Islands of Truth A MATHEMATICAL MYSTERY CRUISE

Ivars Peterson

W. H. Freeman and Company New York

LLRC 1 ! MCAs; / C

Contents

Portraits of Mathematics xi

Preface xv

1 Beginnings 1

Tokens of Plenty 2 New Math 9

Stretching Exercises 17 A Mystery 28

2 New Twists 33 The Etruscan Venus 35

Inside Out 46 Knot Physics 52

A New Dimension 61

C CONTENTS

3 Fitting Arrangements 71

Paving the Plane 73 From Here to Infinity 86 Spheres in a Suitcase 96

Computers in Kindergarten 103

4 Snowflake Curves 111

Fractal Forgeries 1 14 Written in the Sky 125

Time to Relax 133 Life on the Edge 140

v

CONTENTS

5 Number Play 151 A Shortage of Small Numbers 153

Tickling the Mind 158 The Formula Man 17 1

Pieces of Pi 178 Musical Numbers 186

6 Hard Times 194

Math of a Salesman 197 Smart Guesswork 203 A Different Sort 2 15

Playing Bit by Bit 222

7 Shadows of Chaos 234

Mixing Magic 238 Taking a Chance 244

Strange Vibrations 247 Stalking the Wild Trajectory 253

Troubling Uncertainty 259

8 Truth and Beauty 268

The Straight Side of Circles 271 Beyond Understanding 273

A Marginal Note 280 Math of the Spheres 285

Figures of Beauty 288

Further Reading 293

Sources of Illustrations 309

Index 315

Index

Abstraction, 3, 8, 10, 16, 17, 31, 34-35, 61

Acoustics: in concert halls, 186, 188 and number theory, 186- 192 reflection phase gratings,

186- 188, 190- 192 Adjacencies (for describing

tilings), 81, 84- 85 a n e transformation, 123 Aggregation, diffusion-limited,

128- 129 Alberti, Leon Battista, 105 - 106 Alexander's horned sphere, 15 Algebra, 8, 35, 60, 283

von Neumann, 57 Algebraic geometry, 186, 284 Algebraic topology, 16 Algorithm, 17, 197

and computational complexity, 197, 202

efficiency, 197 - 203, 215, 22 1 - 222

genetic, 209 -215 optimization, 203 - 214

Algorithmic complexity theory, 263, 266

Amorphous materials, 135 - 136 defects, 136

Analysis, 10 Andrews, George, 173 Angle of repose, 141, 144 Apery, Fran~ois, 40, 41 Arbelos, 167, 170 Archimedes, 9, 167, 170,

180-181, 271 twin circles, 168, 170

Architecture: and computers, 104 grammar, 105 - 107 machine, 104

and mathematics, 106 - 107, 109

rules, 104- 106 Arakelov, S., 284 Area, 115, 163-165, 271-273 Arithmetic, 270, 292

algebraic geometry, 284 modular. See Modular

arithmetic Atiyah, Michael, 59

Babylonian mathematics, 8 Bak, Per, 140- 150 Banach, Stefan, 272 Bankoff, Leon, 167, 170 Barnsley, Michael, 123 Beauty of mathematics, 288 - 292 Bendler, John, 137 Berliner, Hans, 23 1 - 232 Binary numbers, 264 - 266 Binary search trees, 219 - 221 Bin-packing problem, 203 - 205 Blichfeldt, H. F., 100 Boy, Werner, 39 Boy's surface, 39-45 Brauer, Richard, 279 Braid theory, 57 Bruijn, N. G. de, 163 Bubbles. See Soap bubbles '

Calculus, 10, 35 Card shuffling, 24 1 - 243

in tournament bridge, 243 Ceulen, Ludolph van, 180 Chaitin, Gregory, 263, 285 Chaos, 247 - 267

definition, 258 and iteration, 254 and predictability, 258 - 259 and rounding error, 254- 259

INDEX

Chaos (continued) in simple physical systems,

247-253 and uncertainty, 263 - 267

Chaotic dynamics, 247 - 253 in bouncing-ball apparatus,

249 - 25 1 Lyapunov exponent, 249 Poincare map, 250 in Space Ball, 247 -249 in spherical pendulum,

251-253 Chess, 222 - 233 Chess computers, 222 - 233

Belle, 224, 228 Deep Thought, 222 - 227,

229-230, 232 Chess strategies, 229 - 233

chunking, 231 -232 Chudnovsky, David and Gregory,

184- 185 Circle, 17, 99, 112, 182

circumference, 178 - 179, 182 equation, 10 1 generating a sphere, 40 inscribed, 168, 170, 180 - 18 1 as a one-manifold, 63 relation to knots, 55, 59 as a simple closed curve, 13 squaring, 271 -273 as an unknot, 56

Classification problems: four-manifolds, 66- 67 simple groups, 277 - 280 surfaces, 64 - 65 three-manifolds, 24 - 26, 65

Classification theorem (simple groups), 277 - 280

Clebsch, Rudolf, 34 Clebsch's diagonal surface, 34 Coastline, fractal, 136 - 137

of eastern United States, 136- 137

length, 136- 137

Coin tossing, 10, 236 randomness, 238 - 239

Combinatorics, 273 Completeness of an axiom

system, 263, 270, 285 Computational complexity,

197 - 203 and randomness, 247

Computer experiments, 18 - 19, 255, 276

Computer graphics, 17 - 18, 26, 35, 44, 48, 289

finding minimal surfaces, 18-24

and fractal geometry, 26, 114- 124

Computer models (simulations), 237, 265

diffusion-limited aggregation, 128- 129

economic systems, 148 forest fires, 146 Penrose tiling growth, 89, 92 sandpiles, 14 1 - 144 snowflake growth, 128

Computers: in architecture, 104 and chess, 222-233 for classifying three-manifolds,

24-26 and "Life" (game), 146 - 147 and optimization problems,

203-215 and randomness, 244 - 247 role in mathematics, 16, 287 in searching and sorting,

215-222 use in proofs, 273 -277

Computer science, 16, 196 - 197, 202-203, 211, 216, 222

Conjecture, 18, 27, 35, 56, 288 Mertens, 14 structural, 283 undecidable, 270, 285

INDEX

Continued fractions, 175 - 176, 184

Conway, John, 146 Cook, Stephen, 202 Coordinate systems and

dimension, 6 1 - 62 Counting, 2-3, 10-11

abstract, 2, 7 concrete, 2- 6, 9 in Mesopotamia, 3 - 8

Counting systems, 2 - 9 Cox, Donna, 41, 44 Cross-cap, 37 - 39 Crystal, 76, 79-80

defects, 135 - 136 growth, 87-88, 91 -92, 207 ice, 128-130 minimal surfaces, 1 8 - 19 quasiperiodic, 86, 88 - 95 relaxation, 134 structure, 77, 79-80, 86-88

Gystallography, 77, 110 and Penrose tilings, 86

Cuboctahedron, 50 - 5 1 Cuneiform notation, 3, 8 Curvature, 273

space, 10, 29, 59-60 Curves:

elliptic, 283 simple closed, 13 snowflake, 114-117

CUSPS, 19-20 and elliptic curves, 283

Dahlke, Karl, 167, 169 Data structures, 215

self-adjusting, 2 18 - 22 1 Deep Thought, 222 - 227,

229-230, 232 Defect diffusion, 136, 139 Descartes, Rene, 168 Determinism, 259, 262, 266 - 267 Diaconis, Persi, 241 - 242 Dice, 239 - 240

Differential equation, 12, 64, 253 Diffraction gratings, 190 Diffusion-limited aggregation,

128- 129 Digital communication, 96,

102- 103 Dimension, 61 - 62, 1 15

coordinate systems, 61 - 62 fourth, 43 -44, 61 - 70 kactal. See Fractal dimension

Diophantus, 280 - 282 DiVincenzo, David, 89 Donaldson, Simon, 67, 69-70 Dubins, Lester, 272 Dynamical systems, 236 - 237,

247 - 267 self-organized criticality,

140- 150

e, 186 Earthquakes, energy release, 143 Economic modeling, 16,

148 - 149 Einstein, Albert, 10, 29, 59, 68 Efficiency of algorithms, 215,

221 -222 Elastic-net algorithm, 205 - 207 Ellipse, 40, 41 Elliptic curves, 283 Equation:

of a circle, 101 differential. See Differential

equation Laplace, 127 roots, 28, 80, 178, 291 of a sphere, 101 Yang-Baxter, 58 Yang-Mills, 67

Equilateral triangle, 50, 74 - 76, 96, 115

symmetry operations, 8 1 Etruscan Venus, 41 -45 Euclid, 9, 17, 271

INDEX

Euclidean geometry, 167 spaces, 62-70, 97

Euler, Leonhard, 176, 282 Euler number, 64 - 65 Euler's formula, 64 - 65 Eversion, sphere, 46 - 53 Evolution (biological), 148,

210-213 Expert systems, 213 -Exponential decay, 134 - 135 Exponential function, 198 - 200 Exponential growth, 196, 198,

201, 236, 258 Exponential time, 20 1 - 202

Face-centered cubic packing, 96-97, 100

Factorial, 175 Factoring, 247 Fair coin, 238 -239 ~ a k e four-space, 67 - 68 Faltings, Gerd, 284 Family, Fereydoon, 13 1 Fermat, Pierre de, 155, 168,

280 - 282 Fermat's last theorem, 280 - 285 Ford, Joseph, 263, 267 Four-color problem, 276 - 277 Fourth dimension, 6 1 - 70 Fractals, 17, 24, 26, 28,

112- 124, 140, 237, 289 and computer graphics,

114- 124 definition, 1 12 dimension, 115, 117 fern, 124 growth, 128-129, 131 landscapes, 1 18 - 120 noise, 144-146, 150 self-similarity, 1 12 surfaces, 115, 117-118 time, 134, 136- 137, 139- 142,

145, 149

trees, 119, 121-122 Fractional dimension, 1 15,

117-118 Fractions, 8, 178, 254, 264, 292

continued, 175 - 176, 184 Francis, George, 4 1 - 42, 44 Freedman, Michael, 66 Frey, Gerhard, 283 Function, 10- 12, 14, 173,

197-200 definition, 11 exponential, 198 - 199 iterated, 26, 254-255 polynomial, 17, 5 1, 198 - 200

Galaxies, distribution in space, 68 Galois, ~variste, 80 Gardner, Martin, 82 Gauss, Car1 Friedrich, 100, 176 Genetic algorithms, 209 - 2 15

application to biology, 212-213

for designing a truss, 213 for designing electronic

circuits, 2 1 1 in machine learning, 2 13 - 2 14

Geometry, 8, 17, 26, 28, 30, 34-35, 44, 46, 59, 112-113, 153, 278, 283, 289

euclidean. See Euclidean geometry

four-dimensional, 6 1 - 70 fractal, 24, 112-113, 119, 140 optimal, 18 - 20 projective, 17 Riemannian, 10, 29, 60

Geometry Supercomputer Project, 26 - 28

Glass, relaxation, 139 Godel, Kurt, 263, 266, 270 Goldberg, David, 2 14 Golomb, Solomon, 165 Gorenstein, Daniel, 280 Gosper, Bill, 184

INDEX

Graph theory, 106 - 107, 153, 160, 165

Gravitation, 10, 59 - 60 Greek mathematics, 8 - 9, 17,

167, 170, 180-181, 271, 280 - 282

Groups: classification of finite simple

groups, 277 - 280 cyclic, 107 definition, 80 - 8 1 exceptional, 278 Lie, 278 monster, 278 simple, 277 sporadic, 278 symmetry, 77, 80- 81

Group theory, 17, 24, 35, 60, 80, 84, 101, 106, 242

Griinbaum, Branko, 73 Guy, Richard K., 153- 154, 156

Harborth, Heiko, 159 - 160 Hardy, G. H., 72, 172-173, 178,

288 Heisenberg, Werner, 267 Hexagons, 74 - 76

convex, 82 - 83 Hex numbers, 155 - 157 Hirsch, Morris, 272 H o h a n , David, 21 Holland, John, 2 1 1 Homotopy, 40

Romboy, 40 - 44 Hughes, John, 5 1 - 52 Hutchinson, John, 123

Icosahedral symmetry, 92 - 94 Icosahedron, 92 Ida, 44-45 Idaszak, Ray, 41, 44

[ Ikeda map, 258

Incompleteness of axiomatic theories, 263, 266, 270, 285

Infinite series, 175 for n, 175, 179

Information content, 263 - 266 Initial conditions, sensitive

dependence on, 249, 253, 255, 260-263

Integer, 192 Interference, wave, 189 - 190 Invariants, 64

in knot theory, 57 - 60 topological, 64 - 67, 69

Irrational numbers, 178 Iteration, 26, 115, 289

functions, 254- 255

Jones, Vaughan F. R., 57 Jones polynomial, 57 - 60 Jordan, Camille, 14 Jupiter, Great Red Spot, 26 1 - 262

Kanada, Yasumasa, 183 - 184 Kant, Immanuel, 286 Karush, Jack, 272 Kasparov, Gary, 222 - 224,

226-227, 229, 233 Kelvin, Lord, 54, 58 Kepler, Johannes, 125 Kernahan, Brian, 205 Kershner, Richard, 82 Klein bottle, 43-44 Kline, Morris, 288 Knots, 11, 52, 54-61

and braids, 57 catalog, 5 6 - 57 classifying knots (origin),

54-55 definition, 1 1, 55 equivalence, 55 invariant, 57 - 60 and three-manifolds, 24 - 25 torus, 25 trefoil, 56

INDEX

Knots (continued) unknot, 56

Knot theory, 54 - 55 use in physics, 54, 57-60

Krull, Wolfgang, 290 Kuen's surface, 34

Laczkovich, MiMos, 27 1 - 273 Lam, Clement, 273 -276 Landscapes, fractal, 1 18 - 120 Laplace equation, 127 Leech, John, 101 Leech lattice, 101 Length:

of coastlines, 136- 137 of a fractal curve, 1 15

Leonardo da Vinci, 17 "Life" (game), 146

self-organized criticality, 146- 148

~ i m a ~ o n , 41 -43 Lin, Shen, 205 Lyapunov exponent, 249

MacWilliams, F. J., 274 Magic squares, 16 1 - 163 Mandelbrot, Benoit B., 28 Manifolds, 24 -27, 62 - 70

classification, 24 - 26 definition, 24 differentiable, 66 four-dimensional, 66 - 70 as a higher-dimensional

surface, 63 three-dimensional, 24 - 27, 65

Map (geographical), four-color problem, 276 - 277

Mapping, 123 - 124 March, Lionel, 104, 107, 109 Marsaglia, George, 245, 247 Matchstick problems, 159 - 16 1 Materials, amorphous, 134 - 139

defects, 136

Mathematical puzzles, 72, 158- 171

brainteasers, 158 magic squares, 16 1 - 1 63 matchstick problems, 159 - 161 polyominoes, 165 - 167 Rubik's cube, 152

Mathematics research: applied versus pure, 16 changing nature, 3 1 as an extensive enterprise, 10 public image, 9 relation to physical world, 16,

29-31, 285-287 role of experiment, 14, 3 1 as search for patterns, 1 1,

30-31, 35, 56, 61, 72, 153 - 158, 288 -289, 292

structure, 11, 15, 28-31, 60, 278, 285-288

unity, 11, 15, 28-31, 60, 287 visualization, 17, 26 - 28,

34-35, 44, 46, 289 Max, Nelson, 48 Maxwell, James C., 10 Maxwell's equations, 10 McDaniel, Wayne, 159 Mechanics, Newtonian, 64,

259-263 Melanchthon, Philip, 292 Mertens conjecture, 14 Minimal surfaces, 18 - 24

for block copolymers, 20 - 24 for crystals, 18 - 19 for soap films, 18 - 19

Mobius, August, 35 Mobius band (strip), 35 - 39

surface, 38 Modular arithmetic, 188 - 189 Montgomery, Hugh, 164 Monster group, 278 Montroll, Elliott, 136 Morgan, John W., 60 Morin, Bernard, 40, 48, 50-

5 1

INDEX

Motion, equations of, 64, 259 - 263

Mountain, fractal, 1 18 - 1 19

Nagel, Sidney, 144 Natural selection, 2 10 - 2 1 1, 2 14 Negroponte, Nicholas, 104 Newton, Isaac, 10, 168, 180 Newton's laws of motion, 64,

259 - 263 Noise, 145

l l f , 144-146, 150 and chaos, 236, 249, 267

Nondeterministic polynomial time, 202

Nonlinear dynamics, 266 NP-complete problem, 202 - 203 Numbers:

abstract, 2, 8- 9 concrete, 2 cuneiform, 3, 8 as expression of quantity, 2 - 9 irrational, 178 prime. See Prime numbers real, 264 Smith, 159 transcendental, 178, 186, 271

Number theory, 16, 153, 186-188, 192-193, 263, 277-278, 283

and acoustics, 186 - 192 and prime numbers, 153 - 155,

159, 247, 284 structural conjecture, 283

Odlyzko, Andrew, 14 One-sided surfaces. See Mijbius

strip Onoda, George, 88 Optimization, 203 - 2 15

bin-packing, 203 - 205 elastic-net algorithm, 205 - 207 genetic algorithms, 209 - 21 5 simulated annealing, 206 - 209

Orbit (dynamical system), 254-258

Orbits (planetary), 259- 260

Packing spheres, 96- 103, 125 Pappus, 168 Parabola, 1 14 Parshin, A. N., 284 Partition theory, 176- 177 Patterns:

in mathematics, 11, 30-31, 35, 56, 61, 72, 153-158, 288-289, 292

in tilings, 17, 72-76 Pendulum, motion, 248, 25 1 - 253

forced spherical pendulum, 251 -253

Penrsse, Roger, 86 Penrose tiling, 81, 86- 92 Pentagons, 86- 87

convex, 82 - 86 Phillips, Anthony, 49, 52 Pi (n), 178- 186, 263, 271

Chudnovsky algorithm, 184- 185

computation, 171, 179- 185 definition, 178, 182 expressed as a continued

fraction. 176. 18A expressed'as infinite series,

175, 179-180, 182 . -

normal. 182 statistical analysis, 183,

185- 186 Plane tilings, 72 - 76 Plato, 286 Pluto, motion of, 259-260 Poincare, Henri, 261, 288 Poincare map, 250 Polycarbonate polymer (Lexan),

137 Polygons:

convex, 82 - 86 nonconvex, 82 regular, 74 - 76

INDEX

Polymers, 20 - 24 and minimal surfaces, 20 - 24 polycarbonate, 137 relaxation, 133, 137- 139

Polynomial, 17, 165 function, 5 1, 198 - 200 Jones, 57-60 invariant, 57 - 60

Polynomial time, 198 - 200, 202 Polyominoes, 165 - 167 Prediction, mathematical, 255 Prime numbers, 153 - 155, 159,

165, 182, 188, 190, 247, 274, 277, 284

Primitive roots, 189 - 190 Probability, 10, 182, 237

theory, 152, 242, 263 Projective geometry, 17 Projective plane, 38 -40

finite, 274 - 276 Proof, 12-16, 28-29, 35, 52,

153, 270 computer-assisted, 16,

273 -280 errors, 35, 278-280 nature of mathematical proof,

12-16, 153, 165, 270-285 Puzzles, mathematical. See

mathematical puzzles Pythagoras, 9 Pythagorean triangle, 282

Quadratic forms, 66 - 67 Quadratic residue sequences, 188 Quantum field theories, 59 -60

topological, 59 Quantum mechanics, 59, 267 Quasiperiodic crystals

(quasicrystals), 86, 88 - 95

Ramanujan, Srinivasa, 158 - 159, 171-178, 184

Randomness, 236 - 247 and algorithmic complexity

theory, 263 - 267

in card shuffling, 24 1 - 243 and chaos, 264-267 in coin tossing, 238-239 in dice. 240 in the digits of n, 182 - 183,

185, 263 in lotteries, 240 -241 and mixing, 240 - 244 and physical laws, 236-237,

259 - 267 Random numbers, 244 - 247

and computational complexity, 247

and physical noise, 246 tests, 245 - 246 use of computers to generate,

245 - 246 Real line, 264 Real numbers, 264 Recreational mathematics, 152,

171 Reflection phase gratings,

186- 188, 190- 192 based on primitive roots, 190 based on quadratic residues,

188 Regular polygons, tilings, 73 - 74 Reinhardt, Karl, 82 Relativity theory, 10, 29 -30, 68 Relaxation, 133 - 139

stretched exponential, 135- 136

Ribet, Kenneth A., 283 Rice, Marjorie, 83, 85 Riemann, Bernhard, 10, 29 Riemann hypothesis, 14 Rochberg, Richard, 164 Romboy homotopy, 40 - 44 Roots of an equation, 28, 80,

178, 291 Rotational symmetry, 76, 81 Routing problems, 197, 200 -

202 Rubik's cube, 152 Ruelle, David, 287

INDEX

Ruler and compass construction for squaring the circle, 271

uss sell, Bertrand, 2, 152, 288

Sallows, Lee, 161 - 162 Sandpiles, 140- 144

angle of repose, 141, 144 avalanches, 140 - 144 computer simulations,

141 - 143 experiments, 143 - 144

Scherk's surface, 2 1 - 22 Schmandt-Besserat, Denise, 2, 3 Schroeder, Manfred, 186 Searching, 215, 218 - 221

in computer chess, 224 - 226 Self-adjusting data structures,

218-221 applied to binary search trees,

219 efficiency, 2 19

Self-organized criticality, 140- 150

Self-similarity, 1 12 Series, infinite, 175,179- 180, 182 Serre, Jean-Pierre, 283 Shanks, Wililiam, 180 Shannon, Claude E., 223 Shapiro, Arnold, 48 Shephard, G. C., 73 Shlesinger, Michael, 136 Shoemaker's knife, 167, 170 Sierpinski gasket, 122- 123 Silk, relaxation, 139 Simple group, 277 Simulated annealing, 206- 209

for designing circuits, 208 for image processing, 209-

210 Sleator, Daniel, 2 19 - 220 Sloane, Neil J. A., 274 Smale, Stephen, 47, 52 Smith numbers, 159 Snowflake curve, 114- 1 17

Snowflakes, 125 - 133 computer simulations,

131- 133 fractal, 131

Soap bubbles, 18 clusters, 18 and crystals, 18-20 model for metal grain

structure, 19 rules, 18

Soap films, 18 as models of minimal surfaces,

18 Sorting methods, 2 16 - 2 18

bubble sort, 2 16 - 2 18 selection sort, 2 17 - 2 18

Sorting problems, 2 15 Space, 10

euclidean, 62, 67 - 68 four-dimensional, 6 1 - 70 multidimensional, 6 1 - 65

Space-time, 29-30, 61, 68 Sphere, 14, 37-42, 76, 112, 118,

272 equation, 63, 101 eversion, 46 - 53 four-dimensional, 99 as a manifold, 24, 63 -64, 66 topological, 37, 44, 63 -66

Sphere packings, 96- 103, 125 in different dimensions,

99- 101 and digital communication,

96, 101 - 103 face-centered cubic, 96 - 97,

100 irregular, 97, 99 tetrahedral, 98 - 99

Sporadic groups, 278 Square, 74, 76, 271 -273 Squaring the circle, 27 1 - 273 Statistical mechanics, 58, 17 1

hard-hexagon model, 176 Steen, Lynn Arthur, 16 Steiner, Jakob, 39

INDEX

Steiner's Roman surface, 39 - 45 Steinhardt, Paul, 89 Stiny, George, 109 Stretched exponential

relaxation, 135 - 136 String theory, 30-31, 59, 177 Structural conjecture, 283 Surface:

Boy's, 39-45 classification, 24, 64 - 65 Clebsch's diagonal, 34 Etruscan Venus, 42 - 45 fractal, 115, 117-118 higher-dimensional. See

Manifold Ida, 44-45 Kuen's, 34 minimal, 18 - 24 one-sided. See Mijbius strip Scherk's, 2 1 - 22 Steiner's Roman, 39 - 45

Surface energy, 18 Surface tension, 130 Symmetry, 17, 24, 76-81, 106,

125, 242, 274 fivefold, 76, 86, 92 icosahedral, 93 in geometrical patterns, 76 - 78 group, 76-78, 80, 84-86 operations, 76 - 77, 80 - 8 1 in physical laws, 79 - 80 rotational, 76, 81

Synge, John L., 153

Tang, Chao, 140 Tarjan, Robert, 219-220, 222 Tarski, Alfred, 271 Taylor, Jean, 18- 19 te Riele, Herman, 14 Tetrahedron, 21, 23, 98

hyperbolic, 26 Thermodynamics, 203 Thomas, Edwin L., 2 1 Thompson, John G., 274

Thompson, Ken, 228 Three-body problem, 259 - 261 Three-manifolds, classification,

24 - 26 Thurston, William, 24 - 26 Tiling a rectangle, 163 - 165

with polyominoes, 166 - 169 Tilings, 17, 72-77

as decoration, 72, 77 Penrose, 81, 86-92 periodic, 76-77 regular, 73-74 semiregular, 74 - 75

Tiling the plane, 72 - 76 with convex hexagons, 82 - 83 with convex pentagons, 82 - 85

Time, fractal, 134, 136- 137, 139-142, 145, 149

Tokens, for counting, 2 - 8 Topological equivalence, 35 -36, 1

64-67 1 Topological invariants, 64 - 67,69

dimension, 64 Euler number, 64 - 65 quadratic forms, 66 - 67

Topological transformations, 35-36, 40, 44

four-manifolds, 66 - 70 Romboy homotopy, 40 - 44 sphere eversion, 46 - 53

Topology, 35, 44, 52, 59, 61 algebraic, 16

Torus, 24-25, 52, 64, 165 eversion, 53

Toms knot, 25 Transcendental numbers, 178,

186, 271 Transformations:

affine, 123 eversion (of sphere), 46 - 53 homotopy, 40 topological, 35 - 36, 40

Traveling salesman problem, 200-202, 205 -207

Trees, fractal, 1 19, 121 - 122

INDEX

Trefoil knot, 56 Tully, R. Brent, 68 Turbulence, 146

and chaos, 236, 255, 261 -262

Unity of mathematics, 11, 15, 28-31, 60, 287

Unknot. 56

Visualization in mathematics, 17, 26-28, 34-35, 44, 46, 289

Volume, 272

Weather forecasting, 236, 255 Weber, Wilhelm, 139

Weyl, Hermann, 288 Wiesenfeld, Kurt, 140 Wigner, Eugene P., 29 Wilansky, Allbert, 159 Witten, Edward, 59, 70 Wolf. Alan. 248 -249 right, rank Lloyd, 103, 107,

109 houses, 107 - 109

Yang-Baxter equations, 58 Yang-Mills equations, 67 Yau, Shing-Tung, 284 Yorke, Jim, 256