Connections: the geometric bridge between art and science

Contents......Page 6
Preface......Page 12
Acknowledgements......Page 16
Credits......Page 18
Preface to the Second Edition......Page 24
1.1 Introduction......Page 29
1.2 Myth and Number......Page 30
1.3 Proportion and Number......Page 35
1.4 The Structure of Ancient Musical Scales......Page 37
1.5 The Musical Scale in Architecture......Page 40
1.6 Systems of Proportion Based on V2 0 and Q......Page 44
1.7 The Golden Mean and Its Application to the Modulor of Le Corbusier......Page 49
1.8 An Ancient System of Roman Proportion......Page 56
Appendix 1.A......Page 60
2.1 Introduction......Page 63
2.2 Similarity......Page 64
2.3 Families of Similar Figures......Page 65
2.4 Self-Similarity of the Right Triangle......Page 66
2.6 A Circle Chopper......Page 69
2.7 Construction of the Square Root of a Given Length......Page 71
2.8 Archimedes Spiral......Page 72
2.9 Logarithmic Spiral......Page 73
2.10 Growth and Similarity in Nature......Page 76
2.11 Growth and Similarity in Geometry......Page 80
2.12 Infinite Self-Similar Curves......Page 83
2.13 On Growth and Form......Page 91
Appendix 2.A......Page 93
Appendix 2.B......Page 95
3.1 Introduction......Page 103
3.2 Fibonacci Series......Page 104
3.3 Some Tiling Properties of Q......Page 109
3.4 The Golden Rectangle and the Golden Section......Page 110
3.5 The Golden Mean Triangle......Page 113
3.6 The Pentagon and Decagon......Page 114
3.7 The Golden Mean and Patterns of Plant Growth......Page 117
3.8 The Music of Bartok: A System Both Open and Closed......Page 125
4.1 Introduction......Page 133
4.2 Graphs......Page 136
4.3 Maps......Page 142
4.4 Maps and Graphs on a Sphere......Page 145
4.5 Connectivity of Graphs and Maps......Page 147
4.6 Combinatorial Properties......Page 148
4.7 Regular Maps......Page 150
4.8 New Graphs from Old Ones......Page 152
4.9 Duality......Page 153
4.10 Planar and Nonplanar Graphs......Page 155
4.11 Maps and Graphs on Other Surfaces......Page 157
4.12 The Torus and the Mobius Strip......Page 162
4.13 Magic Squares......Page 165
4.14 Map Coloring......Page 166
4.15 Regular Maps on a Torus......Page 169
4.16 Szilassi and Csaszar Maps......Page 170
4.17 Floor Plans......Page 173
4.18 Bracing Structures......Page 182
4.19 Eulerian Paths......Page 187
4.20 Hamiltonian Paths......Page 191
5.1 Introduction......Page 195
5.2 Polygons......Page 197
5.3 Regular Tilings of the Plane......Page 201
5.5 Semiregular Tilings......Page 205
5.6 Symmetry......Page 206
5.7 Duality of Semiregular Tilings......Page 209
5.8 The Module of a Semiregular Tiling......Page 210
5.10 Transformations of Regular Tiling......Page 211
5.11 Nonperiodic Tilings......Page 222
5.12 Origami Patterns......Page 226
5.13 Islamic Art......Page 228
6.2 Planar Soap Films......Page 237
6.3 Random Cellular Networks......Page 242
6.4 Rural Market Networks......Page 245
6.5 Dirichlet Domains......Page 248
6.6 Spider Webs Dirichlet Domains and Rigidity......Page 252
6.7 Lattices......Page 258
6.8 Pattern Generation with Lattices......Page 262
6.9 Dirichlet Domains of Lattices and Their Relation to Plant Growth......Page 266
6.10 Quasicrystals and Penrose Tiles......Page 271
Appendix 6.A Projective Geometry......Page 276
7.1 Introduction......Page 283
7.2 The Platonic Solids......Page 285
7.3 The Platonic Solids as Regular Polyhedra......Page 287
7.4 Maps of Regular Polyhedra on a Circumscribed Sphere......Page 289
7.5 Maps of the Regular Polyhedra on the Plane—Schlegel Diagrams......Page 291
7.6 Duality......Page 292
7.7 Combinatorial Properties......Page 296
7.8 Rigidity......Page 298
7.9 The Angular Deficit......Page 301
7.10 From Maps to Polyhedra—The Dihedral Angle......Page 303
7.11 Space-Filling Properties......Page 305
7.12 Juxtapositions......Page 307
7.13 Symmetry......Page 310
7.14 Star Polyhedra......Page 316
Appendix 7.A Duals......Page 319
Appendix 7.B A Proof of Descartes Formula......Page 320
Appendix 7.C......Page 322
8.1 Introduction......Page 323
8.2 Intermediate Polyhedra......Page 324
8.3 Interpenetrating Duals Revisited......Page 327
8.4 The Rhombic Dodecahedron......Page 329
8.5 Embeddings Based on Symmetry......Page 331
8.6 Designs Based on Symmetry Breaking......Page 334
8.7 Relation to the Golden Mean......Page 336
8.8 Tensegrities......Page 338
8.9 The Tetrahedron—Methane Molecule Molecule and Soap Bubble......Page 341
8.10 Tetrahedron as the Atom of Structure......Page 343
8.11 Packing of Spheres......Page 345
8.12 Geodesic Domes and Viruses......Page 351
9.2 Archimedean Solids......Page 355
9.3 Truncation......Page 357
9.4 The Truncated Octahedron......Page 360
9.5 The Snub Figures......Page 362
9.7 Maps on a Sphere......Page 363
9.8 Combinatorial Properties......Page 365
9.9 Symmetry Revisited......Page 367
9.10 Prisms and Antiprisms......Page 369
10.2 Close Packing of Spheres......Page 375
10.3 The Shape of Space......Page 378
10.4 Packing Ratios......Page 381
10.5 Three-Dimensional Lattices......Page 383
10.6 Dirichiet Domains......Page 384
10.7 Crystal Structure......Page 385
10.8 Networks......Page 388
10.9 Infinite Regular Surfaces......Page 390
10.10 The Diamond and Graphite Nets......Page 393
10.11 Soap Froths......Page 396
10.12 A Unified Look at Nets Related to Cubic Lattices......Page 397
10.13 Zonohedra......Page 399
10.14 Golden Isozonohedra......Page 405
11.1 Introduction......Page 411
11.2 Mirrors......Page 412
11.3 Sets......Page 414
11.4 Mappings......Page 415
11.5 Translations......Page 418
11.6 Rotations......Page 419
11.7 Reflections......Page 420
11.8 Glide Reflection......Page 421
11.9 Proper and Improper Transformations......Page 422
11.10 Isometries and Mirrors......Page 423
11.11 Some Reflection Exercises......Page 430
11.12 Some Additional Relations Involving Isometries......Page 431
12.1 Introduction......Page 433
12.2 The Mathematics of Symmetry......Page 436
12.3 Symmetry Groups......Page 438
12.4 Subsets of a Group......Page 439
12.5 Kaleidoscope Groups......Page 441
12.6 Pattern Generation and the Kaleidoscope......Page 443
12.7 A Colored Kaleidoscope Symmetry......Page 445
12.8 Some Other Examples of Pattern Generation......Page 447
12.9 Pattern Generation in Hyperbolic Geometry......Page 448
12.10 Line Symmetry......Page 450
12.11 The Two-Dimensional Ornamental Symmetry Groups......Page 453
12.12 Symmetry and Design......Page 458
12.13 A Fundamental Postulate......Page 460
12.14 Interaction of Two Rotocenters Implies a Third......Page 463
12.15 Nets......Page 465
12.16 Enantiomorphy......Page 466
12.17 Aesthetics of Wallpaper Patterns......Page 471
12.18 The Symmetry of Islamic Tilings......Page 473
12.19 Symmetry of Similarity......Page 474
Epilogue......Page 481
References......Page 483
Index......Page 491
Supplements......Page 501
New References for the Second Edition......Page 515