A Fuller Explanation The Synergetic Geometry of R. Buckminster Fuller Design Science Collection Series Editor Arthur L. Loeb Department of Visual and Environmental Studies Harvard University Amy C. Edmondson A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller, 1987 In Preparation Marjorie Senechal Shaping Space: A Polyhedral Approach and George Fleck (Eds.) Arthur L. Loeb Concepts and Images Amy C. Edmondson A Fuller Explanation The Synergetic Geometry of R. Buckminster Fuller A Pro Scientia Viva Title Birkhauser Boston . Basel . Stuttgart Amy C. Edmondson A Fuller Explanation The Synergetic Geometry of R. Buckminster Fuller Coden: DSCOED First Printing, 1987 Library of Congress Cataloging in Publication Data Edmondson, Amy C. A Fuller explanation. The synergetic geometry of R. Buckminster Fuller. (Design science collection) "A Pro scientia viva title." Bibliography: p. Includes index. 1. System theory. 2. Thought and thinking. 3. Mathematics-Philosophy. 4. Geometry-Philosophy. 5. Fuller, R. Buckminster (Richard Buckminster). 1895- . I. Title. II. Series. Q295.E33 1986 003 86-14791 CIP-Kurztitelaufnahme der Deutschen Bibliothek Edmondson, Amy c.: A Fuller explanation: the synerget. geometry of R. Buckminster Fuller / Amy C. Edmondson-I. print. -Boston; Basel; Stuttgart: Birkhauser, 1986. (Design science collection) (A pro sci entia viva title) ISBN 978-0-8176-3338-7 ISBN 978-1-4684-7485-5 (eBook) 001 10.1007/978-1-4684-7485-5 Frontispiece photograph courtesy of Phil Haggerty. © Birkhauser Boston, Inc., 1987 Softcover reprint of the hardcover 1s t edition 1987 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recorded or otherwise, without prior written permission of the copyright owner, Birkhauser Boston, Inc., 380 Green Street, P.O.B. 2007, Cambridge, MA 02139, U.S.A. ISBN 978-0-8176-3338-7 Typeset by Science Typographers, Inc., Medford, New York. To my parents Mary Dillon Edmondson and Robert Joseph Edmondson Contents Series Editor's Foreword. . . . . . . . . . . . . . . . . . . . . . . . . . . .. xv Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. XIX Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xxi Note to Readers ................................. XXlll Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xxv 1. Return to Modelability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 From Geometry to Geodesics: A Personal Perspective. . . . . . 3 Operational Mathematics ........................... 6 Experimental Evidence ........................... 7 Nature's Coordinate System. . . . . . . . . . . . . . . . . . . . . . . . 9 Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Generalized Principles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Return to Mode1ability ........................... 14 2. The Irrationality of Pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 "Nature Isn't Using Pi" . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Finite Accounting System ......................... 18 Which Way Is "Up"? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Visual Literacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Peaceful Coexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3. Systems and Synergy .............................. 25 Conceptual and Real Systems . . . . . . . . . . . . . . . . . . . . . . . 28 Limits of Resolution as Part of the Whole-Systems Approach ................................... 30 Synergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 x Contents 4. Tools of the Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Plato's Discovery ................................ 37 Triangles ..................................... 38 Squares ...................................... 40 Pentagons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 A Limited Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Euler's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Truncation and Stellation ........................... 46 An Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 "Intertransformability" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Symmetry ...................................... 52 5. Structure and "Pattern Integrity" ..................... 54 Pattern Integrity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6. Angular Topology ................................ 65 Frequency and Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Units of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Time and Repetition: Frequency versus Continuum ..................... 67 Topology and Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Vector Polyhedra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 68 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72 Planes of Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Other Applications of Dimension . . . . . . . . . . . . . . . . . . . . 74 Angular Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Angular Takeout: An Example . . . . . . . . . . . . . . . . . . . . . . 78 Angle Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7. Vector Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 "Nature's Own Geometry" . . . . . . . . . . . . . . . . . . . . . . . . . 84 Spatial Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Planar Equilibrium ................... . . . . . . . . . . . 87 Cuboctahedron as Vector Equilibrium . . . . . . . . . . . . . . . . . 90 VE: Results ................................... 91 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8. Tales Told by the Spheres: Closest Packing .............. 100 Equilibrium: Equalization of Distances . . . . . . . . . . . . . . .. 100 Symmetry versus Specificity of Form. . . . . . . . . . . . . . . . .. 101 Contents xi Organization of Identical Units. . . . . . . . . . . . . . . . . . . . .. 101 New Level of Focus. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 102 Background: Closepacking .......................... 102 Planes of Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 106 Fuller Observations ............................... 109 Tetrahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 110 Vector Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 114 Frequency .................................... 114 Icosahedron ........................ . . . . . . . . . .. 117 Further Discoveries: Nests. . . . . . . . . . . . . . . . . . . . . . . . . .. 120 "Interprecessing" ............................... 121 A Final Philosophical Note ........................ 124 9. Isotropic Vector Matrix ............................ 127 A Quick Comparison: "Synergetics Accounting" ......... 130 Cells: "Inherent Complementarity" .................. , 131 A Complete Picture ............................... 133 Angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 134 Locating New Polyhedral Systems. . . . . . . . . . . . . . . . . . .. 135 Duality and the IVM .............................. 136 Angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 136 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 137 Domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 138 Framework of Possibility. . . . . . . . . . . . . . . . . . . . . . . . . . .. 140 Invention: Octet Truss. . . . . . . . . . . . . . . . . . . . . . . . . . .. 141 10. Multiplication by Division: In Search of Cosmic Hierarchy. .. 143 Volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 143 Results: Volume Ratios. . . . . . . . . . . . . . . . . . . . . . . . . .. 144 Shape Comparisons: Qualities of Space. . . . . . . . . . . . . . .. 146 Volume: Direct Comparison. . . . . . . . . . . . . . . . . . . . . . .. 147 Multiplication by Division. . . . . . . . . . . . . . . . . . . . . . . . . .. 149 Tetrahedron as Starting Point. . . . . . . . . . . . . . . . . . . . . .. 149 Cube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 150 Vector Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 152 Rhombic Dodecahedron .......................... 153 Multiplication by Division . . . . . . . . . . . . . . . . . . . . . . . .. 154 Cosmic Hierarchy (of Nuclear Event Patternings) .......... 157 Volume Reconsidered ............................ 157 11. Jitterbug....................................... 159 Folding a Polyhedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 160 Volume and Phase Changes ....... . . . . . . . . . . . . . . . .. 163 xii Contents Icosahedron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 163 Single Layer versus IVM .......................... 164 "Trans-Universe" versus "Locally Operative" ........... 165 Fives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 165 "S-Modules" .................................. 167 Icosahedron and Rhombic Dodecahedron .. . . . . . . . . . . .. 167 Pentagonal Dodecahedron. . . . . . . . . . . . . . . . . . . . . . . .. 168 Four Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 169 Complex of Jitterbugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 170 Other Dynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 172 Topology and Phase ............................. 172 12. All-Space" Filling: New Types of Packing Crates ......... 175 Ii Plane Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 176 Filling Space .................................... 177 Complementarity ............................... 178 Other Space Fillers .............................. 179 The Search Continues ............................ 180 The Dual Perspective. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 181 Duality and Domain in Sphere Packing. . . . . . . . . . . . . . .. 183 Truncated Octahedron. . . . . . . . . . . . . . . . . . . . . . . . . . .. 184 Two to One: A Review ........................... 185 13. The Heart of the Matter: A- and B-Quanta Modules. . . . . . .. 189 A-Quanta Modules ........ . . . . . . . . . . . . . . . . . . . . .. 190 B-Quanta Modules .............................. 190 Energy Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . .. 193 Mite .......................................... 195 Mirrors ...................................... 197 Cubes into Mites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 198 Rhombic Dodecahedra ........................... 198 Coupler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 199 Volume and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 201 Review: All-Space Fillers. . . . . . . . . . . . . . . . . . . . . . . . . . .. 203 14. Cosmic Railroad Tracks: Great Circles . . . . . . . . . . . . . . . . .. 206 Why Are We Talking About Spheres? ................. 207 New Classification System . . . . . . . . . . . . . . . . . . . . . . . .. 208 Great-Circle Patterns .............................. 209 Least Common Denominator . . . . . . . . . . . . . . . . . . . . . .. 213 LCD: "Intertransformability" ...................... 215 LCD of 31 Great Circles .......................... 216 VE's 25 Great Circles. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 217