The Problems of Mathematics

IAN STEWART

Second Edition

Oxford New York

OXFORD UNIVERSITY PRESS

Contents

List of Figures Interview with a mathematician Interview to the second edition

1 The nature of mathematics 2 The price of primality 3 Marginal interest 4 The neglected book of Euclid 5 Parallel thinking 6 Sphereful symmetry 7 The miraculous jar 8 Ghosts of departed quantities 9 The duellist and the monster

10 Much ado about knotting 11 The purple wallflower 12 Squarerooting the unsquarerootable 13 Squaring the unsquarable 14 Strumpet Fortune 15 The mathematics of nature 16 Oh! Catastrophe! 17 The patterns of chaos 18 The two-and-a-halfth dimension 19 The lonely wave 20 More ado about knotting 21 Dixit Algorizrni 22 The limits of computability 23 The ultimate in technology transfer

Further Reading Index

ar-Rashid, Harun, 291 Artin, Emil, 274-7 Ashley, ClifEord W., 130, 267 l Askey, Rrchard, 165 astronomy, I r !

Atiyah, Michael, 286, 287 I attractor, 225 l

chaotic, 309 fam~liar, 225 I( fractal, 227 111 Lorenz, 230

, 279, strange, 226, 227 axiomatic system, 66 1 1

Bachelier, LOUIS, 188 111

Balley, James A., rgo ' l Banach, Stefan, 173, 176 I ' 1 1 Banach-Tarsk~ paradox, 173-6 Barenblatt, G . I., 87 '11 iil Barlow, William, 79

1-Khorezmi, Muhammad, 291, 293, Barnum, Phineas Taylor, 130 Barth, John, rgo

[;!l

l-Mamun, Abd-Allah, 291 basic move, 270 1 ' I basin of attraction, 248 Bass, Hyman, 142

"1 / I ( non-standard, 17, 107- 14 Belavin, A. A., 264 ) l )

Bell, Eric Temple, 33 1 1 Bell, John, 31 Beltrami, Eugenio, 63

l 11; Bender, C., 83 Benedicks, Michael, 309 Bennett, Charles C., 28, 29 ill Benoit, Erlc, r 1 1 Berkeley, Bishop George, 102, 103

; 1 ll 1 1

Berlekarnp, Elwyn, 87 1 1 1 0 ) 209 Bernal, J. D., 76 I ( (

k area, 172, 182 Bernoulli, Daniel, 197 Argand, Jean-Robert, 158 Bernoulll, John, 13, 103, 104, 157

p Asrstarchus, 194 Berra, Yog~, Foreword to end edition J Aristotle, 59, go, 193 Berry, Michael, 216, 217

arithmetic, I I Besicovlch, A. S., 250 $- arithmetico-geometric mean, 248 Bessel, Friedrich, 159

Arnauld, Antoine, 155 Bessette, Fransois, 28 Amol'd, Vladlmir, 206, 210, 217 Bietserbach, Ludwia. 160. 164

338 Index biology, molecular, 267, 285 cat, 30 Birkhoff, George, 199, 203 Schrodinger's, 28, 30-2 Birman, Joan, 277, 283, 285 catastrophe, 208-21,318 Bishop, Enett, 292 cusp, 211, 212, 214 Blake, WiUiam, 87 fold, 211 blancmange function, 241, 244 theory, 209-21, 215, 226 Blum, Leonora, 309 catflap, Schrodinger's, ($30

I Bob, 31, 32 Cauchy, Augustin-Louis, 105, 108, Bohm, David, 30 121, 131, 160 Boa-Reymond, Paul du, 93, 183 caustic, 216

I Boltzmann, Ludwig, 281 Cayley, Arthur, 123, 147

l Bolyai, Jinos, 62, 63 S celestial mechanics, 192 Bolyai, Wolfgang Farkas, 62, 171 Chaitin, Gregory, 306-7 Bolzano, Bernard, 91 chaos, 222-38, 308,309,318

l Bombelii, Raphael, 155, 156 symmetric, 235 Boussinesq, J., 256 charge, topological, 262 Boyle, Robert, 85 Chebyshev, Pafnuti, 162 bracket polynomial, 282-5 Chevalier, Auguste, I 15 Bragg, Lawrence, 85 Chevalley, Claude, 125 Brahe, Tycho, 194 Chossat, Pascal, 234 Brahmagupta, 155 Clifford, Willlam Kingdon, 68 braid, 272-7 Cobham, A., 295

elementary, 275 Cohen, Henri, 58 Branges, Louis de, 164-6 Cohen, Paul, 94 Brassard, Gilles, 28, 29 Cole, F N , 26,32 Brauer, Richard, 126, 285 Coleman, Sydney, 263 Bravais, Auguste, 85, 121, 123 Colgan, P. W., 214 Brickell, Emest, 299 colourability, 272 Broglle, Louis de, 255 complexity, 14, 15, 236, 307, 308 Brown, Robert, 187 component, 268 Brown, Scott, 202 Compton, Arthur, 255 Brownian motion, 187, 188, 246, 247 computability, 300- I I buffer zone, 215 computer, 147,151,153,165,313-15 Burau, W., 276 infinites~mal, I I I Burnslde, William, 125 configuration, 149-5 I

reducible, 149-51 calculus, 89, 159 unavoidable, 150, 151

of variations, 14 conformal, 164 Callot, Jean-Louis, I I I congruence, 24 canard, 111 conic section, IOI Cantor, Georg, 91-5, 241, 292, 305 conjecture

i set, 241 abc, 46

Cardano, Girolamo, 155, 156, 179, Bieberbach, 160, 164-6, 314 1 I 80 Euler, 7,39

cardinal, 92 flyping, 284, 285 inaccessible, 94 four-colour, 146-54 large, 94,96 Heawood, 149

Carleson, Lennart, 309 local density, 81

1 Carmichael, R. D., 27 Mertens, 162, 163 I

Cartan, eke, 124,293 Milin, 165 l

I

Index

Mordell, 42, 43, 45 ~Ielauney, V., I 15 Poincark, 138-40, I42 Deligne, Pierre, 4 prime number, 162 de l'Isle, Jean Romt, 85 Shafarevich, 45 Demjanenko, V. A., 39 Smith, I42 Democntus, IOI Tait, 283 De Moivre, Abraham, 159 Tate, 45 De Morgan, Augustus, 146, 147, 190 Weil, 43-5 density, 75

connected sum, 136 local, 81 consistency, 66 semiglobal, 82

strength, 94 Descartes, Rent, 16, 17, 19, 131, constructibility, 292 156, 210 content, 183 determinism, 223, 237, 309 continued fraction, 52 de Vries, Hendrik, 256, 257 continuous, 11 diagonal argument, 92, 305 continuum hypothesis, 94, 95, 297 dice, 222 converge, 90 Dickinson, Emily, I Conway, John Horton, 84, 87, 126, Dickson, Leonard, 12.5

279, 282 Diener, Francine, I I I Coolidge, Julian, 35 Diener, Marc, 111 Copernicus, Nicolas, 194, 195 Diffie, Whitfield, 25 corn circle, 248 dimension, 242 Cotes, Roger, 157, 159 fractal, 243, 2 4 Couette, M. Maurice, 236 Hausdorff-Besicovich, 244 countable, 92 inductive, 243 Crick, Francis, 285 similarity, 244 Crofton, Morgan, 182 topological, 242, 243 cryptography: Diophantus of Alexandria, 33-6, 42

public-key, 25, 299 Dirichlet, Peter Lejeune, 36, 38 quantum, 6 dischargigg, 150

crystal, 84, 121 discrete, 11 crystallography, 71, 115 dissection, 171 Curie, Pierre, 218 DNA molecule, 285, 286 curvature, 68, 140, 141 dodecahedron, 69,76 Cusa, Nicholas of, IOI rhombic, 72, 73 cycloid, 13, 14 Dodson, M. M., 246 cylinder, 133 Donaldson, Sirnon, 14, 15, 14.5, 265,

286, 288, 290 da Costa, Newton, 308, 310 Donne, John, 189 d7Alembert, Jean le Rond, 60, 197 Doria, F. A., 308, 310 Dantzig, George, 298 Douady, Adnen, 317 Darwin, Charles, 107, 108 Dubins, L., 176 Davenport, Harold, 87 duplicating the cube, 54, 57 Davies, Roy, 251 Durell, Clement V., 115 Davis, Martin, 306 Diirre, Karl, 150 Davis, Philip, 87, 152, 153 dynamical system, 226, 238, 247,308, Deep Thought, 224 309 Dehn, Max, 173 continuous, 226

invariant, 173 discrete, 226 Delaunay, Charles, zoo Hamiltonian, 232, 258

340 Index dyslexicon, 176 event, 184 Dyson, Freeman, 312 exhaustion, 50, 51

existence, 292 Edmonds, I., 295 experimental mathematics, 313-17 effect:

Dog-in-the-manger, 18 Doppler, 231

Einstein, Albert, 30,31, 68, 188,205, 206, 237, 266

Einstein-Podolsky-Rosen paradox, see EPR paradox

Ekert, Artur, 28 Eliot, T. S., 30 Elkies, Noam, 7, 39, 40 elliptic curve, 41 epicycle, 194, 261 EPR paradox, 28,30, 31 epsilontics, 105, 109 equation:

Cauchy-Riemann, 160 differential, 191 Diophantine, 34, 35, 41, 43, 44,

306,308 EuIpr. 40 Fermat, 36, -42 heat, 197 KdV, 256-8, 263, 264 Korteweg-de Vries, 2j6-8, 263,

264 Maxwell, 197, 286 Navier-Stokes, 87, 230, 231 partial differential, 197 Pythagorean, 42 quintic, 116, 119, 120, 124 sine-Gordon, 263 wave, 197 Yang-Baxter, 288 Yang-Mills, 288

equivalent, 135, 137, 268 Eratosthenes, 21 Erlangen Programme, 122, 123, 141,

I45 Euclid, 11, 19, 22, 34, 47-68, 88, 94,

107, 293,300 EU~OXUS, 50, 51, I93 Euler, Leonhard, 7, 26, 36, 39, 53,

54, 57, 131, 134, 157, 159, 160, 162, 163, 197, 210

number, 134, 141 Eutocius, 55, 56

factor, 277 fake four-space, 1 4 , 145, 286 Falconer, Ken, 251,252 Faltings, Gerd, 45 Fatou, Pierre, 248 Fedorov, Evgraf, 85 Feigenbaum, Mitchell, 231 Feit, Walter, 125 Feller, William, 187 Fermat, Pierre de, 24,33, 35-7, 180,

215 - Last neorem, 24, 33 37-9, 60,

69,310,314 little theorem, 24-6

Ferrari, Lodovico, 13 Ferro, Scipione del, 13 Feynman, Richard, 289 Fibonacci, 294 fibre bundle, 265 Field, Mike, 234 finite, 108

mathematics, 209 Fischer, Bernd, 126 Fitzgerald, CarI, 166 fluent, 101, 191 fluxion, 101, 103, 104, 191 flype, 284,285 Fontana, Niccolo, 13 four-colour problem, 146-54,315 Fourier, Joseph, gr, 197

series, 91 Fowler, David, 56 fractal, 7, 227, 239-53, 318 Frankel, Abraham A., 96 Frankenstein, Dr, 75 Franklin, Philip, 149 Freedman, Michael, 14, 140, 145 Freyd, Peter, 278, 279 Frobenius, F. Georg, 44

map, 4.4 Frye, Roger, 40 Furth, H. P., 260

galactic mouse, 206 Galilei, Galileo, 13, 89, 191, 194

Index (jalols, Evariste, 13, 15, 116, 118- soluble, 119

23. 125 sporadic, 126 Garabedian, P. R., 165 symmetry, 118, 234 Gardner, C. S., 258 Guckenheimer, John, 230 Gardner, Richard, I77 Giinther, S., 83 Gasper, George, 165 Guthrie, Francis, 146 gauge field theory, 11, I45 Guthrie, Frederick, 146 Gauss, Car1 Friedrich, 10, 22, 24, 36, Guy, Richard, 87

38, 62, 63, 67, 78, 90, 115, 122, 131, 158, 159, 162, 190,248,314, Hadamard, Jacques, 22, 93, 163 315 Haken, Wolfgang, 151-4, 315

Gautschi, Walter, 165 Hales, Stephen, 69, 76 genus, 42, I31 Hall, Marshall, 46 geodesic, 63 Halley, Edmund, 190 geometry, 11, 122, I41 halo, 109

algebraic, 25 j Hamilton, William Rowan, 146, 158, finite, 125 159, 216, 232 hyperbolic, 122, 141-3 handle, 132, 139 non-Euclidean, 63-8, 94, 122, 140, Hardie, Colin, 71

309 Harpe, Pierre de la, 277 spherical, 66 Harriot, Thomas, 155

Gerver, Joseph, 201, 202 Harvey, William, 245 Gerwien, P., 171 Hatt, D., 277 Girard, Albert, 155 HausdorEf, Felix, 175 Glashow, Sheldon, 266 Haiiy, Rent Just, 85 global, 82 Heath-Brown, D. R., 45 Godel, Kurt, 301, 305, 308 Heawood, Percy, 149 Godiva, Lady, 176 Heesch, Heinrich, 150 Golubitsky, Martin, 218, 220, 234 Heiberg, J. L., 97 Gordon, Walter, 263 Hein, Piet, 137 Gorenstein, Daniel, 126 Heisenberg, Werner, 29 Grauert, H., 45 Held, Dieter, 126 great escape, 199 Hellman, Martin, 25 Green, John, 258 HCnon, Michel, 309 Gregory, David, 83 system, 309 Gregory, James, 104 Herakleides, 193 Griess, Robert, 126 Hermite, Charies, 54, 242 group, 116, 117, 120, I41 Hersh, Reuben, 87, 152, 153

abstract, 123 Higman, Donald, 126 braid, 274-7 Hilbert, David, 64, 65, 78, 88, 94, continuous, 121 172,301,305,306,308 Conway, 84 hotel, 88, 94 exceptional, 124 Hippias, 56 finite, 125, 126 Hirsch, Morris, 176 fundamental, 138, 139 Hodge, William, 44 Galois, 119, 120, 126, 127 Hoffmann, Banesh, 254 Lie, 11, 121, 124, 125, 145 Hofstadter, Douglas, 300 Lie type, 125 hole, 132 property, 117, 123 Holmes, Philip, 230 simple, 84, 124, 125, 126, I53 HOMFLY polynomial, 279

Index

homotopy, 138 honeycomb, 218 Hooke, Robert, 85, 190 Hopf, Eberhard, 1x1, 219, 231

bifurcation, 11 I , 219 Hoppe, R., 83 Hoste, Jim, 279 Hsiang, Wu-Yi, 69, 74, 80, 81, 82,

84,86 Hubbard, John, 249, 317 Huntington, Edward, 123 Hunvitz, Adolf, 93 Huygens, Christiaan, 16, 85, 180,

190, 210, 255 hypenvebster, 174, 175

icon, 234 impossible device, 310 incompressible, 76 Infantozzi, Carlos, I 15 infinite, 108

descent, 36,45 series, 89

infinitesimal, 98, 100-5, to8 infinity, 87-96

actual, 90, 91 potential, 90

information, 307 instanton, 263, 264 integer, 48

algebraic, 36, 37 cyclotomic, 38 Gaussian, 36

integration, 183 Iooss, GCrard, 87

Jacobi, Carl, 115, 190 Janko, Zvonimir, 126 Johnson, Lee, 87 Jones, James, 306 Jones, Vaughan, 6, 276-90

polynomial, 277-90 Jordan, Camille, 121, 123, 125 Joseph, Daniel D., 87 Julia. Gaston, 248

set, 248-50, 309

Kac, Victor, 258 Kainen, Paul, 147 Kalippus, 193 Kaluza-Klein theory, 266

KAM theory, 207 Karmarkar, Narendra, 298 Karush, J., 176 Kaufmann, Louis, 281-4 Kelvin, Lord, 77 Kempe, Arthur, 147-50

chain, 148, 149 Kepler, Johannes, 5, 69-79, 84-6,

195. 196, 198, 200 problem, 69-86, 314

Khachian, L. G., 298 Kilhng, Wilhelm, 124 kink, 262 Kirby, Robion, 144 kissing number, 82-4 Klein, Felix, 65, 122, 123, 132, 141

bottle, 132-4, 141, 149 Klein, Oskar, 263, 266 Kline, Morris, 35, 47, 155 knot, 130, 134-7. 262. 267-90,318

alternating, 283, 284 diagram, 270 figure-eight, 274 granny, 136, 268, 274, 276, 278 invariant, 135, 136 Kinoshita-Terasaka, 283 prime, 136, 142 reef, 136,268, 274,276, 278 trefoil, 135, 136, 272-4, 276, 277 wild, 136 Wilder, 136

Knuth, Donald, 15 Koch, Helge von, 241, 242, 247 Koch, John, 152 Koebe, Paul, 164

function, 164 Kolmogorov, Andrei, 184, 206 Konigsberg bridges, 131 Korteweg, Diederik, 256 Kronecker, Leopold, 93, 96 Kruskal, Martin, 257, 258 Kummer, Ernst, 38, 39

Labouriau, Isabel, 220 Laczkovich, Miklds, 5, 177, 178 Lagrange, Joseph-Louis, 78, 121, 198 Lambert, Johann, 61 LamC, Gabriel, 36, 38 Landau, Lev, 231 Lander, L. J., 39 Langford, Wiiliam, 220

Index Laplace, Pierre-Simon de, 180-2, MacLaurin, Colin, 103

197, 198, 223, 225 Maddox, John, 127 lattice, 78, 112, 113, 127, 280 Mahler, Kurt, 78

cubic, 74 Malgrange, Bernard, 210 face-centred cubic, 74 Malthus, Thomas, 295 hexagonal, 72,74,75 Mandelbrot, Benoit, 6, 7, 238, 239, Leech, 84 245, 248-50 packing, 78 set, 6 ,7 , 248-50,309,316, 317 square, 72, 75 manifold, 14, 67, 131, 137, 138, 143. symmetry, 7-36 144,265,289

~ a w t o n , J. H., 246 Manin, Yuri, 45 ~ a x , Peter, 258 Mao Tse-tung, 146 Lebesgue, Henri, 173, 183, 184, 242 Markov, A. A., 276, 277 Lebon, Napoleon, I15 Marvin, 3. W., 76 Leech, John, 83, 84 Mason, R. C., 46 Lefschetz, Solomon, 44, 204 Mather, John, 201, 210

trace formula, 44 Mathieu, Emile-LConard, 125 Legendre, Adrien-Marie, 36, 197 Matijasevich, Yuri, 306 Leibniz, Gottfried, 13, 19, 101, 102, Matzke, E. B., 76

103, 155,157, 159, 192 Maxwell, James Clerk, 197, 265 Lenstra, Henrik, 21, 26, 28,58 Mayer, Jean, 152 Leonardo of Pisa, 294 measure, 173, 183 Leray, Jean, 231 Meeks, William H. 111, 142 Levine, D., 127 Menasco, William, 285 L'HBpital, Guillaume, 13 Mert, Chevalier de, 180 Libri, Guillaume, 294 Merkle, Ralph, 25, 299 Lickorish, Raymond, 279 Mertens, Franz, 162 Lie, Sophus, 11, 121, 123-5 Milin, I. M., 165 Lieb, E. H., 281, 288 Miller, G. L., 27, 28 light, 255 Millett, Kenneth, 279 limit, 105 Mills, Robert, 145,263,265,288,289

cycle, 225 Milnor, John, 14, 80, 144, 145 L~ndemann, Ferdinand, 54, 56, 57, minimal criminai, 148

1 69 Minkowski, Hermann, 146 linear programming, 298 Mirimanoff, Dimitri, 39 link, 138, 268, 283, 284 Mises, Richard von, 184

alternating, 283-5 Miura, R. M., 258 Liouville, Joseph, 38, 54, 58, 120 mixing, 232 Listing, Johann, 132 Mobius, Augustus, 132, I34 Ljunggren, W., 41 band, 132 Lobachevskii, Nikolai Ivanovich, 62 model, 63, 66, 107, 224

63 ice, 281 local, 81 Potts, 281 logarithm, 157, 160 modulus, 24 Lorenz, Edward N., 229, 230 Mohs, Friedrich, 85 Lowner, Karl, 165 moment, 217 Lyons, R., 126 monster, 126, 127, 240

Moody, Robert, 258 McGehee, 201 Moore, Christopher, 309 MacKay, Alan, 127 Mordell, L. J.. 41, 42, 4 McLaughlin, John, 126 morphogenesis, 210

C-- - * 344 Index

l Morse, D. R., 246 l

parallel, 60 Morse, Marston, 204 axiom, 60, 94, 297

, Moser, Jurgen, 206 Parkin, T. R., 39 I Motel, Sttphanie du, 115, 120 Parmenides, 98, 193 1 Murasugi, Kunio, 284 Parshin, A. N., 45

Pascal, Blaise, 180 Neumann, John von, 276, 301 pathological, 240

algebra, 276, 277 Peano, Giuseppe, 94, 183, 241 1 1 neusis-construction, 55 curve, 241 11 Newton, Isaac, 13, 19, 70,83,89,97, Pederson, R. N., 165 101, 102, 103, 104, 109, 110, 156, Peitgen, Heinz-Otto, 249 11 190-2, 196-8, 200, 205, 206, Penrose, Roger, 127, 308, 311

l 1 222, 255, 314 periodic, I gg 1 1 Nimzovich, Aron, 167 permutation, 118, 121 Niven, Larry, 105 Perrin, Jean, 247 1 noise reduction, 232 perturbation theory, I 1 I , 113 11 normal, I 19 Petersen, J . , 149

1 Nowell, W., 214 phase: NP, 296, 297 portrait, 225

1 NP-complete, 297 space, 199 1

number, 11, 155 space reconstruction, 232 algebraic, 36, 53 transition, 280

11 Carmichael, 27 pigeonhole principle, 94

l complex,48,108,155-67,248-50 pixel,112 golden, 49 Plain, Virg~nia, 4-7

I hyperreal, 108 Plato, 193

l ideal, 38 Podolsky, Bor~s, 30, 31 irrational, 49,51,53,57 PoincarC, Henri, 64, 93, 107, 138, kissing, 82-4 139,144,198,199,203,206,224, natural, 48 242, 243 non-standard, 108 poke, 271, 283 prime, 15, 21-32,36, 162 Pblya, George, 187 rational, 48, 108 Polyakov, Alexander, 264 real, 48, 108 pomegranate, 72, 74, 81, 84 transcendental, 3 6 , 5 3 5 93, 292 Pomerance, Carl, 28

Pommerenke, Christian, 166 Ocneanu, A., 279 Pontryagin, Lev, 258 Odlyzko, Andrew, 28, 84, 162 Pope, Alexander, 97, 222 Olbers, Heinrich, 36 potential theory, 197 O'Nan, Michael, 126 Poussin, Charles-Jean de la VallCe, one-to-one correspondence, 88, 92 22, 163 oracle, 310 primality test, 23 order, 118 probability, 179-88 Oscar I1 of Sweden, 198, 199, 206 function, 184 outcome, 184 problem, 12 Ozawa, M., 165 assignment, 296

bin-packing, 296, 297 P, 296,297 combinatorial, 296 PainlevC, Paul, zoo halting, 303, 305, 310 paradox, 88 Hilbert's tenth, 306

Index 345 - 67, 122, 132, 134, 137, 144, 160,

optimization, 296 162 Rolls-Royce, 18 hypothesis, 27,44, 160, 162-4, 314 routing, 296 surface, 132 sorting, 296 Riess, Dean, 87 travelling salesman, 296 rigidity, 172

proclus, 60, 88 Ringel, G., 149 projective plane, 141 ring size, 151 proof, 125, 152, 164, 166, 190, 292, Rivest, Ted, 26

301, 308, 310 Robinson, Abraham, 108 pseudoprime, 26, 28 Robinson, Julia, 306 pseudosphere, 63 Rogers, C. A., 80 Ptolemy, 16, 60, 194, 261 Rosen, Nathan, 30, 31 Putnam, Hilary, 306 Rothman, Tony, 115 Pythagoras, 10, 12, 34, 48, 49, 98, RSA system, 26

I93 Rubik, Erno, 116 Pythagorean triple, 34, 36 cube, 116

Rudvalis, Arunas, 126 quantum Ruelle, David, 231

field theory, 254, 288 Ruffini, Paolo, I 16 mechanics, 14, 28, 86, 160, 200, Rumely, Robert, 28

2.5631 1 Russell, John Scott, 254, 256 physics, 267 Ryan, Matthew, 59

261 quasicrystal, 127 quasiperiodic, 225 quilt, 234, 235

radical, 13, 116, 119 rainbow, 216 random, 237,306,308

packing, 7 6 7 7 sequence, 306 walk, 186

ratio, 50 Rayleigh, Lord, see Strutt, John

William reciprocity, higher, 28 Ree, Rimhak, 125 Reeb, Georges, 106, 108 Reidemeister, Hans, 135, 270, 272,

274,276 relativity, 14, 67, 237, 261, 288 Reveillts, Jean-Pierre, I 12 Reynolds, Osborne, 77 Richard paradox, 308 Richardson, Lewis Fry, 244 Richter, Peter, 249 Riele, Herman te, 162 Riemann, Georg Bernhard, 27, 57,

Saari, Donald, 201 Saaty, Thomas, 147 Saccheri, Girolamo, 61, 63, 65 Salam, Abdus, 266 Salvail, Louis, 28 sample space, 184 sand, 77 Scarpellini, B., 310 Schaeffer, David, 218 Schechtman, Daniel, 127 Schiffer, M., 165 Schoenflies, Arthur, 85 Schrodinger, Envin, 6 Schiitte, K., 83 Schwartz, S., 264 Scott, G. D., 76 Seeber, L. A., 78 Segre, B., 78 SeEdge, John, 27 set, 107 shadow, 19 Shafarevich, Igor, 44, 126 Shakespeare, William, 179, 222 Shamir, Adi, 26, 299 Shannon, Claude, 307 Shelah, Saharon, 96

346 Index Shiskikura, Mitsuhiro, 250 local, 265 Shockwave, 232 global, 264 Shub, Michael, 309 Siebenmann, Larry, 144 Tait, P. J., 149, 283, 284 Siegel, Car1 Ludwig, 306 Takens, Floris, 231 Sims, Charles, 126 Tampach, Godfrey, 70 singularity, 200 tangent, IOI slide, 271, 283 Tarski, Alfred, 5, 6, 168, 169, 173, Sloane, Neil, 84 176 Smale, Stephen, 139, 224,298, 309 Tartaglia, 13, 156 Smith, Barnabas, 97 Tate, John, 45 Smith, Henry, 241 Taubes, Clifford, 14 Smith, Paul, 142 Taurinus, Franz, 63 Smith, Sydney, 59 Taylor, Geoffrey Ingram, 220, 236 Smolin, John, 28 Taylor-Couette flow, 236 smooth structure, I technology transfer, 320 snowflake, 70, 71, 84 Teller, Edward, 260

curve, 241, 242, 244 Temperley, H. N. V., 281, 288 soliton, 254-66 Temperley-Lieb algebra, 281, 288 Solomonoff, Ray, 307 Thales, 193 Solovay, R. M., 96 Theodosius, 34 Solzhenitsyn, Alexander, 204, 206 theorem: space, four-dimensional, 14 Bolyai-Genvien, 171-3 Sperling, H. J., 201 Cauchy's, 160 sphere, 141 Euler's, 131, 134

exotic, 14, 144 hairy ball, 205 packing, 69-86 marriage, 177, 178

squaring the circle, 54-6, 60, 168-78 Pythagoras', 34, 47 Stallings, John, 139 two squares, 35 standard part, 108, 109 Thirring, Walter, 263 starlight, 216 Thistlethwaite, Morwen, 285 statistical mechanics, 281 Thom, RenC, 208-11, 213 statistics, 180 Thompson, John, 125, 127 Steinberg, Robert, 125 t'Hooft, Gerhard, 264 Steinhardt, P. J., 127 three-body problem, 137 Stevin, Simon, 155 Thue, Axel, 78 stochastic process, 184 Thurston, William, 140-2 Stokes, W., 214 time: Strange Attractions, 249 exponential, 295 Strutt, John William, 256 nondeterministic polynomial, 295 subgroup, 119, 120 polynomial, 28, 295, 297 sundial, 7, 250 topology, 130-44, 173, 198, 204,258, sunfish, pumpk~nseed, 213 7-67 superstring, 266 algebraic, 138, 255 suspension, 227 differential, 14 Suzuki, Michio, 125, 126 torus, 133, 141 Swinney, Harry, 231 Tdth, L. Fejes, 78 Sylvester, James Joseph, Foreword trace, 277 symmetry, I 16, 124, 127, 202, 264 transformation, I 18

breaking, 218, 219 birational, 121

Index

trisecting the angle, 54, 57, 60 turbulence, 230, 231 Turing, Alan, 301 -7

machine, 301, 303, 304, 307 Turing-Post program, 301-4, 307 twist, 270, 283

number, 283 Tymoczko, Stephen, 152 Tyupkin, Yu. S., 264

Uhlenbeck, Karen, 14 Ulam, Stanlislaw, 2 57 Ullman, Jeffrey, 298 uncomputable, 309 undecidable, 304, 308 unified field theory, 266 univalent , 164 unknot, 268 unpoke, 271, 283 int twist, 271, 283

Weierstrass, Kari, 105, 107, 160, 240, 7-42> 247

Weil, Andrt, 42, 43 Weinberg, Steven, 266 Weiss, Christian, 85 Wells, Herbert George, 168 Wenzl, Hans, 285 Werner, Abraham, 85 Wessel, Caspar. I$7 Westfall, Richard, 97, 189 Weyl, Hermann, 93, 116, 265, 286 Whitehead, Henry, 144 Whitney, Hassler, 139

trick, 139 Qriener, Norbert, 188, 7-47 Wiesner, Stephen, 28 Wigner, Eugene, 319 Wilder, R. L., 136 Williamson, M. H., 246 witness, 27 Witten, Edward, 5, 288-90 writhe, 283

van de Lune, J., 162 Vandermonde, Alexandre-Thtophile, xia, z., zo2

I59 van der Poorten, Alf, 58 van der Waerden, B. L., 83 Verma, D. N., 258 Vojta, P., 45, 46 Voltaire, 223 volume, 172, 174, 182

Wacker, John Matthew, 70, 71 Waddington, C. H., 210 Wagstaff, S. S., 27, 39 Wallace, William, 171 Wailis, John, 104, I57 wallpaper, 127, 142 Wantzel, Pierre, 17 Watson, James, 285 wave, 255

of translation, 254, 256 weather, 229 Wehrfritz, B. A. F., 87

Yang, Chen Ning, 145,263,265,288, 289

Yang-Mills gauge field, 145, 263, 26a

Yau, shing-~ung; I42 Yetter, David, 278 Young, Thomas, 255 Young, William, 183 Youngs, E., 149

Zabusky, Norman, 257 Zagier, Don, 22, 58 Zeeman, Christopher, 139, 214, 215 Zeno, 98

paradoxes, 99 Zermelo, E., 94 Zermelo-Frankel set theory, 96 zero, 291 zeta function, 43,44,57. 162, 163