the right.

If the sum of the angles at every node is greater then

360°, then the tiling is hyperbolic. Plane and Solid Geometry. : 1 37 SPHERICAL, EUCLIDEAN AND HYPERBOLIC TESSELLATIONS. Plane and Solid Geometry. :

3D Spherical Tessellations (Polyhedra) 2D Euclidean Tessellations 3D Hyperbolic Tessellations 1 Fig. 1.23: Spherical, Euclidean & Hyperbolic Tessellations 38 POLYHEDRA. “In geometry, a is simply a three- PLATONIC SOLIDS. These are the only 5 polyhedra dimensional solid which consists of a collection that are convex, closed shapes created using the of polygons, usually joined at their edges. same regular polygon (regular = equilateral sides), The word derives from the Greek poly (many) with congruent vertices. plus the Indo-European hedron (seat)”.1 It can 1 Tetrahedron – 4 triangular(3) faces be considered as a three dimensional closed 2 – 6 square(4) faces tessellation of polygons 3 Octahedron – 8 triangular(3) faces 4 Dodecahedron – 12 pentagonal(5) faces A study of polyhedra may be valuable since 5 Icosahedron – 20 triangular(3) faces ‘building blocks’ that this thesis seeks to define could be some three-dimensional polyhedron or set of polyhedra.

REGULAR POLYHEDRA: “A polyhedron is said to be regular if its faces and vertex figures are regular (not necessarily convex) polygons.” 2 Using this definition, there are a total of nine Fig. 1.24: Platonic Solids. regular polyhedra, five being the convex Platonic solids and four being the concave (stellated) Kepler-Poinsot solids. Most often, the term “regular polyhedra” is used to refer to the

Platonic solids. Fig. 1.25: Study illustrating why there cannot be more than five regular

polyhedra. Plane and Solid Geometry. 1 Eric W. Weisstein. “Polyhedron.” From MathWorld--A Wolfram Web Resource. http:// : mathworld.wolfram.com/Polyhedron.html

2Coxeter, H. S. M. , 3rd ed., p. 16. New York: Dover, 1973. 1 39 DUALS OF THE PLATONIC SOLIDS: Duals are obtained by connecting the midpoints of each of a shape. The dual polyhedra of the Platonic solids are the Platonic solids themselves. The tetrahedron is its own dual, the cube and the octahedron are duals of each other and the dodecahedron and the Icosahedron are duals of each other.1

ARCHIMEDIAN SOLIDS: 13 polyhedra can be generated by truncating the Platonic Solids at the various points along each edge, making solids with 2 or more different polygonal faces. There are 6 stages in the progression of each Fig. 1.26: Archimedean Solids. set of duels. The tetrahedron makes up only one truncated shape.1

QUASI-REGULAR POLYHEDRA :These 2 “quasi- regular” polyhedra are made by truncating the Platonic Solids at the midpoint of each edge. Each set of dual solids makes the same Fig. 1.27: Quasi Regular Polyhedra. ; truncating the tetrahedron at the midpoint makes another tetrahedron.1 Plane and Solid Geometry.

1Williams, Robert. Natural Structure: Toward a Form Language.pg 55, : 56. Moorpark California: Eudaemon Press, 1972. 1 40 CATALAN SOLIDS: The 13 dual polyhedra of the Archimedean solids are called the Catalan solids in honor of the French mathematician who first published them in 1862 .

Fig. 1.28: Catalan Solids paired with their duals, the Archimedean Polyhedra Plane and Solid Geometry. : 1 41 STELLATED POLYHEDRA: Stellation is the process of constructing polyhedra by extending the facial planes past the polyhedron edges of a given polyhedron until they intersect1 . The set of all possible polyhedron edges of the stellations can be obtained by finding Fig. 1.31: Stellated all intersections on the facial planes. Since the Dodecahedron.. number and variety of intersections can become unmanageable for complicated polyhedra, additional rules are sometimes added to constrain allowable Fig. 1.29: Process illustrating stellation of a Dodecahedron. stellations. Fig. 1.32: Stellated There are no stellations of the cube or tetrahedron. . The only stellated form of the octahedron is the octangula, which is a compound of two tetrahedra There are three dodecahedron stellations: the small stellated dodecahedron, , and great stellated dodecahedron1. 58 icosahedron Fig. 1.30: Process illustrating 2 stellations exist subject to certain restrictions . stellation of a Cuboctahedron.

The Kepler-Poinsot solids consist of the three Fig. 1.33: The Kepler-Poinsot Solids. dodecahedron stellations and one of the icosahedron stellations, and these are the only stellations of Platonic solids which are uniform polyhedra. Plane and Solid Geometry.

Fig. 1.34: Other Stellated Polyhedra 1Wenninger, M. J. Polyhedron Models. New York: Cambridge University Press, 1989. : 2

Coxeter, H. S. M.; Du Val, P.; Flather, H. T.; and Petrie, J. F. The Fifty-Nine Icosahedra. Stradbroke, England: Tarquin Publications, 1 1999.They include the icosahedron itself in their count, for a total of 59 42 POLYHEDRAL PACKING: A space-filling polyhedron is a polyhedron which can be used to generate a tessellation of space. The cube is the only that can tessellate space with itself only. A combination of tetrahedra and octahedra do fill space In addition, octahedra, , and cubes can also fill space .1

“There are eight known self-packing allspace-fillers: The cube. (6 faces) Discoverer unknown. Fig. 1.35: Polyhedral The . (12 faces) Discoverer Packing of Cubes unknown. This allspace filler is the one that occurs most frequently in nature. Lord Kelvin’s tetrakaidecahedron. (14 faces) Keith Critchlow’s -cornered tetrahedron. (16 faces) The truncated octahedron. ( 14 faces) The trirectangular tetrahedron. (4 faces) Described by Coxeter, “Regular Polytopes,” p. 71. The tetragonal . (4 faces) Described by

Coxeter, “Regular Polytopes,” p. 71. Fig. 1.36: Polyhedral The irregular tetrahedron (Mite). (4 faces) Discovered Packing of Lord Kelvin’s Fig. 1.37: Polyhedral tetrakaidecahedron 2 Packing of sphenoid and described by Fuller.“ hendecahedra. Plane and Solid Geometry.

1Eric W. Weisstein. “Space-Filling Polyhedron.” From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Space- : FillingPolyhedron.html 1 2 http://www.rwgrayprojects.com/synergetics/s09/p4300.html#950.12 1999. 43 CURVED SURFACES. Curved surfaces are derived by carrying out the operation of rotation or revolution around a point, an Fig. 1.39: Curved Surfaces Derived by Revolving an Open Curve Around an Axis. axis or along a curve. While an Conical Funnel Catenoid Hyperboloids. Barrel Surface. operation of revolution is being Surface. Surface. Surface. carried out, a simultaneous operation of translation, and/or a parametric change may be added on to produce a more complex surface. Sphere. Apple Surface. Torus. Elliptical Figure of Eight Toroid. Surface. All regular curved surfaces, including Fig. 1.40: Curved Surfaces Derived by Revolving a Closed Curve Around an Axis. the complex parametric surfaces shown on the opposite page, can be specified by an equation.

Hennenbergs Surface. Hypar Surfaces. Scherks Surfaces.

Keun’s Corkscrew Sine Surface. Seashell Surfaces. Surface. Surface.

Fig. 1.41: Curved Surfaces Derived from a Parametric Equation. Plane and Solid Geometry. Fig. 1.38: Curved Surfaces of Revolution. : 1 44 DIMENSIONAL STABILITY. Dimensional Stability is an important consideration for the structural stability of a frame. If the frame is dimensionally stable, then theoretically, all the connections could be pin-jointed. If not, then the connections need to be rigid 1 Tetrahedron moment connections. The other necessary condition for structural stability, is proper design of the members for strength.

A geometric configuration may be defined as an arrangement of line segments Octahedron and nodes. These may be two dimensional as in the case of the 2D tessellations described earlier or three dimensional as in the case of polyhedra. The configuration is stable if the nodes are unable to change their relative positions when all the connections are assumed to be pin jointed, i.e. when Fig. 1.42: Dimensionally Stable rotation at the joints is possible, but not translation. Icosahedron Platonic Solids.

Only three of the five polyhedra are dimensionally stable as shown in fig 1.42 opposite: the tetrahedron, the octahedron and the icosahedron, all of which have triangular faces. The following general formula bay be used to establish dimensional stability:

For a planar frame exhibiting dimensional stability: Hexhedron 2g = n +3 For a space frame exhibiting dimensional stability: 3g = n +6 where, g = the number of nodal connections. Pentagonal 1 Fig. 1.43: Other Stable Deltahedra.

n= the number of bars. Plane and Solid Geometry. 1

F. Stussi Baustatik 1 ( Birkhauser Verlag, Basel and Stuttgart, 1962, p-118, 159) : 1 45 3 Tetrahedron In addition to possessing the required number of bars, an additional condition must be satisfied, and that is that the bars should be in the correct position to complete of the network of segments and nodes. 4 Tetrahedron

The only two dimensional tessellations that are dimensionally stable are those that are composed of triangles. These may be regular or irregular.

Of the platonic solids, the tetrahedron, the octahedron and the icosahedron 5 Tetrahedron have triangular faces and are stable. These are not the only polyhedra that are stable. Fig 1.44 and fig. 1.45 opposite illustrates some more examples of stable polyhedra. All convex deltahedra1 are stable. Fig. 1.44: Stable Configurations Based on Tetraheda. The geodesic domes and twisted domes presented later, are also examples of stable polyhedra. Johnsons Solid 17

When stable a polyhedra is joined to another stable polyhedra they form a stable network of bars and nodes. The Octet Truss that was discovered independently by Alexander Graham Bell and Buckminister Fuller is an Johnsons Solid 58 example of such a stable network that is composed of tetrahedrons and .

The next few pages describe the stable configurations of the octet truss, Twisted Dome geodesic domes that have been used in architecture, and twisted domes that

could potentially be used. Plane and Solid Geometry. 1

A deltahedra is a polyhedra that has faces made up of only equilateral triangles. :

. 1 Fig. 1.45: Other Stable Deltahedra 46 DIMENSIONALLY STABLE CONFIGURATIONS IN ARCHITECTURE. It is advantageous to use dimensionally stable configurations in architecture as the framework for such structures becomes lighter, and the joints need not be rigid. The first dimensionally stable configuration that was used in architecture was the octet truss.

THE OCTET TRUSS: The tetrahedron and the octahedron are theFig. only 1.46: two Alexander Graham Bell and the Octet Truss. platonic solids that are dimensionally stable configurations. These two shapes combine to form the octet truss. The octet truss can be extended to fill space, providing a lattice known to crystallographers as the face-centered cubic (fcc), and to Fuller as the isotropic vector (IVM). It is an ‘isotropic’ vector matrix as its hubs correspond to the centers of spheres of equal radius, packed tightly together. The lattice so formed therefore contains rods of equal length with equal distances between each point and its 12 nearest neighbors. It is quite widely used in architecture and engineering. Fig. 1.47: The first built Octet Truss.

The truss was probably first discovered and used in architecture by Alexander Graham Bell in an observation tower structure at Beinn Bhreagh, Nova Scotia that was completed in 1907.1 The geometry was discovered again by Buckminister Fuller who called it the isotropic vector matrix (IVM) and deemed it to be nature’s own structural coordinate system. In the 1940’s he patented a construction system which makes use of this geometry. Plane and Solid Geometry.

1

Alexander Graham Bell and the Octet Truss. Retrieved May 12, 2005 from http:// : www.grunch.net/synergetics/bell.html 1 Fig. 1.48: The Geometry of the Octet Truss. 47 OTHER CONFIGURATIONS OF STABLE POLYHEDRA. Tetrahedra and octahedra can be Another stable configuration of octahedra put together in a different manner has been illustrated in fig 1.50. This as illustrated in fig 1.49 below, to configuration has recessed ‘voids’ that are form a planar corrugated surface. In created by the geometry. The voids are all a manner similar to the octet truss, the same and could be turned into windows. this geometry can be used for This geometry could be used for planar wall and wall surfaces. surfaces.

Fig. 1.49: Stable configuration with a corrugated profile.

Fig. 1.50: Stable configuration with voids. Plane and Solid Geometry. : 1 48 GEODESIC DOMES. A can be defined as a triangulation of a Platonic solid or other polyhedron to produce a close approximation to a sphere (or hemisphere).1 Geodesic spheres and domes come in various frequencies. The frequency of a dome relates to the number of smaller triangles into which it is subdivided. A high frequency dome has more triangular components and is more smoothly curved.

A geodesic dome can be constructed by using any regular convex polyhedra as a framework. First ensure that all of the polyhedron’s faces are triangular by triangulating each non-triangular face. Fig. 1.51: Architect Buckminister Fuller and a dome with his This is done by connecting its vertices to a new patented geodesic system. vertex placed at the center of the face. Next make all the vertices of the polyhedron equidistant from the center of the polyhedron by moving them directly away from or towards the center. This creates a ‘1-frequency’ geodesic sphere.

R. Buckminster Fuller is credited with the design of the first geodesic dome (i.e., Fuller’s dome was

constructed from an icosahedron) Plane and Solid Geometry. 1

Eric W. Weisstein. Geodesic Dome. From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ : GeodesicDome.html 1 49 A one-frequency icosahedron consists of 20 equilateral triangles. It is the most useful polyhedron for dome building. Each vertex is at the same distance from the center of this polyhedron and is on the surface of an imaginary sphere.

Higher frequency geodesic spheres may be constructed by replacing each face with a regular triangular mesh and then ensuring all of the new vertices are equidistant from the center. 2-frequency geodesics bisect each edge of their Original Cuboctahedron. associated 1-frequency geodesics by using 4 triangles for each mesh; 3-frequency geodesics trisect each edge and add one vertex to the center of each face of their associated 1-frequency geodesic by using 9 triangles for each mesh.”

In fig 1.52 below a 1-frequency, 2-frequency and 3-frequency geodesic sphere have 1-Frequency Geodesic from a been constructed from a . In fig. 1.53 opposite a 1-frequencyCuboctahedron.. and 2-frequency geodesic has been constructed from a cubeoctahedron.

2-Frequency Geodesic from a Cuboctahedron.. Fig. 1.53: Geodesic domes Original Icosahedron. 1-Frequency Geodesic from a Cuboctahedron.

from Icosahedron. Fig. 1.52: Geodesic domes from an Icosahedron. 2-Frequency Geodesic Plane and Solid Geometry. from Icosahedron. 3-Frequency Geodesic : from Icosahedron. 1 50 TWISTED DOMES: Twisted objects are usually regarded as visually interesting. Attempts have been made to created twisted polyhedra. George Heart has produced a series of twisted polyhedra which he has called propellohedra.

Stephan Werbeck is credited with the following construction method of creating ‘twisted domes.’ He created this process while trying to reconstruct the shape of a particular type of . The construction is illustrated in fig 1.54 opposite. The process is as follows: Fig. 1.54: Figure illustrating construction process of a ‘Twisted *Start out with a polyhedron. Dome’. *Replace each face of the polyhedron with a made up of equilateral triangles *Divide each triangular face in smaller equilateral triangles. *Twist the complete relative to each other while making sure that the smaller triangles are kept edge to edge. *This process creates holes that must be filled with extra faces.1

Fig. 1.55: Twisted Domes derived from non convex deltahedra. Plane and Solid Geometry.

1Gailiunas, Paul. Twisted Domes, Retrieved May 12, 2005 from http://web.ukonline.co.uk/polyhedra/Twist.pdf : 1 51 Paul Gailiunas studied this process and described a family of ‘Dual Goldberg Polyhedra’ that can be created by applying this process to deltahedra1 (Polyhedra that are composed of equilateral triangles only). They are illustrated in fig 1.56 opposite. The diagrams were created using a software called ‘HEDRON’

If the same process is applied to a non convex deltahedra, the derived ‘Dual Goldberg Polyhedra’, will inherit valleys from the concave edges. Figure 1.55 shows an example of such a dual Goldberg polyhedra derived from a stella octangula.1

Such twisted polyhedra are visually interesting and could be exploited architecturally in the same way as geodesic domes. An added advantage in this case would be that in most cases all the faces of twisted domes or ‘Dual Goldberg Polyhedra’ are composed of equilateral triangles. This would make construction economical. In some cases the ‘holes’ that result from the twisting process are square or in the shape of other polygons. Plane and Solid Geometry.

1 the polyherda that is formed by replacing each face of an : octahedron with a triangular pyramid. Fig. 1.56: Family of Twisted Domes derived from a

tetrahedron, an Octahedron and an Icosahedron.. 1 52 CONCLUSIONS. This chapter was a brief introduction to some of Dimensional stability is another requirement for the the concepts in Euclidean Geometry. It is by no units. None of the dimensionally stable objects means a comprehensive study and should not be illustrated in this chapter are space filling. The only regarded as such. For the interested reader, Keith combination of objects having the qualities of being Critchlow has written an excellent comprehensive ‘dimensionally stable’ and ‘space filling’, was the text on geometry, called ‘Order In Space’. Another configuration of the Octet Truss, which yields planar comprehensive text is ‘Natural Structure’ by Robert surfaces and folded planes only. Williams.

The purpose of this study is to derive a set of Although Geodesic domes and Twisted domes are basic units that are both modular, and can be dimensionally stable, they are ‘singularities’, and can combined to create variety. This chapter illustrated be used to define form by themselves. The whole cases of ‘regular geometry’: geometry that has some of the object needs to be used in order for the symmetry and that is predictable. dimensional stability condition to be fulfilled. Parts of these objects can be used, by themselves or in Most of the objects(polyhedra) that were defined conjunction with other parts, if the ‘open edges’ are and illustrated here, were not space filling objects framed and stabilized by other construction. by themselves. Some of them could be combined with others fill up space without leaving any gaps Since a study of predictable geometry with fixed or holes. However, all the configurations formed by rules, does not yield the desired units, a study these objects were regular: i.e. they were planar, of variable geometry with ‘flexible’ rules has been folded planes or surfaces with constant curvature undertaken in the next chapter. like a sphere or a cylinder. Plane and Solid Geometry. : 1 53 Chapter 2: Aperiodic Geometry

INTRODUCTION In the previous section of this document geometric arrangements for two and three dimensional tessellations were explored. All these arrangements were symmetric and predictable by rules that defined a ‘fundamental unit’ which was repeated throughout the pattern. Such patterns are called periodic. Periodic means that there exists a certain fundamental unit or region, that defi nes the entire geometry. The fundamental unit may be repeated by direct translation (sliding) or by operations like rotation, revolution etc. in two or three dimensions.

This chapter examines ‘aperiodic geometry’ which is inherently flexible. Sir Roger Penrose was the first to define aperiodic ‘penrose tessellations’. Haresh Lalvani later discovered an infinite family of aperiodic tessellations that were projections from higher dimensional space on to a two dimensional plane. This chapter explores these tessellations and their properties. In the end it cites the fivefold aperiodic tessellation as a special case because of some of its properties, that are probably responsible for its being the most prevalent symmetry in the natural world.

Fig. 2.1: Pinwheel aperiodic tessellation. 54 PROJECTIONS FROM HIGHER DIMENSIONAL SPACES. Understanding higher dimensional space is very difficult, given the fact that we can only experience three spatial dimensions. But, it is possible to ‘visualize’ higher dimensional space by understanding ‘projections’ from it. Architects and designers are familiar with the concept of drawing two dimensional plans and elevations of three dimensional buildings. Fig 2.2: In ‘Flatland’ a sphere takes on the appearance of circles of different sizes as it passes through a plane. A ‘plan’ is a ‘projection’ of a three dimensional object, on to a two dimensional plane. It makes it possible to understand a higher dimensional object, The sphere showed its form to the flat-landers by by studying its projections on to a lower dimension. raising its body through the Flatland surface. The flat-landers first saw a point that quickly grew to This concept has been clearly and humorously a circle, which continued increasing in size, and explained in a hundred year old book called then started decreasing in size until it became a ‘Flatland’ written by Edwin A. Abbot. Flatland is a point, and then it disappeared. So the inhabitants story about two-dimensional creatures-- triangles, of flatland, perceived the sphere to be an infinite squares, circles and other polygons--that live on collection of circles pieced together. a plane. They are visited by a three dimensional object- a sphere, which they do not understand, Observing projections of higher dimensional objects because they live and experience only the two in two and three dimensional space, makes it dimensions of ‘Flatland’. The sphere tries to explain to possible to gain some kind of an understanding of the flat-landers, the existence of higher dimensional higher dimensional space. This is one of the methods objects like itself, and ways in which they can of understanding higher dimensional space.1 understand the form of such objects. Aperiodicity.

1Hesse, Bob. Viewing Four-dimensional Objects in Three Dimensions. retrived May 12, 2005 from http://www.geom.uiuc.edu/ : docs/forum/polytope/ 2 55 Another important lesson from this story, is that the sphere (object with uniform curvature in three dimensions), appeared to be circles of different (non-uniform) sizes as it passed through a lower dimension. Similarly a ,1 appears to be different polyhedra as is passes through three dimensional space. It starts out as a tetrahedron, then becomes a , then an octahedron, and reverses back to a tetrahedron before it disappears.

In the same way, penrose and other tilings that are aperiodic in the second and third dimension, are periodic arrangements in their own dimension. In the next few pages, the infinite family of aperiodic tilings, discovered by Haresh Lalvani is presented.

4

Fig 2.3: A Four Dimensional object passing through three dimensional space, takes on 5 the form of various polyhedra as it passes through space. Fig 2.4: Increasing Dimensions Aperiodicity.

1 A Hypercube is one of the six basic units in four dimensional space, similar to the platonic solids in third dimensional space. : 2 56 THE INFINITE FAMILY OF APERIODIC TESSELLATIONS IN TWO & THREE DIMENSIONAL SPACE. Dr. Haresh Lalvani, an architect- morphologist, is known for his use of higher dimensional mathematics to create structures based on new geometries. He is credited with the method of creating an infinite family of aperiodic tessellations in two dimensional space. Each aperiodic tessellation uses a set of rhombi and is Fig 2.5: Aperiodic a projection from a higher dimensional Rhombic Tiling for space on to a two dimensional plane. n = 4. Fig 2.6: Aperiodic Rhombic Tiling for The tessellations also belong to different n = 5. Fig 2.7: Aperiodic Modified Rhombic symmetry groups. Tiling for n = 6.

He has called his method for creating these tessellations, ‘the vector star method.’ It is described in his book ‘Space Structures’. He uses the regular polygons as generators for the rhombi that are used in the tessellations. Each regular polygon can generate one aperiodic tessellation. Since there exist an Fig 2.8: Variations of the Aperiodic Rhombic Tiling for n = 6. infinite number of regular polygons, hence there exists an infinite family of a periodic tessellations that can be created from them. Aperiodicity. : 2 57 THE VECTOR STAR METHOD. The Vector Star Method can be explained by the process of creating the aperiodic tessellation based on the regular Septagon: a seven sided polygon with equal sides and angles. The process of determining the rhombi for the tessellation is as follows: * Begin with a regular polygon. A seven sided septagon has been selected for this example. The number of sides of the polygon denotes the dimension from which the tessellation is projected. It is denoted as ‘n’. So, n = 7. * The symmetry of the generating polygon is called as ‘P’. In this case P = 7 * Draw out a polygon with 2n sides, i.e. a 14 sided regular polygon.

* Draw out the vectors from the center of the polygon to its vertices. Fig 2.9: Derivation of the three rhombi * The central angle of the polygon is to be considered as a unit angle for for the n=7 case from a regular 14-sided polygon. Their face angles are distinct the rhombi. pairs which add up to seven. * Each distinct pair of vectors from the n polygon, determine the edges of a rhombic tile that can be drawn out as stated below. The number of vectors in the tiling is denoted as ‘i’. * Figure out the different combinations in which pairs of whole numbers having a positive value < P, can add up to P. In this case, there are three such combinations: i.e. 1) 1 + 6 = 7 2) 2 + 5 = 7 3) 3 + 4 = 7 Each combination forms a of the tessellation. The angles of the rhombus can be determined by multiplying the whole numbers of each

pair with the central angle of the generating polygon. Aperiodicity. : 2 58 CONCLUSION: THE FIVEFOLD CASE.

From the infi nite family of aperiodic tilings that has been described, the fi vefold case is stated to be a ‘special case’ because P=n=i. This case has has been elaborated further in the next chapter.

Table 2.1 : A table showing the infinite family of rhombi projected from dimension ‘n’. Aperiodicity. : 2 59 SECTION 1: GEOMETRY, STRUCTURE & FORM.

Chapter 3: The Golden Ratio and Fivefold Symmetry.

INTRODUCTION In the previous chapter, it has been stated that the aperiodic tiling with fivefold symmetry is a special case because P=n=i, that is, the tiling is a projection from the fifth dimension, it has five fold symmetry and five distinct vectors are used in the tiling. This chapter seeks to understand why that particular case is special.

The tiling is based on the pentagon and on five fold symmetry, which has been important since antiquity, because it contains the ‘golden ratio’ number (Phi) in its geometry. Since its first clear definition around 300 BC by Euclid of Alexandria, through its appearance in the Fibonacci series, and more recently in Quasicrystals, this number has arrested the interest of numerous mathematicians, physicists, philosophers, architects, artists and musicians.

This chapter describes the history, properties and appearances of the golden section that exists in nature. It is observed in the proportions of humans, plants, animals, DNA and the . Since it is the most common proportion seen in the physical , it is intuitively appealing that the use of it would yield the desired ‘bricks’ for the system.

60 HISTORY. The first known mathematical occurrence of the Golden Following Euclid, a few Greek geometers contributed Section dates as far back as , the Greek minor theorems to the development of the golden geometer(560-480 BC), and his school of thinkers. They ratio. They were, Hero (in the 1st century AD), adopted the pentagram as the symbol of health of their (in the 2nd century AD), and Pappus of brotherhood. Most researchers claim the Pythagorean Alexandria in the 4th century AD.1 preoccupation with the pentagram makes it plausible that they were the first to discover this ratio. of During the dark ages western culture lost interest Metapontum, a pythagorean, is generally credited with in the Golden Ratio. It gained fame again with the the discovery. work of Leonard Fibonacci of Pisa and his famous Fibonacci series mentioned in his book ‘Liber Abaci’ Eudoxus and are among the other early Greek (1202). In this series every number is the sum of geometers who are associated with this number. Eudoxus the previous two numbers before it, and the ratio of can be given credit for several theorems about the successive elements approaches the golden ratio. ratios found in the Golden Section. Plato, in his views presented in his ‘’, considered the golden section It became popular again during the renaissance to be the most binding of all mathematical relationships, period and has been used extensively in art and and the key to the physics of the cosmos. architecture, and in paintings and sculptures to achieve balance and beauty. In 1509, Luca Pacioli Euclid was the first geometer to specifically define the wrote the first known book dedicated to this ratio: ratio in his treatise, ‘The Elements’. He called it the ‘ratios ‘De Divina Proportione’ (The Divine Proportion). It of means and extremes’ (which later came to be known contains drawings made by Leonardo da Vinci of as the golden section). He used the basic properties of the 5 Platonic solids. It is possible that Leonardo this ratio, discovered by his predecessors, to construct (da Vinci) may have been the first to call it ‘the

regular pentagons, and . sectio aurea’ (Latin for the golden section). (Ø). The Golden Ratio-Phi

1Livio, Mario. The Golden Ratio. New York, New York: Broadway Books, 2002. : 3 61 The two other men who played a significant In the of music too, role in the history of the Golden Ratio, are the claims have been made by tenth century Mathematician Abu’l-Wafa and the several researchers, as to the German painter, Albrecht Durer. use of the Golden proportion in musical composition. The music In Botany, the connection between the Golden of the Hungarian pianist, Bela Ratio and the arrangements of leaves (phyllotaxis) Bartok has been analyzed and was first observed by , an Italian found to have been based on astronomer, who is well known for his discoveries the laws of the golden ratio. about planetary motion. He was also the first to notice that the ratio of successive Fibonacci More recently this ratio has numbers approaches the Golden Ratio. Fig 3.1: Pentagonal plan of a made its appearance in church ‘Penrose Tilings’ discovered by Numerous articles and books have been written Roger Penrose in the 1970’s on the topic since then, that make connections and in the geometry of between the ratio, and manifestations of it in quasicrystals discovered in the the physical world. “On Growth and Form’ by Sir 1980’s. D’Arcy Wentworth Thompson(1917) is one such book.

The French architect and painter, Le Corbusier, made the ratio popular in the modern era, in his proportioning system called, ‘Le Fig 3.2: Vitruvian Man:Sacred proportions of the human body by Modulor’(illustrated opposite) (Ø). The Golden Ratio-Phi Leonardo Da Vinci.

1Livio, Mario. The Golden Ratio.pg. 173. New York, New York: Broadway Books, 2002. : 3 Fig 3.3: Le Modular: a 62 Harmonic Proportioning system based on the Golden Ratio. MATHEMATICAL PROPERTIES OF THE GOLDEN SECTION. The Golden Section is both a ratio and a proportion. It is We can solve this in the same way as for Phi(φ) the relationship one part to another part, and that part and we find that to the whole. It is an incommensurable, g = –1/2 +√5/2 or g = –1/2 – √5/2 that can never be fully known. One definition of Phi(φ) Phi = (√5 + 1)/2 phi = (√5 – 1)/2 (the golden section number) is that to square it, add 1: 2 MM 1 The numerical values of these two numbers are 1·6180339887... and –0·6180339887... In Book 6 of ‘The Elements’, Proposition 30, Euclid shows They are designated as Phi (uppercase P) how to divide a line in ‘mean and extreme ratio’. This ratio = 1.6180339.. and phi (lowercase p) = was later called the ‘Golden Section Ratio’. 2 0.6180339.. 1 <------1 ------> g 1–g A G B OTHER MATHEMATICAL PROPERTIES OF PHI: Euclid used the phrase ‘mean and extreme ratio’ to mean It is the solution to the ‘continuous fraction’ the ratio of the smaller part of this line, GB to the larger equation below: part AG (i.e. the ratio GB/AG) is the SAME as the ratio of the larger part, AG, to the whole line AB (i.e. is the same as the ratio AG/AB). If we let the line AB have unit length and AG have length g (so that GB is then just 1–g) then the definition means that GB/AG = AG/AB, i.e. using the lengths of the sections, 1-g/g = g/1, which we rearrange to get 1 – g = g2. It is also the solution to the ‘continuous square root’ equation. 2 This is similar to Phi which is defined as MM 1

2 2 and in this case we have: g = 1–g or g +g=1. (Ø). The Golden Ratio-Phi

1SolidKnott, (Three-dimensional) Ron(Dr) . Two-dimensional Geometrical Geometry Facts aboutand the the Golden Golden section: Section. Fascinating Flat Facts about Phi, Some : Retrived May 12, 2005 from http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi2DGeomTrig.html 3 63

PHI AND THE PENTAGRAM ISOSCELES TRIANGLES. The Value of Phi upto 1000 decimal places The golden ratio appears in the pentagon in the Phi is an irrational number. Its digits after the decimal relationship between its sides and the do not form and repeating sequence of numbers. 1

(joining two non-adjacent points). It also appears Dps..: 1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 50 in the decagon where the ratio of the side to the 28621 35448 62270 52604 62818 90244 97072 07204 18939 11374 100 84754 08807 53868 91752 12663 38622 23536 93179 31800 60766 circumscribed radius is 1: Phi. 72635 44333 89086 59593 95829 05638 32266 13199 28290 26788 200 06752 08766 89250 17116 96207 03222 10432 16269 54862 62963 13614 43814 97587 01220 34080 58879 54454 74924 61856 95364 300 86444 92410 44320 77134 49470 49565 84678 85098 74339 44221 25448 77066 47809 15884 60749 98871 24007 65217 05751 79788 400 34166 25624 94075 89069 70400 02812 10427 62177 11177 78053 15317 14101 17046 66599 14669 79873 17613 56006 70874 80710 500

13179 52368 94275 21948 43530 56783 00228 78569 97829 77834 78458 78228 91109 76250 03026 96156 17002 50464 33824 37764 86102 83831 26833 03724 29267 52631 16533 92473 16711 12115 88186 38513 31620 38400 52221 65791 28667 52946 54906 81131 71599 34323 59734 94985 09040 94762 13222 98101 72610 70596 11645 62990 98162 90555 20852 47903 52406 02017 27997 47175 34277 75927 78625 61943 20827 50513 12181 56285 51222 48093 94712 34145 17022 37358 05772 78616 00868 83829 52304 59264 78780 17889 92199 02707 76903 89532 19681 98615 14378 03149 97411 06926 08867 42962 26757 56052 31727 77520 35361 39362 1000

Some interesting relationships between Phi and phi 1 Phi x phi = 1 Phi - phi = 1 Phi + phi = √5 Fig 3.4: The Pentagram Isosceles Phi = 1 + phi Triangles with the pentagram and decagon form which they can be phi = Phi – 1 derived. Phi = 1/phi phi = 1/Phi The Golden Ratio-Phi (Ø). The Golden Ratio-Phi Phi2 = Phi + 1

Flat Facts about Phi, Some Solid (Three-dimensional) Geometrical Facts about the : 1Knott, Ron(Dr) . Two-dimensional Geometry and the Golden section: Fascinating Golden Section. (-phi)2 = -phi + 1

Retrived May 12, 2005 from (phi)2 = 1 – phi 3 http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi2DGeomTrig.html 64 OBSERVATIONS OF FIVEFOLD SYMMETRY & THE GOLDEN SECTION, IN THE NATURAL WORLD. Fivefold symmetry manifests itself quite obviously in the geometry of flower petals and leaves. The symmetry is seen again in the arrangement of leaves that subtend the golden angle as the grow up on the stalk. It is also quite obvious in the spiral arrangements of buds of a cauliflower, pine cone, broccoli and the seeds of a sunflower.

Studies have been done on this topic by several people throughout history. In 1904, Botanist A. H. Church published a book called, ‘On the Relation of Phyllotaxis to Mechanical Laws’ in which he described the spiral pattern that leaves arrange themselves in, on a stem. In 1907, The German mathematician, G. Van Iterson, showed that the spiralFig 3.6: Phyllotaxis and arrangement of seeds in a sunflowerthe Golden angle. can be obtained by packing successive points, separated by 137.5 degrees, (the golden angle) in a tightly wound 1

spiral. (Ø). The Golden Ratio-Phi : Fig 3.5: The appearance of Fivefold Symmetry in Flowers.Fig 3.7: Other appearances of

the Golden angle in plant life. 3 1Livio, Mario. The Golden Ratio.pg. 173 New York, New York: Broadway Books, 2002. 65 The proportions of the golden ratio have been claimed to be exhibited, in the ratio of various parts of bodies of animals and even people. Illustrations on this page, show the golden ratio seen in parts of a dolphin and a moth and a human hand. Although this seems plausible, there is insufficient statistical evidence to support the hypothesis.

In conclusion, there are examples in nature where there can be no doubt that fivefold symmetry is operational. In other cases it may seem to be contrived, especially when the golden ratio is measured out. The degree of accuracy of such measurements, cast a shadow of doubt, on the factual presence of the golden ratio in these cases.

Fig 3.8: Fivefold symmetry and the Golden ratio as observed in human and animal life. Images courtesy of http:// www.goldennumber.net, Gary B. Meisner, Copyright 2005. (Ø). The Golden Ratio-Phi : 3 66 CONCLUSION.

The golden ratio has some truly unique mathematical properties, as described in this chapter, that cannot be paralleled by any other known number. There is also evidence that these mathematical properties manifest themselves in the physical world.

Indeed, this assumption is confirmed by the fact that a large number of entities in the physical world, observe this symmetry and proportion.

With such a long history, and frequent appearance in the natural world, It seems only logical, that the mathematical proportion that has played such a prominent role in the physical universe could be imitated to achieve beauty in architecture.

With this in mind the next chapter, explores the two dimensional Penrose Tessellations and their three dimensional versions: Quasicrystals, with the hope that this geometry could be used to create surfaces with a natural and non uniform curvature. The Golden Ratio-Phi (Ø). The Golden Ratio-Phi : 3 67 Chapter 4: Penrose Tilings & Quasicrystals

INTRODUCTION In the previous chapter, the uniqueness of the golden ratio, and its ubiquitousness in the natural world has been demonstrated. This chapter examines the aperiodic penrose tilings discovered by Sir Roger Penrose, and their three dimensional versions: Quasicrystals.

The use, and potential use of quasicrystal geometry in architecture is explored. The sculptural work done with these geometrical forms by artist Tony Robbin, from New York is described next.

In the end, the different ways in which these units can be arranged are examined in great detail, for a potential to define architectural form with these units. As stated earlier, the goal is to arrive at a ‘natural’ form.

Fig 4.1: Aperiodic Rhombic Penrose Tilings.

68 PENROSE TILINGS. Penrose Tilings are based on the golden ratio and have a five fold symmetry. They were discovered in 1974 by the British mathematical physicist, Sir Roger Penrose. He found a set of shapes which tile a surface without generating a repeating pattern (known as quasi-symmetry). There are several different sets of Penrose tiles. The original Penrose tile sets consisted of numerous shapes. He later reduced his set down to six shapes and finally just two shapes: kites and darts.

In 1984, he demonstrated that, when fit together according to certain simple rules, these shapes can infinitely cover a plane, in an uncountable infinite number of arrangements. The patterns may display certain local symmetries, but no fundamental patch is ever repeated consistently; that is, the system is aperiodic.

Fig 4.2: The Second set of Penrose Tiles, made up of six different shapes and a tessellation made from them Fivefold Symmetry : Penrose Tilings & Quasicrystals. : Penrose Symmetry Fivefold :

4 69 There are two different sets of penrose tile i.e. i) the kite and dart and ii) flat and sharp rhomb. They can be made from the two pentagram isosceles triangles as illustrated below.

The kites and darts can be modified to form the set of 2 different rhombs. The angles of the rhombs are one tenth of a circle (36 degrees) times {2,2,3,3} and {1,1,4,4}.So, one rhomb has four corners with Figthe 4.4: The Kite and Dart Penrose Tiling. angles {72, 72, 108, 108} degrees , and the other has angles of {36, 36, 144, 144} degrees. These sets can tile an infinite plane aperiodically.1

Fig 4.3: The two Pentagram Isosceles Triangles. Fig 4.5: The Rhombic Penrose Tiles, made up of two different rhombi. Fivefold Symmetry : Penrose Tilings & Quasicrystals. : Penrose Symmetry Fivefold

1Edwards, Steve. Aperiodic tiling and Penrose tiles. Retrived May 12, 2005 from http://www2.spsu.edu/math/tile/ : aperiodic/penrose/penrose1.htm 4 70 QUASICRYSTALS. Quasicrystals are based on the golden section or Phi. They are three dimensional versions of Penrose Tilings. They were first discovered by Dan Shechtman in April of 1982. He discovered a metal alloy with five-fold symmetry in its diffraction pattern. Since this is not a possible symmetry for the diffraction pattern of a crystalline structure, he assumed that the alloy must have a different structure. His discovery was met by much disbelief. Though it took more than two and a half years to be accepted into scientific literature, it eventually gave rise to the new research area of quasicrystals.

Unlike periodic crystals, quasicrystals possess which is incompatible with periodicity. To this date, quasicrystals have been found that have the symmetry of a Fig 4.6: An Electron Diffraction pattern with fivefold tetrahedron, a cube, an icosahedron and that of 5-sided, symmetry, that would be exhibited by Quasicrystals. 8-sided, 10-sided and 12-sided prisms. They also possess unique physical properties that are currently being researched. One notable property is that even though they are all alloys of two or three metals they are very poor conductors of electricity and of heat.1

Quasicrystals are space filling units. Their geometry uses two that can tile an infinite three dimensional space

aperiodically. Tilings & Quasicrystals. : Penrose Symmetry Fivefold

1Lifshitz, Ron . Quasicrystals - Introduction. Condensed Matter Physics. California Institute of Technology. Retrived May12 : from http://www.cmp.caltech.edu/~lifshitz/quasicrystals.html 4 71 THE GEOMETRY OF QUASICRYSTALS. The geometry of quasicrystals is based on a single rhombus that has sides of equal length and diagonals that are in the ratio 1: Phi. Six of these rhombs can join up in two different ways to form the two basic rhombohedron of quasicrystalline geometry: a flattened oblate one, and a pointed prolate rhombohedron. Like the two Penrose rhombs, these two rhombohedron lend themselves to a great variety of arrangements and combinations. They are space filling units and can tile three dimensional space aperiodically.1 This makes it possible to create a variety of forms and patterns, with just these two basic shapes.

When translated into frame construction, the basic rhombus Fig 4.7: Quasicrystal shape used to form these quasicrystals is inherently not Geometry and methods of space filling with dimensionally stable.2 This problem can be resolved by adding Quasicrystals. a strut to each rhombus that forms the face of the quasicrystals as shown in fig. 4.8.

This geometry is suitable for use in design of architectural space- frame structures for the following reasons: * It is possible to economically factory manufacture the two different unit types and two lengths of struts. * A great variety of form is possible with these two basic shapes, Fig 4.8: The original Quasicrystals, and their including standard flat surfaces. dimensionally stable versions, with a diagonal

member added on to every rhombic face. Tilings & Quasicrystals. : Penrose Symmetry Fivefold

2 Inchbald, Guy. A 3-D Quasicrystal Structure, retrieved from http://www.queenhill.demon.co.uk/polyhedra/quasicr/quasicr.htm : 2

The concept of dimensional stability has been explained in chapter one of this text. 4 72 SURFACES THAT CAN BE PRODUCED FROM QUASICRYSTAL GEOMETRY. A great variety of surfaces can be produced from just these two units. However they most easily form planar surfaces.

PLANAR SURFACES: The different types of planar Fig 4.9: Flat Planar surfaces formed by quasicrystal surfaces formed by the units have been illustrated opposite.

Fig 4.9: Both the oblate and the prolate units can form flat planar surfaces when repeated in a regular manner to form a grid.

Fig 4.10: Corrugated Planar surfaces formed by Quasicrystals. Fig 4.10: When the rows of units are alternately mirrored, they form corrugated surfaces.

Fig 4.10.a: The profile of the corrugations can be altered by different combinations of the rows of the two types of units.

Fig 4.11: Various other patterned planar surfaces are possible using the two units.

Fig 4.11: Patterned Planar surfaces formed by Quasicrystals. Tilings & Quasicrystals. : Penrose Symmetry Fivefold : 4 73 Fig 4.12: PENROSE TILE PATTERNS: Any penrose tile pattern can be translated into a three dimensional space-frame made with quasicrystal units. The resulting frame would be planar and the resulting depth of the frame would depend upon the pattern used.

Fig 4.13: OPEN SEEDED PATTERNS: Open Fig 4.12: Transformation of a two dimensional ‘Penrose Tiling’ to a Patterned Planar surface formed by quasicrystal units. seeded or defective seeded patterns can be created with these units. They result in planar frames with greater depth. These patterns too can be extended infinitely.

The seed (center of the pattern) is called defective because it cannot be filled with similar quasicrystal units. Architecturally, the open seeds present an opportunity for a different material or framework.

Fig 4.13: An ‘Open-Seeded’ Penrose Tiling and its associated a Patterned

Planar surface formed by quasicrystal units. Tilings & Quasicrystals. : Penrose Symmetry Fivefold : 4 74 NON PLANAR SURFACES.

FORCING THE PATTERN OUT OF THE FLAT PLANE:The pattern can be forced out of a plane by changing its scale. Changing the scale of the pattern can be achieved by repeating the symmetry around a central node for several layers.

This would result in forms made out of space-frames that are composed of ‘folded planes’.

The differences between these space frames and their more conventional counterparts are as follows: * The geometry and hence symmetry of these folded planes is not arbitrary, but is determined by the proportion of the golden section that is used to construct the units. * The whole system of folded plane surfaces can be constructed from the basic two rhombohedra. There is no need for ‘special units’ or different lengths of struts at the points where the planes fold. Fig 4.14: Changing the scale of the * The flat planes can be replaced by a penrose pattern for pattern, by repeating the symmetry more visual interest. around a node, * The flat planes can be replaced by ‘curved arch-like’ non planar surfaces. This operation usually requires opening of the central node or seed. Fivefold Symmetry : Penrose Tilings & Quasicrystals. : Penrose Symmetry Fivefold : 4 75 SYMMETRIES AROUND A NODE: CLOSED SEED PATTERNS. The symmetries around a node can be classified by *the number of planes meeting at the central node (seed) *the curvature i.e. synclastic (concave) or anticlastic

The most obvious symmetries that can be created by these units are the 5-fold and 10-fold symmetries. The 5-fold symmetry arises from joining 5 of the prolate (pointed) units to form a and the 10-fold symmetry arises from joining ten of the oblate (flat) units to form a ten planned anticlastic folded Table 4.1: The 17 different symmetries surface. possible around a node

There are a total of 17 different symmetries around a node or ‘closed seed patterns’ as illustrated in the table opposite. Fivefold Symmetry : Penrose Tilings & Quasicrystals. : Penrose Symmetry Fivefold : Fig 4.15: Closed seed pattern with five and ten planes meetin at a node. 4 76 CLOSED SEED PATTERNS.

CLOSED SEED PATTERN ONE is characterized by a CLOSED SEED PATTERN ONE closed seed with: * three planes meeting at the central node * synclastic curvature It forms a shallow pyramid with a triangular base that has bilateral symmetry.

VIEW PLAN ELEVATION

CLOSED SEED PATTERN TWO is characterized by a closed seed with: CLOSED SEED PATTERN TWO * three planes meeting at the central node * anticlastic curvature It forms a shallow anticlastic surface.

VIEW PLAN ELEVATION CLOSED SEED PATTERN THREE is characterized by a closed seed with:

* four planes meeting at the central node CLOSED SEED PATTERN THREE * synclastic curvature It forms a shallow pyramid with a trapezoidal base that has bilateral symmetry.

VIEW PLAN ELEVATION Fivefold Symmetry : Penrose Tilings & Quasicrystals. : Penrose Symmetry Fivefold

Fig 4.16: Closed Seed Patterns : 4 77 CLOSED SEED PATTERN FOUR is characterized by a CLOSED SEED PATTERN FOUR closed seed with: * five planes meeting at the central node * anticlastic curvature It forms a shallow pyramid with a pentagonal base that has five-fold symmetry.

VIEW PLAN ELEVATION

CLOSED SEED PATTERN FIVE is characterized by a CLOSED SEED PATTERN FIVE closed seed with: * five planes meeting at the central node * anticlastic curvature It forms an anticlastic surface that has bilateral symmetry.

VIEW PLAN ELEVATION

CLOSED SEED PATTERN SIX is characterized by a CLOSED SEED PATTERN SIX closed seed with: * six planes meeting at the central node * anticlastic curvature It forms an anticlastic surface that has bilateral symmetry.

VIEW PLAN ELEVATION Tilings & Quasicrystals. : Penrose Symmetry Fivefold Fig 4.16: Closed Seed Patterns : 4 78 CLOSED SEED PATTERN SEVEN is characterized by a CLOSED SEED PATTERN SEVEN. closed seed with: * six planes meeting at the central node * anticlastic curvature It forms an anticlastic surface that has bilateral symmetry.

VIEW PLAN ELEVATION

CLOSED SEED PATTERN EIGHT is characterized by a CLOSED SEED PATTERN EIGHT. closed seed with: * six planes meeting at the central node * anticlastic curvature It forms an anticlastic surface that has bilateral symmetry.

VIEW PLAN ELEVATION

CLOSED SEED PATTERN NINE is characterized by a CLOSED SEED PATTERN NINE. closed seed with: * seven planes meeting at the central node * anticlastic curvature It forms an anticlastic surface that has bilateral symmetry.

VIEW PLAN ELEVATION Tilings & Quasicrystals. : Penrose Symmetry Fivefold Fig 4.16: Closed Seed Patterns : 4 79 CLOSED SEED PATTERN TEN is characterized by a CLOSED SEED PATTERN closed seed with: * seven planes meeting at the central node * anticlastic curvature It forms an anticlastic surface that has bilateral symmetry.

VIEW PLAN ELEVATION

CLOSED SEED PATTERN ELEVEN is characterized by CLOSED SEED PATTERN a closed seed with: * seven planes meeting at the central node * anticlastic curvature It forms an anticlastic surface that has bilateral symmetry.

VIEW PLAN ELEVATION

CLOSED SEED PATTERN TWELVE is characterized by CLOSED SEED PATTERN a closed seed with: * eight planes meeting at the central node * anticlastic curvature It forms an anticlastic surface that has bilateral symmetry.

VIEW PLAN ELEVATION Fivefold Symmetry : Penrose Tilings & Quasicrystals. : Penrose Symmetry Fivefold

Fig 4.16: Closed Seed Patterns : 4 80 CLOSED SEED PATTERN THIRTEEN is characterized CLOSED SEED PATTERN THIRTEEN. by a closed seed with: * eight planes meeting at the central node * anticlastic curvature It forms an anticlastic surface that has bilateral symmetry.

VIEW PLAN ELEVATION CLOSED SEED PATTERN FOURTEEN is characterized CLOSED SEED PATTERN FOURTEEN. by a closed seed with: * eight planes meeting at the central node * anticlastic curvature It forms an anticlastic surface that has bilateral symmetry.

VIEW PLAN ELEVATION

CLOSED SEED PATTERN FIFTEEN is characterized by CLOSED SEED PATTERN FIFTEEN. a closed seed with: * eight planes meeting at the central node * anticlastic curvature It forms an asymmetrical anticlastic surface.

VIEW PLAN ELEVATION Tilings & Quasicrystals. : Penrose Symmetry Fivefold

Fig 4.16: Closed Seed Patterns : 4 81 CLOSED SEED PATTERN SIXTEEN is characterized by CLOSED SEED PATTERN SIXTEEN. a closed seed with: * nine planes meeting at the central node * anticlastic curvature It forms an anticlastic surface that has bilateral symmetry.

VIEW PLAN ELEVATION

CLOSED SEED PATTERN SEVENTEEN is characterized CLOSED SEED PATTERN SEVENTEEN. by a closed seed with: * ten planes meeting at the central node * anticlastic curvature It forms an anticlastic surface that has ten-fold symmetry.

VIEW PLAN ELEVATION

Fig 4.16: Closed Seed Patterns Fivefold Symmetry : Penrose Tilings & Quasicrystals. : Penrose Symmetry Fivefold : 4 82 BREAKING UP THE FLAT PLANES. OPEN SEEDED PATTERNS The seventeen different symmetry patterns that can be created from the closed seeds result in folded planar surfaces. These flat planes can be broken up by opening up the seed and introducing new Closed Seed Pattern 17. units into the pattern of the seed while maintaining its symmetry pattern. The larger the opening of Open Seeded Pattern One. the seed, the more ‘curved’ the resulting surfaces become. In some cases where the opening of the seed is large enough, a different pattern of the same two units can be infilled to create a different geometry within the seed. However it is not possible to ‘fully close’ the open seeds with more of the same two basic units.

Open Seeded Pattern Two. In the following illustrations, the seventeenth closed seed pattern shown opposite has been broken up in three ways. The open seed of the third way is large enough for a new pattern to infil it

Surfaces formed using these open seeds as generators are illustrated on the following page.

Open Seeded Pattern Three. Fivefold Symmetry : Penrose Tilings & Quasicrystals. : Penrose Symmetry Fivefold

Fig 4.17: Open Seed Patterns : 4 83 Open seeded pattern Two. Open seeded pattern One.

VIEW VIEW

ELEVATION ELEVATION Open seeded patterns Three.

VIEW ELEVATION

Fig 4.18: Surfaces derived from Open seeded patterns. Tilings & Quasicrystals. : Penrose Symmetry Fivefold : 4 84 REPLACING THE FLAT PLANES BY A PATTERN. REPLACING A PLANE OF CLOSED SEED PATTERN Another way of breaking up the flat planes is to FOUR BY TWO PLANES WITH A PATTERN. replace them by a surface with a pattern. In the two In the example shown opposite, a single plane of a examples illustrated on the opposite page, some or pentagonal pyramid has been replaced by two planes all of the planar surfaces of the closed seed patterns of a patterned pentagonal pyramid. four and six have been replaced by a pattern.

Fig 4.20: Replacing a planes of closed seed Fig 4.19: Replacing the planes of closed seed pattern four by two planes with a pattern. pattern four and six with a pattern. Fivefold Symmetry : Penrose Tilings & Quasicrystals. : Penrose Symmetry Fivefold : 4 85 JOINING THE MODULES TOGETHER.

The different modules that have been forced out of the plane can be joined together, according to the same symmetry that they followed in the original tiling. This is illustrated in fig. xx opposite.

In the same way, it is possible to join modules with a pattern together, as illustrated in fig. xx below.

Fig 4.21: Joining the Modules together.

VIEW

Fig 4.22: Joining Modules with a pattern together.

ELEVATION Fivefold Symmetry : Penrose Tilings & Quasicrystals. : Penrose Symmetry Fivefold : 4 86 It is also possible to spilt up modules with a pattern, A part of the patterned and then join up parts of these modules together, module has been cut out as illustrated in fig. xx. In come cases additional and repeated, with the quasicrystal units may be required to fill in the necessary rotation. Filler gaps. units are added to cover up gaps. This concludes the study of the possible surfaces that can be created from quasicrystal units. Next, a prominent artist who uses this geometry in his work has been described.

VIEW PLAN

ELEVATION

Fig 4.23: Joining Parts of Modules with a pattern together. Fivefold Symmetry : Penrose Tilings & Quasicrystals. : Penrose Symmetry Fivefold : 4 87 USE OF QUASICRYSTALS IN ART AND ARCHITECTURE BY ARTIST TONY ROBBINS.

Artist Tony Robbin from New York, is the patent holder for the application of quasicrystal geometry to architecture and the author of a computer program for generating Quasicrystals.

He was involved with the proposed design for a building that was to be built with quasicrystal geometry. It was to be an extension to the Technical University at Lingby, Denmark. Due to cost restraints, the project was not carried out, instead a sculptural form was built in the atrium.

Figure 4.24 illustrates the sculpture and other examples of his art work.

Fig 4.24: Art work by Tony Robbin, portraying higher dimensional space. Fivefold Symmetry : Penrose Tilings & Quasicrystals. : Penrose Symmetry Fivefold : 4 88 CONCLUSIONS.

Two dimensional Penrose Tilings and three dimensional quasicrystals use the golden ratio to create a great amount of variety in pattern and form with a limited number of shapes/blocks. This concept can be translated into architecture by mimicking the geometry and creating space-frames from it. Some of the examples of quasicrystal symmetry, studied in this chapter have a potential to form architecturally usable spaces. These examples could be converted into space frames. Such space-frames would be economic because they contain a few basic shapes and lengths that can be standardized and mass manufactured. Thus variety could be introduced into architectural form, without expensive custom manufactured components.

However, even so, the natural curves that this thesis sought to defi ne with the help of basic ‘bricks’ is not possible with quasicrystal geometry.

In the next chapter, the Golden Ratio has been explored further, inorder to arrive at the desired blocks. Fivefold Symmetry : Penrose Tilings & Quasicrystals. : Penrose Symmetry Fivefold : 4 89 Chapter 5: The Golden Tetrahedrons

INTRODUCTION In the previous chapter of this document, it has been demonstrated that, the geometry of Quasicrystals offers potential for the creation of a variety of symmetries and forms. However, the surfaces that the geometry most naturally creates are planar or surfaces with folded planes. The folded planes can then be forced out of a plane by opening out the central node or seed and introducing a pattern into the plane as demonstrated. While this breaks the planes up, it does not do so in a smooth manner.

This original aim of this work was to find a geometry that could create variable smooth flowing surfaces with a relatively limited kit of parts.

The exploration into Penrose Tilings and Quasicrystals demonstrated that by taking cue from nature, variety is possible when the Golden Section Ratio is used to generate the parts.

In this chapter the Golden section Ratio that is built into quasicrystals is used to determine four lengths, that are used to create four different types of triangles. Combinations of these triangles make up the six basic tetrahedra that can be combined together in different ways to yield curved surfaces that are relatively smooth.

The system has been named ‘The Golden Tetrahedral System’ and has been developed and will be patented by the author.

90 THE GOLDEN TETRAHEDRA SYSTEM. THE FOUR BASIC LENGTHS USED IN THE SYSTEM: The four basic lengths of the system are derived from the geometry of YELLOW LENGTH quasicrystals.

The ‘yellow’ length of the system is the length of side of the rhombs that form the faces of quasicrystals. The other lengths can be derived assuming the yellow length to be of unit value. BROWN LENGTH

A rhombus is drawn with sides of unit lenghth (Yellow Length) and diagonals that are in the ratio 1: Phi(φ). This is the same rhombus that makes up the quasicrystals. The shorter diagonal of this rhombus becomes the next length of the system: the ‘Brown’ length. Its numerical value is 1.0514622... BLUE LENGTH

The next two lengths are derived from the 3 dimensional diagonals of the two types of quasicrystal units i.e. the oblate and the prolate quasicrystals.

The ‘Blue’ length is the 3 dimensional diagonal of the prolate(pointed) quasicrystal unit. Its numerical value is 1.4510592... GREEN LENGTH

Fig 5.1: The four Basic Lengths of The ‘Green’ length is the 3 dimensional diagonal of the oblate(flat) the Golden Tetrahedral System. quasicrystal unit. Its numerical value is 0.5627774... The Golden Tetrahedrons. : 5 91 MATHEMATICAL FORMULAE . The formulae for the lengths have been derived in Triangle ACB is a . three ways. The first method assumes the yellow length to be of unit value. The second method 22 2 ?abc ()1 assumes the ‘Brown’ length to have a value of two units. The third method assumes the ‘Brown’ length Substituting thr values for lengths a,b and c we get, to have a value of 2φ units. This results in simpler

2 2 2 formulae to derive the other lengths. 11..xx M 222M xx . 1 METHOD 1: Assumption: The ‘Yellow length’ has a 22 2 xxM11 MM 1 value of one unit.  222 xxx. M 1 21xx22. To derive the ‘Brown’ Length: Draw a rhombus M 2 with side of unit length and diagonals that have a 21 x  M ratio 1: Phi(φ). 1 x 2 2  1 M x 2

From the figure opposite, weM know that the ‘Brown’ length of the system has a value of 2x, 1 Therefore the ‘Brown’ Length = 22x 2 M

= 1.0514622... In Triangle ABC, Length AC = b =1x, The Golden Tetrahedrons.

Length CB = a = φx , : Length AB =c =1 5 1In a right angled triangle, the square of the length of the hypotenuse, is equal to the sum of the squares of the lengths 92 of the two remaining perpendicular sides. To derive the ‘Blue’ Length: In the prolate Triangle ACB is a right triangle. quasicrystal unit shown below, ABCD is a . o ?‘ACB 90

D ? B abc22 2 (Pythagorean Theorem)

§ ·  ¨ 2 ¸ ©  ¹ 2 1 2 C 12§ · c  ¨ 2 M ¸ A ©  ¹

22§ 1 · 14 ¨ ¸ c © 2 ¹ PROLATE QUASICRYSTAL M

4 2 1 c§ · ?2 ¨ ¸ M©  ¹ In Triangle ACB,

4 Length AC = b =1(yellow length) c 1 2

M 1 Length CB = a = 2 (Brown Length) 2  M § 4 · Therefore the ‘Blue Length’= 1 ¨ ¸ ©2 M ¹ Length AB =c =? (Blue Length)

= 1.4510592... The Golden Tetrahedrons. : 5 93 To derive the ‘Green’ Length: The figure below shows a single prolate quasicrystal to thr right and a pair of oblate quasicrystals. D

A D A C B B

PROLATE QUASICRYSTAL PAIR OF OBLATE QUASICRYSTALS The two oblate quasicrystals can be aligned with the prolate quasicrystal to match up the corresponding points A,D and C. In the resulting figure, point B also In the figure above, lies in the same plane as Triangle ACD. Let point O Length AB = c=1 (yellow length) be the intersection of line AB and line CD. 1 Length AC = Length BD = 2 (brown length) 2  M D D Point O is the mid point of line BD. A Therefore, Length BO = 1/2 Length BD C A

O C 1 B = 2  M B

PAIR OF OBLATE QUASICRYSTALS ALIGNED Length AO + Length OC = Length AC

WITH THE PROLATE QUASICRYSTAL. Therefore, The Golden Tetrahedrons. 1 The ‘Green Lengths’ CB and DB can be derived from x + y = 2 : 2  M this figure using the Pythagorean Theorem. 5 94

First consider Triangle AOB Therefore, Next consider Triangle BOC,

o ‘AOB 90 1 2 2 b§ · 2 yx 2  ?a ¨ ¸ y  2 M 2© ¹ § 2· ?b¨ ¸ § · (Pythagorean Theorem) 2 © ¹ 2 1 1 ¨ ¸ x c  ©  ¹ 2 y 2 1 § 2 · (Pythagorean Theorem) 2 2 ¨ ¸ MM ¨ 1  ¸  § · 2  ¨ ¸  ¨ 2 MM¸441 1 ¨  2 ¸  2 ¨ ¸  1   a © ¹ ¨2 ¸ 1 21 2 2 ¨ ¸ y 2  © 2 M ¹ § ·  x 2 12 2  2 ¨ ¸  M   ©  ¹  § 2 · M M 2 M  ¨ ¸ 1 1 1 441 1 ©  ¹ y 2 2   M a  § · 2 2 2 2  ¨ ¸ M 221 ©  ¹ M   MM x 1 21 M 441 1 y 2 1 2 M M a  2 2 2 M § · M   ?2 1¨ ¸ ? M MM x 1 ©  ¹ 641 2 Squaring both sides, M 2 a MMM M 2 2 §21M · 641 1 2  M y ¨ ¸ a x 1 © 2 M ¹ 2 2 MM  441 1 Length AO + Length OC = Length AC y 2 MM The ‘Green Length’ = a M 2 M   Therefore, M 1 541MM 641MM  x + y = 2 y 2 = 2  M 2 2  M M = 0.5627774... The Golden Tetrahedrons.

: 5 95 METHOD 2: Assumption: The ‘Brown length’ has a To derive the ‘Blue’ Length: In the prolate value of two units. quasicrystal unit shown below, ABCD is a rectangle.

To derive the ‘Yellow’ Length: Draw a rhombus D with diagonals that have a ratio 1: Phi(φ). Let the B length of the shorter diagonal (Brown Length) be two units. C A

PROLATE QUASICRYSTAL In Triangle ACB, Length AC = b = 2  M (Yellow Length) Length CB = a = 2 (Brown Length)

In Triangle ABC, Length AB =c =? (Blue Length) Length AC = b (Brown Length/2)=1, Triangle ACB is a right triangle. Length CB = a = φ , 22 2 Length AB =c =?(Yellow Length) ?abc (Pythagorean Theorem) Triangle ACB is a right triangle. Substituting values for lengths a,b and c we get, 22 2 2 ?abc (Pythagorean Theorem) 22 c 22 M  Substituting values for lengths a,b and c we get, 2 M 22 2 c 42 c 1 M  c2 6 M c2 11M  ?   c 6 c2 2 M ? M Therefore the Blue Length = The Golden Tetrahedrons. c 2 6  M

Therefore the Yellow MLength = 2  M = 2.7600786.... : = 1.902113... 5 96

To derive the ‘Green’ Length: As First consider Triangle AOB Next consider Triangle BOC, demonstrated earlier the figure below can o o ‘AOB 90 ‘BOC 90 be obtained by aligning a pair of § 2 · 2 ?22 b¨ ¸ b§ · oblate quasicrystals to mach up with the cx © ¹ ?2 ¨ ¸ 2 2 a © ¹ y § · 2 corresponding vertices of a single prolate  2¨ ¸ 2 © ¹ 2 2 § · quasicrystal. 2 M x ¨ ¸   2 © 2¹ 2 2 2 ? 2  a 2 M 21M x 2  ? 2 2  ?x 1 a 22 M 12   22M x 22 a 144 MM M x 2 MM M a 54 1 we know that, ?a 2 63 M x + y =2 a 2 32 y = 2 - x M y = 2 - φ a 32 M The ‘Green Length’ = a

In the figure above, = 32  M Length AB = c= 2  M (Yellow Length) Length AC = Length BD =2 (Brown Length) = 1.0704663... Point O is the mid point of line BD. Therefore, Length BO = 1/2 Length BD = 1 Length AO + Length OC = Length AC The Golden Tetrahedrons. Therefore, x + y = 2 : 5 97 METHOD 3: Assumption: The ‘Brown length’ has a To derive the ‘Blue’ Length: In the prolate value of 2φ units. quasicrystal unit shown below, ABCD is a rectangle.

To derive the ‘Yellow’ Length: Draw a rhombus D B with diagonals that have a ratio 1: Phi(φ). Let the length of the shorter diagonal (Brown Length) be 2φ units. C A

PROLATE QUASICRYSTAL In Triangle ACB,

Length AC = b = 43 M  (Yellow Length) Length CB = a = 2φ (Brown Length) Length AB =c =? (Blue Length) In Triangle ABC, Triangle ACB is a right triangle. Length AC = b (Brown Length/2)=φ , 22 2 ?abc (Pythagorean Theorem) Length CB = a = φ2 ,

Length AB =c =?(Yellow Length) Substituting values for lengths a,b and c we get,

Triangle ACB is a right triangle. 2 2 2 22 2 c 24MM 3 ?abc (Pythagorean Theorem) 22  Substituting values for lengths a,b and c we get, c 443MM 22 22  c MM c2 4143 MM 2 2 2 c 11 c 4443 MM MM    2 c221212 ?c 87  MMM Therefore the Yellow Length M 2 c 112 1 = 43M  c 87 The Golden Tetrahedrons. MM M ?2  Therefore the Blue LengthM = 87M  c 43 = 3.0776834... : = 4.4659010.... c 43M 5 M 98

To derive the ‘Green’ Length: As First consider Triangle AOB Next consider Triangle BOC, demonstrated earlier the figure below can o o ‘AOB 90 ‘BOC 90 be obtained by aligning a pair of § 2 · § 2 · ?22 b¨ ¸ ?2 b¨ ¸ 2 oblate quasicrystals to mach up with the cx © ¹ a © ¹ y 2 2 § · corresponding vertices of a single prolate  ¨2 ¸ 2 2 2 © ¹ § · quasicrystal. 43M x ¨ ¸     2 ©2M 2¹ 2  22 M  a 76432  43MMx 2 MMM ?     43x 2 1 a 22MM76432 MM ?MM    2 2 ?x 431  a 17MMMM 64 3 2 2 x 32MM ?a 2 87432   MMM x 32M ? we know that, M a 87432

x y 2M a 3 MMM ?yx 2M  ?   The ‘Green Length’ = a y 232MM ? 2  2 ?y 232   In the figure above, MM = 3 ?y 22 443232   Length AB = c= 43 M  (Yellow Length) ?2 MMM  M Length AC = Length BD =2φ (Brown Length) y 414323 2 = 1.7320508... ?   Point O is the mid point of line BD. y 2 4443232MMMM Therefore, Length BO = 1/2 Length BD = 1 y 2 76432MMMM Length AO + Length OC = Length AC

MMM The Golden Tetrahedrons. Therefore, x + y = 2φ : 5 99 TABLE 5.1: COMPARISON OF LENGTHS OBTAINED WITH DIFFERENT STARTING ASSUMPTIONS .

Assumption Yellow Length (YL) Brown Length Blue Length Green Length (GL) (BrL) (BL)

(YL) =1 1 § 4 · 641MM  1 2 1 ¨ ¸ 2  M ©2 M ¹ 2  M 1.0000 1.051462224… 1.451059202… 0.562777422…. (BrL) = 1 2  M 6  M 32  M 1 2 2 2 0.951056516… 1.0000 1.38003931… 0.535233134…

*(BL) = 1 0.68915176… 0.72461704… 1 0.38783905… *(GL) = 1 1.77690142… 1.86834472… 2.57838916… 1

(BrL) = 2 2 2  M 6  M 32  M

1.902113033… 2.0000 2.76007862… 1.070466269…

(BrL) = M 43M  M 87M  3

2 2 2 1.538841769… 1.618033989… 2.232950509… 0.866025403…

(BrL) = 2M 43M  2M 87M  3

3.077683537… 3.236067977… 4.465901019… 1.732050808…

The Golden Tetrahedrons. *The values in these rows were calculated by using the process of ‘scaling’ on a computer model. : 5 100 RATIOS OF THE DIFFERENT LENGTHS .

Yellow Length Brown Length Blue Length Green Length (GL) (YL) (BrL) (BL) RATIOS 1.538841769… 1.618033989… 2.232950509… 0.866025403…

Yellow Length (YL) 1 1.051462224… 1.451059202… 0.562777421… 1.538841769…

Brown Length (BrL) 0.951056516… 1 1.38003931… 0.535233134… 1.618033989…

Blue Length (BL) 0.689151758… 0.72461704… 1 0.387839049… 2.232950509…

Green Length (GL) 1.776901421… 1.868344722… 2.578389157… 1 0.866025403…

The Golden Tetrahedrons. : 5 101 THE GEOMETRY OF THE SYSTEM . THE FOUR (+2) BASIC THE FOUR BASIC TETRAHEDRONS USED IN THE TRIANGULAR FACES USED IN SYSTEM: THE SYSTEM: The ‘’ tetrahedron has three The ‘Pink’ triangular face is ‘Aqua’ Faces and one ‘Pink’ face. an that has PINK FACE Its faces are drawn with a green three ‘Brown’ lengths forming EARTH TETRAHEDRA central triangle. the sides.

The ‘Water’ tetrahedron has two The ‘Aqua’ triangular face is ‘Purple’ Faces and two ‘Aqua’ faces. an that has Its faces are drawn with a blue central AQUA FACE two ‘Yellow’ lengths and one triangle. ‘Brown’ length. WATER TETRAHEDRA

The ‘’ tetrahedron has two The ‘Orange’ triangular face ‘Orange’ Faces, one ‘Pink’ face and has one ‘Yellow’ length, one one ‘Aqua’ face. Its faces are drawn ORANGE FACE ‘Brown’ length and one ‘Blue’ with a red central triangle. length. FIRE TETRAHEDRA The ‘’ tetrahedron has two ‘Orange’ The ‘Purple’ triangular face has Faces and two ‘Aqua’ faces. Its faces one ‘Yellow’ length, one ‘Brown’ are drawn with a yellow central PURPLE FACE length and one ‘Green’ length. triangle.

Fig. 5.2: The four The Orange and the Purple AIR TETRAHEDRA Basic Triangular The Water and the Air tetrahedra are Faces of the faces are enantiomorphic, that Fig. 5.3: The four The Golden Tetrahedrons. enantiomorphic, and both their right system. is, they have right and left Basic Tetrahedra of the system and left handed forms are part of the : handed forms.

system. 5 102 COMBINATIONS OF THE FOUR BASIC TETRAHEDRA. The four tetrahedra can combine in different ways to from symmetric and asymmetric polyhedra with conditions ranging from four to ten triangles meeting at a point. All the figures illustrated, are three dimensional and have a depth to them. In some cases ‘hidden’ tetrahedra exist a b c below the visible surface.

Figure 5.4.h illustrates that an icosahedron can be created from the earth tetrahedra. d e

f g

j h i

h The Golden Tetrahedrons.

lm

k : Fig. 5.4: Combinations of the four Basic Tetrahedra. 5 103 SURFACES WITH UNIFORM CURVATURE DESCRIBED BY THE SYSTEM.

DOMES: The geometry can be used to form a dome with thickness as illustrated below.

PLAN VIEW

ELEVATION The Golden Tetrahedrons. Fig. 5.5: Domes formed by the System. :

VIEW 5 104 VAULTS: This vaulted surface is curved version of the ‘Octet Truss’ that has been described in chapter one. The octahedron shape is made up of 2 fire and 2 air tetrahedra. The tetrhedron shape used is the ‘earth’ tetrahedron. In this case the ratio of the depth of the frame to the diameter of the circle is 1:14. The circle that gets described by this GENERATING CURVE. geometry does not close up fully. There is either a gap or an overlap.

CYLINDERS: The ‘earth’ tetrahedra can be used to build a cylinder that may form a column.

BARREL VAULT The Golden Tetrahedrons. CYLINDER : Fig. 5.6: Circles, Cylinders and Vaults formed by the System. 5 105 SURFACES WITH RANDOM CURVATURE DESCRIBED BY THE SYSTEM. The Four Golden Tetrahedra can also be combined to form random, even anti-clastic curves, that begin to approximate natural ones.

VIEW PLAN The Golden Tetrahedrons. SIDE ELEVATION FRONT ELEVATION Fig. 5.7: Random Curved Surfaces Fig. 5.8: An Anticlastic Curved Surface : Described by the System. Described by the System. 5 106 CONFORMING THE SYSTEM TO ANY RANDOM PREDETERMINED FORM.

Fig. 5.9: A wire mesh model of the System is seen going above and below the surface it is attempting to conform to.

One of the methods of designing a roof or building with this system, is to first design it using some other Fig. 5.11: The Golden Tetrahedral roof, without the mesh methodthat was that used allows to design the easyit. creation and modification of a free flowing surface mesh. The system can then be made to approximately conform to this designed mesh. The smaller the size of the units, the more accurately it can conform to the original mesh.

This is the method that was used by the author for the design of the roof of the parachute pavilion. Figures

5.9 to 5.11 illustrate part of the design process. The Golden Tetrahedrons. Fig. 5.10: The same model seen shaded with solid faces. : 5 107 CONCLUSION. These six tetrahedra can be combined to form three dimensional frames that take on different forms.

ADVANTAGES. * A great variety of form is possible using just these six tetrahedra: from flat surfaces and folded planes as demonstrated in the previous chapter to cylindrical surfaces, spherical surfaces and random anticlastic surfaces. It is possible to ‘sculpt’ out forms with these units. * The basic unit is a tetrahedron which is dimensionally stable and the strongest possible unit form. The frame thus formed from these units Fig. 5.12: An Anticlastic Curved Pavilion Described by the System. is therefore dimensionally stable. * The system is modular, it consists of four different lengths and four different triangular faces that could be mass produced. * The patterns that result have local symmetries and in some cases a larger symmetry can exist in the form. These aperiodic patterns can be extremely attractive. The local symmetries present give a sense of order to the geometry, yet its aperiodic nature creates variety and disorder in the pattern.

DISADVANTAGES. * The ‘number count’ for the components of such a frame would be very high. * The number of connections that have to be made would make Fig. 5.13: Shadow Patterns. construction of the frame a long and laborious process. The Golden Tetrahedrons.

* The complexity of the patterns could make the assembly difficult. : * The number of connections would increase the cost of the frame. 5 108 For the interested reader, a brief study of currently available space frame construction technology is presented in Appendix 1 of this document. This is in order to determine the best possible solution for the construction of such frames.

In Appendix 2, The Design Brief for the Parachute Pavilion Design Competition, that has been used as a test case for the Golden Tetrahedral System, is presented. The Golden Tetrahedrons. : 5 109 His Life and Complete Graphic Work BIBLIOGRAPHY.

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113 Appendix 1: Space Frame Structures.

INTRODUCTION

Geometry has been used to define a system of modular components in the form of quasicrystals and irregular ‘Golden Tetrahedrons’. These can be assembled in different patterns and symmetries (as demonstrated in chapters four and five) to result in a variety of forms ranging from flat planar surfaces to folded planes and complex faceted surfaces that approximate curved surfaces.

This geometric framework defined by the tetrahedrons is dimensionally stable and is suitable for translation into a structural space frame system composed of hubs and struts. The modular units (or parts of them) can be factory manufactured and site assembled according to a predetermined design. The architectural challenge then lies in the detailing of the joinery and connections and choice of materials for the system.

This appendix deals with the design of the joints and connections of the system. It begins with a study of space frame construction and the advantages and disadvantages of such systems. A few examples of commercially available patented systems are examined for their joinery details.

114 SPACE FRAMES ‘A space frame is a three-dimensional framework for STRUTS: They are the linear elements that enclosing spaces in which all members are interconnected span from node to node in the space frame. and act as a single entity. A benefit of this type of Generally tubular or circular hollow sections structure is that very large spaces can be covered, are preferred for use in space frames because uninterrupted by support from the ground.’1 The structure of their good compressive and local bending usually depends upon full triangulation. resistance.

A Space frame is defined by its geometry, the overall SKIN : This is the exterior cladding of the design or form and methods of construction. Most space space frame that requires to be weather proof. frames are based on the manufacture and assembly It is advantageous for the skin to be light of three components: hubs, struts and skin which are weight and easy to assemble. often standard components and their shape and length is determined by the geometry of the space frame. There SPAN TO DEPTH RATIO are many ways of assembling these components. Often The overall span: depth ratio of space frames the hubs and struts are assembled first to form the depends upon the geometry of the system framework which is then clad with a skin. used, the method of support and the type of loading. These factors vary from project to COMPONENTS OF A SPACE FRAME project and this makes it difficult to generalise HUBS: These are the node joint connections. They are a span: depth ratio. typically engineered to receive the struts at the precise However a range of ratios can be generalized. angle determined by the geometry of the space frame. For For flat planar grids the ratios vary from 12: Space Frame Structures Space Frame smaller space frame structures that are fully triangulated, 1 to as much as 35:1 with 20:1 being an : the connection between the hubs and the struts may be average ratio. For cantilever spans the ratios designed to be flexible or pin jointed. are usually around 9:1.

1Chilton, John . Space Grid Structures. Oxford: Architectural Press, 2000. (Chilton back cover) 2Borrego, John . Space Grid Structures: Skeletal Frameworks and Stressed-Skin Systems. Cambridge: The M.I.T. Press, 1968.(Borrego 17-19) Appendix 1 115 ADVANTAGES OF SPACE FRAMES Some of the advantages of space frames are listed below: * Loads are distributed more evenly to the supports. * Deflections in the three dimensional space frames are less than in comparative two dimensional structures of equivalent span, size and loading. *The open nature of the structure allows installation of mechanical and electrical services and air handling ducts within the structural depth of the frame. *Assembly and fixing details are standard and often simplified. * Secondary elements such as purlins may not be required. If required, secondary members can be attached at the nodes * ‘The structural indeterminacy of space frames means that in general failure of one element does not lead to overall failure of the structure.’1 *Accurate components can be factory manufactured in bulk as they are usually standard components. *Space frame components tend to be smaller in size as compared to standard beam and column elements. This makes them easily transportable and simple to assemble on site. *The modular nature of the components makes it easy to extend the surface or form. *The frame may even be dismantled and re-erected at some other place with

out difficulty. Structures Space Frame *Large column free spaces allow for considerable freedom in space planning. : *Space frames can be assembled on the ground and hoisted into position.

1Trebilcock, Peter, and Mark Lawson. Architectural Design in Steel. New York: Spon Press, 2004. (Trebilcock et al. 129) Appendix 1 116 DISADVANTAGES OF SPACE FRAMES The disadvantages of space frames include the following:

* In some cases, particularly for short spans, or in cases where there is no benefit of two way spanning action, space frame structures may be more expensive than alternate structural systems.1 *The geometry of the space frame defines the overall shape or form and hence the overall shape or form is limited to those that can be defined by the geometry. This imposes a limitation on the possible forms. *Irregularly shaped buildings require special solutions at points where the geometry of the space frame cannot fit the shape. *Visually space frame structures may appear too ‘busy’. The perceived of the structure is affected by the grid size, depth, and configuration of the geometry. *A space frame with complex geometry may require longer time and greater skill for erection on site. *When space frames are used to support floors, some form of fire protection is required. This may be more expensive to achieve economically because of the large number of relatively small 2

components. Structures Space Frame :

1Trebilcock, Peter, and Mark Lawson. Architectural Design in Steel. New York: Spon Press, 2004. (Trebilcock et al. 129) 2Chilton, John . Space Grid Structures. Oxford: Architectural Press, 2000. (Chilton 20-21) Appendix 1 117 TYPES OF SPACE FRAMES ERECTION METHODS Space frame systems may be classified according to the *Scaffold Method hub and strut connection type . Individual Components are assembled in Types of hub and strut connections include:1 place at the actual elevation. In cases where * Connector Plates (Single preformed and Multiplanar the frame is a continuous surface that rises Gusstet Type) up from the ground, the part of the space * Spherical Nodes (Hollow and Solid Types) frame that has already been assembled * Cylindrical (Extruded) below may act as scaffolding for the part * Prismatic above.. * Nodeless *Lift up Method Components are pre assembled on the ground A Few examples of the above types of space frames and crane or hoist assembled into place. as seen in commercially available systems have been described in the next few pages. Some systems employ a combination of different strut to hub connections for the top and bottom layers of the frame to cater to the different requirements of cladding materials. Space Frame Structures Space Frame : Appendix 1 118 CONNECTOR PLATES HUB TYPES In this space frame system, connector plates are either pre formed or welded/bolted together to receive the struts at the required angle. The connection to the struts is agaiin made either by bolting or welding.

The ‘UNISTRUT’ sysyem illustrated in fig xx opposite is an example of the preformed connector plates type. The struts are site bolted to the connector plates. MULTIPLANAR GUSSER PLATES TYPE : In this system the node consists of gusset plates welded together so that connections can be made to the struts at the required angle. Fig. xx opposite shows details of this type of a system from an auditorium sports pavilion at the University of California at Los Angeles, designed by Welton Beckett and Associates. Space Frame Structures Space Frame

Fig. 6.1: The UniStrut Spaceframe System. :

1Chilton, John . Space Grid Structures. Oxford: Architectural Press, 2000. (Chilton 31) Appendix 1 119 SPHERICAL HUB TYPES In this type the node may either be hollow as in the case of the Nodus system or solid as in the Mero system.

THE NODUS SYSTEM: The Nodus system was developed by the tubes division of The British Steel Corporation and introduced commercially in the early 1970’s. Since 1985 it has been owned by Space Decks Ltd.. The node is composed of two half casings which are joined together with a central bolt. The members are clamped between the two half casings and the diagonal bracing members are attached to the casing via forked connectors Fig. 6.2: The Nodus Hub. and steel pins.

THE MERO SYSTEM: Mero Structures Inc. has been a full service company since 1986. It is based in Wisconson. They supply services for product development, engineering, fabrication, installation and project management for three dimensional structural systems. The mero hub is a solid ball that is engineered to receive the struts at the precise angle determined by the geometry of the frame. Threaded holes are drilled into the solid ball at the required angle to Space Frame Structures Space Frame receive the struts. :

Fig. 6.3: The Mero System. Appendix 1 120 EXTRUDED HUB TYPES This system is made up a cylindrical node that is GEOMETRICA: Geometrica was founded in 1992 by Dr.. extruded with the required dovetailed notches in Wright. The company specializes in designing, detailing and its cross section. manufacturing components for large column-free enclosures for a range of commercial uses. THE TRIODETIC SYSTEM : It was developed The design of the geometrica hub and strut joint is during the 1950’s by Fentiman Brothers of based on the Triodetic system with an extruded cylindrical Ottawa, Canada. It consists of an extruded hub. The ends of the tubular memembers forming the struts aluminum section with longitudinal slots into are flattened out and are dovetailed into a matching slots which the cramped ends of hollow tubular bars in the connector. A variety of geometric configurations can are slotted. They are held in position between be built with this system by varying the end-angles of the two end plates held by a single bolt passing tubes. through the center of the node. As the material Geometrica has improved on Fentiman’s original at the ends of the tubes is not removed but only idea by optimizing the connector’s engagement patterns, displaced, the strength of the cross section is resulting in full transmission of the tube material strength maintained in this system. through the joint.1 Space Frame Structures Space Frame :

Fig. 6.4: The Triodetic System.

1This claim is made by the company on their web site - http://www.geometrica.com Appendix 1 Fig. 6.5: The Geometrica System. 121 THE N55 SPACE FRAME. N55 is a design group based in Copenhagen. Their philosophy is to ‘‘work with art as a part of everyday life’. They have designed the N55 space frame and it functions as their work space.

The original N55 space frame is designed as a living unit for 3-4 persons, but the geometry that is used for the space frame can be adapted to respond to variable spatial requirements. The entire unit is constructed from small lightweight components which can be assembled by hand. All components are materially minimized and have a low degree of manufacturing. The components take up very little space when stacked. Fig. 6.6: The N55 Spaceframe System.

GEOMETRY: The N55 Space frame is based on the geometry of the octet truss.1 A single octahedron combines with two tetrahedra to form a rhombohedron (as shown in fig. XX ) NUMBERS COUNT: The geometry of octet space frames which serves as the basic building block for the requires a larger number of struts than most -other Space Frame Structures Space Frame space frame. This geometry distributes loads types of space frames. The N55 space frame was made : evenly throughout the construction to delivers from approximately 3000 pc 110° angle struts and greatest strength with the least use of material. 4500 pc 70° angle struts. Appendix 1 122 STRUTS: The equal length struts are economically produced from highly durable acid resistant stainless steel. They are bent in 70° angles for tetrahedra and 110° angles for octahedra. They are assembled by hand using stainless acid resistant bolts and nuts.

ASSEMBLY: In this type of construction manufactured ‘hubs’ are not required. The angles of the geometry are built into the struts. 6 pc of the 70° angle struts are assembled to form a tetrahedron. 12 pc of the 110° angle struts form an octahedra. Two tetrahedra are then attached to one octahedra to form the basic building block. The building blocks are fastened together to form the desired frame. The struts overlap each other in a regular pattern (as shown in fig XX opposite). Fishplates are bolted at the node points to increase the rigidity of Fig. 6.7: The N55 Spaceframe: Assembly and Details. the inner and outer skin.

The framework is clad with overlapping acid-resistant triangular and square steel plates. For the windows polycarbonate has been used instead of glass as this Space Frame Structures Space Frame material does not break with small amounts of pressure. : This provides for movements in the structure due to stress and temperature variation. Appendix 1 123 AURODYN: HYPERFRAME TM (Solid ball Hub Type) Aurodyn is in the process of developing ‘Hyperframe’ - a node and strut building structural system and its related products. They are also designing a 3-D Modeling software program to be used for designing structures based on their geometry. Some of the features of Hyperframe include: * The system is designed with the golden ratio geometry, which enables it to adapt to a broad range of applications. It can be used to design an array of structural forms including flat surfaces, simple curves, complex curves, domes, multi layered tetrahedral networks, polyhedra, pyramids and other meshes. * The geometry is inherently triangulated with planar triangulation in single layers or surfaces, and tetrahedral cells in any structure of two or more layers. * The modularity of hyperframe facilitates easy assembly and disassembly. *It provides opportunity to add onto or evolve permanent structures over time. Fig. 6.8: The Aurodyne ‘Hyparframe’ System. * The nature of the geometry enables unlimited scalability.

Aurodyn is also developing ‘Smart Panels’ that are pre assembled modular units that can be used to clad ‘Hyperframe’ structures. Space Frame Structures Space Frame They are energy efficient and are made from sustainable materials. : The system also includes panels with integrated solar collectors for radiant heating and cooling. Appendix 1 124 BAMBUTEC International. BAMBUTEC International (Germany) has developed ASSEMBLY AND BONDING: Assembly involves fitting technology to make the use of bamboo possible, the rods into the hubs and bonding them with with great precision, in and other frame a pre-injected superstrong adhesive. Additionally structures. Their aim is to improve the economy threaded dowels are used to tighten the joint. The of people who earn their living through growing, high axial tension induced increases the strength of processing and marketing bamboo. They also desire the joint which is stressable after 24 hours. to promote the use of bamboo as a sustainable building material.

BAMBUTEC TECHNOLOGY: Bambutec technology facilitates precise production of the joints or hubs that would receive the bamboo rods. It also mills out the rods so that they fit precisely into the hubs. Fig. 6.9: A Bambutec Bridge.

PRODUCTION OF THE JOINTS: The joints are plotted full-scale according to the CAD-. The CAD plots are glued onto the board-material and sawed out. The fittings for the rods are machined with a special milling machine for BAMBUTEC-joints.

PRODUCTION OF THE RODS: The milling machine Space Frame Structures Space Frame for BAMBUTEC-rods mills out the ends of the rods : precisely to ensure that they are in an exact axial alignment.

Fig. 6.10: Bambutec Technology. Appendix 1 125 CONCLUSIONS

Thousands of individual members are required for the The Geometrica system utilizes three components construction of space frames. Economy can be achieved in the joint: the extruded hub, end plates and with the use of factory manufactured modular components. a central bolt. However in this system a higher However, the sheer number of connections to be made degree of manufacturing is required for both the can escalate the cost of the project. The complexity of the hubs and ends of the tubular members so their connections also affects the time required for assembly. slots match up precisely.

Available technology for space frames construction ranges Both the Aurodyne and Mero systems, utilize solid from the fairly complex systems with a high degree of hubs that have the required angles of the geometry manufacturing, to extremely low-tech solutions. Each has factory built into them. The ends of the struts too advantages and disadvantages. have to be built up with threaded bolts that can be screwed into the hubs The ‘Nodus’ system is an example of a complex system with multiple manufactured components required for the The most ideal and economic solution would be construction of a single joint. The advantage of this the simplest one, involving the least number of system is that the node is hollow and lighter in weight components that go into the joint. Also the degree compared to a system that utilizes solid hubs. of manufacturing required for each component should be minimized. The N55 system on the other hand, is a simple one that utilizes only two components for the construction of the

frame: angle sections and bolts. In this system each Structures Space Frame

angle section is bolted to the neighboring one with two : bolts, such that a total of at least 6 bolts are used at every node. Appendix 1 126 Appendix 2: The Parachute Pavilion Competition Brief.

INTRODUCTION This section deals with the application of the system described in the earlier chapters. The ‘Parachute Pavilion’ design competition has been selected as an appropriate site and program.

Design Competition Background: This is an Open Design Competition seeking innovative proposals to contribute to the 21st-century vision for Coney Island. The competition is directed by the Van Alen Institute (VAI), together with the Coney Island Development Corporation’s (CIDC) planning initiatives. The site is a designated landmark in the shadow of the famed Parachute Jump—an iconic reminder of the history of Coney Island.

Design Competition Objectives for the Pavilion The Pavilion design is to play a key role in the ongoing revival of Coney Island, with implications for urban recreation in waterfront communities and beyond. The Parachute Pavilion is planned to be an all-season generator of activity, drawing the public onto the boardwalk, the beach and Surf Avenue, and to a new recreational destination. The recreational and commercial activities accommodated by the pavilion are to serve as a catalyst for animating the park and waterfront. It is to be a paradigm of innovative design that connects to both the history and future of Coney Island. 127 CONEY ISLAND HISTORY

“Brooklyn: The Borough of Brooklyn has a population of 2.5 million and occupies more than 71 square miles. Brooklyn is minutes from Manhattan by subway, ferry, water taxi or car, and is connected to Manhattan by numerous subway lines and bridges. Brooklyn plays an important role in New York’s economy: it is home to some 37,000 companies covering a wide range of industries, two of the City’s functioning port facilities and several industrial . Since January 2002, Marty Markowitz has served as Brooklyn’s Borough President.

Coney Island: Coney Island is not actually an island, but a small peninsula that hangs from the southernmost edge of Brooklyn. It is easily accessible from the city’s other boroughs and points inside Brooklyn by car and by subway. Fig. 7.1: Map of New York. The peninsula contains several distinct neighborhoods: the private residential community of Sea Gate on the western tip, Manhattan Beach at the far eastern edge, and Brighton Beach and Coney Island in the middle. The Coney Island neighborhood itself comprises a mix of uses, including high-rise residential development as well as two-family and single-family dwellings; Fig. 7.2: Map of Coney Island. some neighborhood retail along Surf Avenue, Mermaid and Neptune avenues; and the centrally located amusement area.

The historic amusement area spans from West 8th to West 24th Street, and The Parachute Pavilion: Competition Brief. Pavilion: The Parachute from Surf Avenue to the Atlantic Ocean. In addition to amusement parks, rides : and concessions, this area contains a three-mile beachfront boardwalk, the New York Aquarium, KeySpan Park—home of the Brooklyn Cyclones minor league baseball team—and Asser Levy Park and Amphitheater.”1

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128 “Since the early 1800s, Coney Island, “playground of the world,” has played many roles in the lives and imagination of New Yorkers and the world. From its beginnings as a quiet seaside town, Coney Island went on to boom years in the 1880s, as entrepreneurs rushed to stake their claims and make their fortunes. The area enjoyed brief stability in the late 1890’s and early 1900’s, the heyday of Luna Park (1903-1946), Dreamland (1904-1911) and

Steeplechase Park (1897-1907, 1908-1964), Fig. 7.3: Historic View of the Parachute Jump Coney Island’s famed amusement parks, but with the Great Depression, Coney Island Today, Coney Island is in the midst of a revival, transformed once again. The area became a spurred by public, private and community initiatives. “Nickel Empire” of cheap amusements; a nickel KeySpan Park remains sold out season after season paid the fare on the new subway line, and and the amusement area has witnessed ever-greater visitors were greeted by the original Nathan’s crowds for both everyday beach activities and events, Famous, home of the five-cent hot dog. The from the annual Mermaid Parade, Siren Festival and amusement parks struggled to stay afloat and rock concerts occasionally held at KeySpan, to mini- Coney Island began to experience hard economic

marathons and summertime concerts held at the Competition Brief. Pavilion: The Parachute times. Nevertheless, Coney Island continued to Asser Levy Park Amphitheater. With the creation of provide an accessible and affordable opportunity : the Coney Island Development Corporation, the area for a diverse population, always looming large is poised for further positive change, in which the in the history of New York. Parachute Pavilion will play a vital part.” 1

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129 SITE HISTORY “The Parachute Pavilion will be located on the boardwalk edge of the former Steeplechase Park site. Today, all that remains of the park is the rebuilt Steeplechase Pier and the landmarked Parachute Jump. Steeplechase was once home to numerous attractions, including the famous Pavilion of Fun—a 270-foot-wide, 450-foot-long and 63-foot-high Art Nouveau steel-and-glass pavilion built in 1907. Steeplechase opened its doors in 1897, and became Coney Island’s longest-lived amusement park.

In 1964, Steeplechase Park was bought by developers who demolished it for a high- rise housing development. When the necessary zoning change could not be obtained, the site was sold to the city and subsequently leased to various amusement park entrepreneurs. As plan after plan failed to come to fruition, the site became a symbol of Coney Island’s economic woes. Finally, in 1998, Mayor Rudolph Giuliani selected it as the home of the Brooklyn Cyclones, Brooklyn’s new minor league baseball team, and KeySpan Park opened in 2001 to sell-out crowds. Steeplechase Pier, 1000 feet out into the Atlantic, remains an active part of the Coney Island shore.

Often referred to as Brooklyn’s Eiffel Tower, the 262-foot-high Parachute Jump, which in 1941 was moved to Steeplechase Park from the 1939-1940 New York World’s Fair in Flushing Meadows, Queens, became Coney Island’s most exciting ride. It was modeled after Army training equipment and featured a 250-foot drop in a The Parachute Pavilion: Competition Brief. Pavilion: The Parachute seat for two. The Jump closed in 1968 amid safety concerns and became a New : York City-designated landmark in 1989. It still stands at the edge of the former park site, poised to receive the next exciting addition to Steeplechase—the Parachute Pavilion.”1

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130 SITE DESCRIPTION. “The Parachute Pavilion will be located between West 16th and West 17th Streets, at the intersection of the Riegelmann Boardwalk, KeySpan Park’s Surf Avenue-to-Boardwalk path, the Parachute Jump and the Steeplechase Pier. The competition site is adjacent to the Parachute Jump’s fence, and is bounded by the Riegelmann Boardwalk to the south, a smaller boardwalk path connecting the Boardwalk to the stadium to the east, KeySpan Park’s parking lot to the west and KeySpan Park itself to the north. The footprint of the competition site is approximately 7,800 SF.

Directly east of the KeySpan Park-to-Boardwalk path are two youth softball/ soccer fields, which are bordered by West 16th Street, now closed to vehicular traffic. Beyond West 16th Street to the east is an empty lot. The heart of the amusement area as it stands today begins at West 15th Street and continues on to West 8th Street.

The site is near the western edge of the amusement area, and is bordered on the west by the heavily used Abe Stark Ice Rink, which together with the boardwalk, beach and Steeplechase Pier, is owned and maintained by the New York City Department of Parks and Recreation. West of the Ice Rink are several large vacant lots, as well as the recently landmarked Child’s Restaurant. The Parachute Pavilion: Competition Brief. Pavilion: The Parachute Across Surf Avenue to the north, there is a cluster of city-owned vacant : parcels of land. For the most part, the activity along Surf Avenue is retail and concessions, although many of the buildings lining Surf Avenue are presently vacant.”1

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131 “KeySpan Park holds 7,500 seats and is accessible north-south street, as well as by the Surf Avenue-to- from the boardwalk directly east of the competition Boardwalk path on the KeySpan Park site. Additional site, as well as from Surf Avenue. The Park is stairs bring visitors from the Boardwalk down to the the site of year-round activity— not only baseball beach. games, but community events and concerts. It also includes a restaurant, Peggy O’Neill’s, and the Vehicular access to Coney Island is via the Shore Brooklyn Baseball Gallery (for more information, visit (Belt) Parkway, Ocean Parkway and Cropsey Avenue, www.brooklyncyclones.net). The Abe Stark Ice Rink a north-south road that connects the Shore (Belt) to the west and Steeplechase Pier, a popular fishing Parkway to Neptune Avenue, two blocks north of Surf destination across the boardwalk from KeySpan Park Avenue. Surf Avenue is the main east-west vehicular and the Parachute Jump, are also used year-round. artery of downtown Coney Island. Stillwell Avenue, east of West 15th Street, is the primary road on the The 262-foot-high Parachute Jump, the tallest north-south axis within the amusement area. The structure in Coney Island, has not been in operation closest subway stop to the Parachute Pavilion site is since 1968. It is now closed off from the public the recently refurbished Stillwell Avenue station at by a low fence. It is the last surviving amusement the corner of Stillwell and Surf avenues, serviced by ride from Coney Island’s Steeplechase Park, and is the D, F, N and W lines. The area also has a D-and a very effective symbol of Coney Island’s heyday. F-line subway stop at West 8th Street.” The jump recently underwent an extensive structural refurbishment and repainting. The Parachute Pavilion: Competition Brief. Pavilion: The Parachute : The Riegelmann Boardwalk is raised approximately 8 feet from street and beach level. The Boardwalk is accessed via stairs and ramps at the end of each

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132 PROGRAM REQUIREMENTS: RELATIONSHIP TO SITE “The following criteria and program requirements “The Parachute Pavilion will be adjacent to a New York should serve as guidelines for designing the City designated landmark and must be sited using the Parachute Pavilion: following criteria:

Restaurant The Pavilion must be sited only within the designated 3,000-4,000 SF (including both indoor and footprint (see Site Diagram) and not extend beyond it. outdoor seating options, kitchen, bar and The total Pavilion height must not exceed 30 feet (from restrooms) street/parking lot level). Only the eastern part of the Pavilion, above 20 feet high Store and adjacent to the Boardwalk extension, may extend 1,000-2,000 SF (for Coney Island/Parachute up to 5 feet beyond the designated footprint (this will Jump souvenirs, surfing gear, fishing supplies, create a canopy or overhang that is 10 feet high above etc.) the boardwalk).”

Multi-use Exhibition/Event Space 1,000-2,000 SF—a flexible and revenue producing space (rentable for private/public temporary exhibits and/or events) The Parachute Pavilion: Competition Brief. Pavilion: The Parachute Office Space : Four offices at 100 SF each (to be occupied by either city agencies/Parks Department or donated to local advocacy groups)” Fig. 7.5: Site View.

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133 The Parachute Pavilion: Competition Brief. Pavilion: The Parachute : Appendix 2

Fig. 7.4: Site Plan. 134 The Parachute Pavilion: Competition Brief. Pavilion: The Parachute :

Fig. 7.6: Aerial View of the Site and its Surroundings. Appendix 2

135 The Parachute Pavilion: Competition Brief. Pavilion: The Parachute :

Fig. 7.7: Site viewed from Riegelmann Boardwalk. Appendix 2

136 The Parachute Pavilion: Competition Brief. Pavilion: The Parachute :

Fig. 7.8: Aerial View of the Site. Appendix 2 Fig. 7.9: A View of The Site and Parachute Tower from the Beach

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