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Divided Spheres : Geodesics and the Orderly Subdivision of the Sphere - Edward S. Popko

Divided Spheres

Geodesics and the Orderly Subdivision of the Sphere

Hardcover Published: 16th July 2012
ISBN: 9781466504295
Number Of Pages: 532

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This well-illustrated book-in color throughout-presents a thorough introduction to the mathematics of Buckminster Fuller's invention of the geodesic dome, which paved the way for a flood of practical applications as diverse as weather forecasting and fish farms. The author explains the principles of spherical design and the three main categories of subdivision based on geometric solids (polyhedra). He illustrates how basic and advanced CAD techniques apply to spherical subdivision and covers modern applications in product design, engineering, science, games, and sports balls.

Edward Popko's Divided Spheres is the definitive source for the many varied ways a sphere can be divided and subdivided. From domes and pollen grains to golf balls, every category and type is elegantly described in these pages. The mathematics and the images together amount to a marvelous collection, one of those rare works that will be on the bookshelf of anyone with an interest in the wonders of geometry. --Kenneth Snelson, renowned inventor of Tensegrity sculpture Dr. Popko's elegant new book extends both the science and the art of spherical modeling to include Computer-Aided Design and applications, which I would never have imagined when I started down this fascinating and rewarding path. His lovely illustrations bring the subject to life for all readers, including those who are not drawn to the mathematics. This book demonstrates the scope, beauty and utility of an art and science with roots in antiquity. Spherical subdivision is relevant today and useful for the future. Anyone with an interest in the geometry of spheres, whether a professional engineer, 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. --Magnus Wenninger, author of Spherical Models Dr. Popko shows you how to solve practical design problems based on spherical polyhedra. Novices and experts will understand the challenges and classic techniques of spherical design just by looking at the many beautiful illustrations. --Steve Waterman, mathematician

Prefacep. xiii
Acknowledgmentsp. xix
Divided Spheresp. 1
Working with Spheresp. 3
Making a Pointp. 3
An Arbitrary Numberp. 4
Symmetry and Polyhedral Designsp. 6
Spherical Workbenchesp. 8
Detailed Designsp. 9
Other Ways to Use Polyhedrap. 10
Summaryp. 11
Additional Resourcesp. 12
Bucky's Domep. 13
Synergetic Geometryp. 15
Dymaxion Projectionp. 17
Cahill and Waterman Projectionsp. 20
Vector Equilibriump. 21
Icosa's 31p. 22
The First Domep. 24
NC State and Skybreak Carolinap. 26
Ford Rotunda Domep. 31
Marines in Raleighp. 33
University Circuitp. 34
Radomesp. 35
Kaiser's Domesp. 38
Union Tank Carp. 40
Covering Every Anglep. 41
Summaryp. 47
Additional Resourcesp. 48
Putting Spheres to Workp. 49
Tammes Problemp. 49
Spherical Virusesp. 51
Celestial Catalogsp. 53
Sudbury Neutrino Observatoryp. 54
Climate Models and Weather Predictionp. 56
Cartographyp. 60
Honeycombs for Supercomputersp. 61
Fish Farmingp. 63
Virtual Realityp. 65
Modeling Spheresp. 67
Dividing Golf Ballsp. 68
Spherical Throwable Panoramic Camerap. 72
Hoberman's MiniSpherep. 73
Rafiki's Code Worldp. 74
Art and Expressionp. 75
Additional Resourcesp. 77
Circular Reasoningp. 79
Lesser and Great Circlesp. 81
Geodesic Subdivisionp. 83
Circle Polesp. 85
Arc and Chord Factorsp. 86
Where Are We?p. 87
Altitude-Azimuth Coordinatesp. 87
Latitude and Longitude Coordinatesp. 89
Spherical Tripsp. 90
Loxodromesp. 91
Separation Anglep. 93
Latitude Sailingp. 94
Longitudep. 94
Spherical Coordinatesp. 95
Cartesian Coordinatesp. 96
, Coordinatesp. 98
Spherical Polygonsp. 99
Excess and Defectp. 114
Summaryp. 125
Additional Resourcesp. 126
Distributing Pointsp. 127
Coveringp. 128
Packingp. 131
Volumep. 133
Summaryp. 135
Additional Resourcesp. 136
Polyhedral Frameworksp. 137
What Is a Polyhedron?p. 138
Platonic Solidsp. 139
Symmetryp. 152
Archimedean Solidsp. 162
Additional Resourcesp. 179
Golf Ball Dimplesp. 181
Icosahedral Ballsp. 182
Octahedral Ballsp. 184
Tetrahedral Ballsp. 185
Bilateral Symmetryp. 186
Subdivided Areasp. 187
Dimple Graphicsp. 188
Summaryp. 189
Additional Resourcesp. 190
Subdivision Schemasp. 191
Geodesic Notationp. 192
Triangulation Numberp. 194
Frequency and Harmonicsp. 195
Grid Symmetryp. 197
Class I: Alternates and Fordp. 199
Class II: Triaconp. 219
Class III: Skewp. 231
Covering the Whole Spherep. 244
Additional Resourcesp. 245
Comparing Resultsp. 247
Kissing-Touchingp. 248
Sameness or Nearly Sop. 251
Triangle Areap. 253
Face Acutenessp. 255
Euler Linesp. 255
Parts and Tp. 257
Convex Hullp. 260
Spherical Capsp. 262
Stereogramsp. 263
Face Orientationp. 267
King Icosap. 272
Summaryp. 273
Additional Resourcesp. 273
Computer-Aided Designp. 275
A Short Historyp. 276
CATIAp. 278
Octet Truss Connectorp. 278
Spherical Designp. 290
Three Class II Triacon Designsp. 294
Panel Spherep. 295
Class II Strut Spherep. 302
Class III Parabolic Stellationsp. 307
Class I Ford Shellp. 315
31 Great Circlesp. 322
Class III Skewp. 326
Additional Resourcesp. 329
Advanced CAD Techniquesp. 331
Reference Modelsp. 331
An Architectural Examplep. 332
Spherical Reference Modelsp. 334
Prepackaged Reference and Assembly Modelsp. 335
Local Axis Systemsp. 336
Assembly Reviewp. 337
Design-in-Contextp. 339
Associative Geometryp. 339
Design-in-Context versus Constraintsp. 340
Mirrored Enantiomorphsp. 342
Power Copyp. 343
Power Copy Prototypep. 344
Macrosp. 346
Publicationp. 349
Data Structuresp. 350
CAD Alternatives: Stella and Antiprismp. 350
Antiprismp. 355
Summaryp. 361
Additional Resourcesp. 362
Spherical Trigonometryp. 363
Basic Trigonometric Functionsp. 363
The Core Theoremsp. 365
Law of Cosinesp. 366
Law of Sinesp. 366
Right Trianglesp. 367
Napier's Rulep. 367
Using Napier's Rule on Oblique Trianglesp. 368
Polar Trianglesp. 369
Additional Resourcesp. 371
Stereographic Projectionp. 373
Points on a Spherep. 374
Stereographic Propertiesp. 374
A History of Diverse Usesp. 375
The Astrolabep. 375
Crystallography and Geologyp. 377
Cartographyp. 378
Projection Methodsp. 379
Great Circlesp. 381
Lesser Circlesp. 382
Wulff Netp. 385
Polyhedra Stereographiesp. 386
Polyhedra as Crystalsp. 386
Metrics and Interpretationp. 387
Projecting Polyhedrap. 388
Octahedronp. 390
Tetrahedronp. 390
Geodesic Stereographicsp. 391
Spherical Icosahedronp. 392
Summaryp. 393
Additional Resourcesp. 394
Geodesic Mathp. 395
Class I: Alternates and Fordsp. 397
Class II: Triaconp. 399
Class III: Skewp. 405
Characteristics of Trianglesp. 407
Storing Grid Pointsp. 408
Additional Resourcesp. 415
Schema Coordinatesp. 417
Coordinates for Class I: Alternates and Fordp. 418
Coordinates for Class II: Triaconp. 420
Coordinates for Class III: Skewp. 422
Coordinate Rotationsp. 423
Rotation Conceptsp. 424
Direction and Sequencesp. 424
Simple Rotationsp. 425
Reflectionsp. 426
Antipodal Pointsp. 428
Compound Rotationsp. 429
Rotation around an Arbitrary Axisp. 430
Polyhedra and Class Rotation Sequencesp. 431
Icosahedron Classes I and IIIp. 432
Icosahedron Class IIp. 434
Octahedron Classes I and IIIp. 436
Octahedron Class IIp. 436
Tetrahedron Classes I and IIIp. 439
Tetrahedron Class IIp. 439
Dodecahedron Class IIp. 442
Cube Class IIp. 443
Implementing Rotationsp. 445
Using Matricesp. 446
Rotation Algorithmsp. 446
An Examplep. 451
Summaryp. 451
Additional Resourcesp. 452
Glossaryp. 453
Useful Websitesp. 479
Bibliographyp. 481
About the Authorp. 501
Indexp. 511
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9781466504295
ISBN-10: 1466504293
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 532
Published: 16th July 2012
Publisher: Taylor & Francis Inc
Country of Publication: US
Dimensions (cm): 23.5 x 19.1  x 2.54
Weight (kg): 1.25
Edition Number: 1

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