A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller

320 Pages·1986·7.427 MB·English

by  Amy C. Edmondson (auth.)

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