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 Geometry 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 laws 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 space frame 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 Golden Ratio & Fivefold Acknowledgements 4 Symmetry. 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 Solid Geometry Introduction 21 Chapter IV: Penrose Tilings and Quasicrystals. 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 Quasicrystal Geometry Polygons 28 Planar Surfaces 72 Two Dimensional Tessellations 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 Truss 46 Chapter V: The Golden Tetrahedral System. Geodesic Domes 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 tessellation{p,q} Chapter II: Aperiodic Geometry Table 2.1 : A table showing the infinite family of rhombi projected from dimension ‘n’. Chapter IV: Penrose Tilings and Quasicrystals. Table 4.1 The 17 Different Symmetries 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 Dimensions. 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: Circle 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 Euclidean Space 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: Hyperbolic Space 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