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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 |
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