Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere

by Popko; Edward S.

• ISBN13: 9781466504295
• ISBN10: 1466504293
• Format: Hardcover
• Copyright: 2012-07-30
• Publisher: A. K. Peters

Summary

This well-illustrated book is the first comprehensive yet accessible introduction to the mathematics of Buckminster Fuller's invention of the geodesic dome fifty years ago, which paved the way for a flood of practical applications in construction and other areas. Geodesic applications now encompass everything from product and packaging design, civil engineering, virology, nanotechnology, computer graphics, climate modeling, supercomputer architecture, wireless mobile networks, virtual reality gaming, astronomy, and computer-aided design (CAD).

Table of Contents

Preface	p. xiii
Acknowledgments	p. xix
Divided Spheres	p. 1
Working with Spheres	p. 3
Making a Point	p. 3
An Arbitrary Number	p. 4
Symmetry and Polyhedral Designs	p. 6
Spherical Workbenches	p. 8
Detailed Designs	p. 9
Other Ways to Use Polyhedra	p. 10
Summary	p. 11
Additional Resources	p. 12
Bucky's Dome	p. 13
Synergetic Geometry	p. 15
Dymaxion Projection	p. 17
Cahill and Waterman Projections	p. 20
Vector Equilibrium	p. 21
Icosa's 31	p. 22
The First Dome	p. 24
NC State and Skybreak Carolina	p. 26
Ford Rotunda Dome	p. 31
Marines in Raleigh	p. 33
University Circuit	p. 34
Radomes	p. 35
Kaiser's Domes	p. 38
Union Tank Car	p. 40
Covering Every Angle	p. 41
Summary	p. 47
Additional Resources	p. 48
Putting Spheres to Work	p. 49
Tammes Problem	p. 49
Spherical Viruses	p. 51
Celestial Catalogs	p. 53
Sudbury Neutrino Observatory	p. 54
Climate Models and Weather Prediction	p. 56
Cartography	p. 60
Honeycombs for Supercomputers	p. 61
Fish Farming	p. 63
Virtual Reality	p. 65
Modeling Spheres	p. 67
Dividing Golf Balls	p. 68
Spherical Throwable Panoramic Camera	p. 72
Hoberman's MiniSphere	p. 73
Rafiki's Code World	p. 74
Art and Expression	p. 75
Additional Resources	p. 77
Circular Reasoning	p. 79
Lesser and Great Circles	p. 81
Geodesic Subdivision	p. 83
Circle Poles	p. 85
Arc and Chord Factors	p. 86
Where Are We?	p. 87
Altitude-Azimuth Coordinates	p. 87
Latitude and Longitude Coordinates	p. 89
Spherical Trips	p. 90
Loxodromes	p. 91
Separation Angle	p. 93
Latitude Sailing	p. 94
Longitude	p. 94
Spherical Coordinates	p. 95
Cartesian Coordinates	p. 96
¿ ¿ ¿, Coordinates	p. 98
Spherical Polygons	p. 99
Excess and Defect	p. 114
Summary	p. 125
Additional Resources	p. 126
Distributing Points	p. 127
Covering	p. 128
Packing	p. 131
Volume	p. 133
Summary	p. 135
Additional Resources	p. 136
Polyhedral Frameworks	p. 137
What Is a Polyhedron?	p. 138
Platonic Solids	p. 139
Symmetry	p. 152
Archimedean Solids	p. 162
Additional Resources	p. 179
Golf Ball Dimples	p. 181
Icosahedral Balls	p. 182
Octahedral Balls	p. 184
Tetrahedral Balls	p. 185
Bilateral Symmetry	p. 186
Subdivided Areas	p. 187
Dimple Graphics	p. 188
Summary	p. 189
Additional Resources	p. 190
Subdivision Schemas	p. 191
Geodesic Notation	p. 192
Triangulation Number	p. 194
Frequency and Harmonics	p. 195
Grid Symmetry	p. 197
Class I: Alternates and Ford	p. 199
Class II: Triacon	p. 219
Class III: Skew	p. 231
Covering the Whole Sphere	p. 244
Additional Resources	p. 245
Comparing Results	p. 247
Kissing-Touching	p. 248
Sameness or Nearly So	p. 251
Triangle Area	p. 253
Face Acuteness	p. 255
Euler Lines	p. 255
Parts and T	p. 257
Convex Hull	p. 260
Spherical Caps	p. 262
Stereograms	p. 263
Face Orientation	p. 267
King Icosa	p. 272
Summary	p. 273
Additional Resources	p. 273
Computer-Aided Design	p. 275
A Short History	p. 276
CATIA	p. 278
Octet Truss Connector	p. 278
Spherical Design	p. 290
Three Class II Triacon Designs	p. 294
Panel Sphere	p. 295
Class II Strut Sphere	p. 302
Class III Parabolic Stellations	p. 307
Class I Ford Shell	p. 315
31 Great Circles	p. 322
Class III Skew	p. 326
Additional Resources	p. 329
Advanced CAD Techniques	p. 331
Reference Models	p. 331
An Architectural Example	p. 332
Spherical Reference Models	p. 334
Prepackaged Reference and Assembly Models	p. 335
Local Axis Systems	p. 336
Assembly Review	p. 337
Design-in-Context	p. 339
Associative Geometry	p. 339
Design-in-Context versus Constraints	p. 340
Mirrored Enantiomorphs	p. 342
Power Copy	p. 343
Power Copy Prototype	p. 344
Macros	p. 346
Publication	p. 349
Data Structures	p. 350
CAD Alternatives: Stella and Antiprism	p. 350
Antiprism	p. 355
Summary	p. 361
Additional Resources	p. 362
Spherical Trigonometry	p. 363
Basic Trigonometric Functions	p. 363
The Core Theorems	p. 365
Law of Cosines	p. 366
Law of Sines	p. 366
Right Triangles	p. 367
Napier's Rule	p. 367
Using Napier's Rule on Oblique Triangles	p. 368
Polar Triangles	p. 369
Additional Resources	p. 371
Stereographic Projection	p. 373
Points on a Sphere	p. 374
Stereographic Properties	p. 374
A History of Diverse Uses	p. 375
The Astrolabe	p. 375
Crystallography and Geology	p. 377
Cartography	p. 378
Projection Methods	p. 379
Great Circles	p. 381
Lesser Circles	p. 382
Wulff Net	p. 385
Polyhedra Stereographies	p. 386
Polyhedra as Crystals	p. 386
Metrics and Interpretation	p. 387
Projecting Polyhedra	p. 388
Octahedron	p. 390
Tetrahedron	p. 390
Geodesic Stereographics	p. 391
Spherical Icosahedron	p. 392
Summary	p. 393
Additional Resources	p. 394
Geodesic Math	p. 395
Class I: Alternates and Fords	p. 397
Class II: Triacon	p. 399
Class III: Skew	p. 405
Characteristics of Triangles	p. 407
Storing Grid Points	p. 408
Additional Resources	p. 415
Schema Coordinates	p. 417
Coordinates for Class I: Alternates and Ford	p. 418
Coordinates for Class II: Triacon	p. 420
Coordinates for Class III: Skew	p. 422
Coordinate Rotations	p. 423
Rotation Concepts	p. 424
Direction and Sequences	p. 424
Simple Rotations	p. 425
Reflections	p. 426
Antipodal Points	p. 428
Compound Rotations	p. 429
Rotation around an Arbitrary Axis	p. 430
Polyhedra and Class Rotation Sequences	p. 431
Icosahedron Classes I and III	p. 432
Icosahedron Class II	p. 434
Octahedron Classes I and III	p. 436
Octahedron Class II	p. 436
Tetrahedron Classes I and III	p. 439
Tetrahedron Class II	p. 439
Dodecahedron Class II	p. 442
Cube Class II	p. 443
Implementing Rotations	p. 445
Using Matrices	p. 446
Rotation Algorithms	p. 446
An Example	p. 451
Summary	p. 451
Additional Resources	p. 452
Glossary	p. 453
Useful Websites	p. 479
Bibliography	p. 481
About the Author	p. 501
Index	p. 511
Table of Contents provided by Ingram. All Rights Reserved.