Divided spheres. Geodesics and the orderly subdivision of the sphere

525 Pages·2012·24.549 MB·English

by  Popko E.S.

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DiviDeD
SphereS
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Geodesics & the orderly
This well-illustrated book—in color throughout—presents a thorough introduction to i
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the mathematics of Buckminster Fuller’s invention of the geodesic dome, which paved subdivision of the sphere
the way for a flood of practical applications as diverse as weather forecasting and
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fish farms. The author explains the principles of spherical design and the three main
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categories of subdivision based on geometric solids (polyhedra). He illustrates how
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basic and advanced CAD techniques apply to spherical subdivision and covers modern
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applications in product design, engineering, science, games, and sports balls.
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“Dr. Popko’s elegant new book extends both the science and the art of spherical modeling
to include Computer-Aided Design and applications….His lovely illustrations bring the
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subject to life for all readers, including those who are not drawn to the mathematics.…
Anyone with an interest in the geometry of spheres, whether a professional engineer, p
an architect or product designer, a student, a teacher, or simply someone curious about
the spectrum of topics to be found in this book, will find it helpful and rewarding. h
—Magnus Wenninger, Benedictine monk and polyhedral modeler
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“Edward Popko’s Divided Spheres is the definitive source for the many varied ways a
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sphere can be divided and subdivided.…The mathematics and the images together
amount to a marvelous collection, one of those rare works that will be on the bookshelf e
of anyone with an interest in the wonders of geometry.”
—Kenneth Snelson, sculptor and photographer S
“Ed Popko’s comprehensive survey of the history, literature, geometric and mathematical
properties of the sphere is the definitive work on the subject.…This book should be in
the library of anyone interested in the orderly subdivision of the sphere.”
—Shoji Sadao, architect, cartographer, and lifelong
business partner of Buckminster Fuller
K14517
Mathematics
Divided Spheres
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Divided Spheres
Geodesics and the Orderly
Subdivision of the Sphere
Edward S. Popko
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2012 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20120409
International Standard Book Number-13: 978-1-4665-0430-1 (eBook - PDF)
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To my mother Frances and father Edward, who gave me the freedom to expore.
Their childhood gift of Fun with Figures probably started it all.1
1 Book cover by Mae and Ira Freeman, © 1946, renewed in 1974 by Random House, Inc. Used by permission of
Random House, Inc.
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Table of Contents
Preface xiii
Acknowledgments xix
1. Divided Spheres 1
1.1 Working with Spheres ..............................................3
1.2 Making a Point....................................................3
1.3 An Arbitrary Number...............................................4
1.4 Symmetry and Polyhedral Designs ....................................6
1.5 Spherical Workbenches .............................................8
1.6 Detailed Designs ..................................................9
1.7 Other Ways to Use Polyhedra .......................................10
1.8 Summary .......................................................11
Additional Resources ..................................................12
2. Bucky’s Dome 13
2.1 Synergetic Geometry ..............................................15
2.2 Dymaxion Projection..............................................17
2.3 Cahill and Waterman Projections.....................................20
2.4 Vector Equilibrium................................................21
2.5 Icosa’s 31 .......................................................22
2.6 The First Dome ..................................................24
2.7 NC State and Skybreak Carolina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26
2.8 Ford Rotunda Dome...............................................31
2.9 Marines in Raleigh................................................33
viii Divided Spheres
2.10 University Circuit................................................34
2.11 Radomes.......................................................35
2.12 Kaiser’s Domes .................................................38
2.13 Union Tank Car .................................................40
2.14 Covering Every Angle ............................................41
2.15 Summary ......................................................47
Additional Resources ..................................................48
3. Putting Spheres to Work 49
3.1 Tammes Problem .................................................49
3.2 Spherical Viruses .................................................51
3.3 Celestial Catalogs.................................................53
3.4 Sudbury Neutrino Observatory ......................................54
3.5 Climate Models and Weather Prediction ...............................56
3.6 Cartography .....................................................60
3.7 Honeycombs for Supercomputers ....................................61
3.8 Fish Farming ....................................................63
3.9 Virtual Reality ...................................................65
3.10 Modeling Spheres................................................67
3.11 Dividing Golf Balls ..............................................68
3.12 Spherical Throwable Panoramic Camera..............................72
3.13 Hoberman’s MiniSphere ..........................................73
3.14  Rafiki’s Code World..............................................74
3.15 Art and Expression...............................................75
Additional Resources ..................................................77
4. Circular Reasoning 79
4.1 Lesser and Great Circles ...........................................81
4.2 Geodesic Subdivision..............................................83
4.3 Circle Poles .....................................................85
4.4 Arc and Chord Factors.............................................86
4.5 Where Are We? ..................................................87
4.6 Altitude-Azimuth Coordinates.......................................87
4.7 Latitude and Longitude Coordinates ..................................89
4.8 Spherical Trips...................................................90
4.9 Loxodromes .....................................................91
4.10 Separation Angle ................................................93
4.11 Latitude Sailing .................................................94
4.12 Longitude......................................................94
4.13 Spherical Coordinates ............................................95
4.14 Cartesian Coordinates ............................................96
4.15 ρ, φ, λ Coordinates ...............................................98
4.16 Spherical Polygons...............................................99
4.17 Excess and Defect ..............................................114
4.18 Summary .....................................................125
Additional Resources .................................................126
Table of Contents ix
5. Distributing Points 127
5.1 Covering.......................................................128
5.2 Packing........................................................131
5.3 Volume........................................................133
5.4 Summary ......................................................135
Additional Resources .................................................136
6. Polyhedral Frameworks 137
6.1 What Is a Polyhedron?............................................138
6.2 Platonic Solids ..................................................139
6.3 Symmetry......................................................152
6.4 Archimedean Solids..............................................162
Additional Resources .................................................179
7. Golf Ball Dimples 181
7.1 Icosahedral Balls ................................................182
7.2 Octahedral Balls ................................................184
7.3 Tetrahedral Balls ................................................185
7.4 Bilateral Symmetry ..............................................186
7.5 Subdivided Areas................................................187
7.6 Dimple Graphics ................................................188
7.7 Summary ......................................................189
Additional Resources .................................................190
8. Subdivision Schemas 191
8.1 Geodesic Notation ...............................................192
8.2 Triangulation Number ............................................194
8.3 Frequency and Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .195
8.4 Grid Symmetry..................................................197
8.5 Class I: Alternates and Ford........................................199
8.6 Class II: Triacon.................................................219
8.7 Class III: Skew..................................................231
8.8 Covering the Whole Sphere........................................244
Additional Resources .................................................245
9. Comparing Results 247
9.1 Kissing-Touching................................................248
9.2 Sameness or Nearly So............................................251
9.3 Triangle Area ...................................................253
9.4 Face Acuteness..................................................255
9.5 Euler Lines.....................................................255
9.6 Parts and T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .257
9.7 Convex Hull....................................................260
9.8 Spherical Caps ..................................................262
9.9 Stereograms ....................................................263