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Mumford D., Wright D., Series C. — Indra's Pearls: The Vision of Felix Klein
Mumford D., Wright D., Series C. — Indra's Pearls: The Vision of Felix Klein

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Название: Indra's Pearls: The Vision of Felix Klein

Авторы: Mumford D., Wright D., Series C.

Аннотация:

Felix Mein, a great geometer of the nineteenth century, rediscovered an idea from Hindu mythology in mathematics: the heaven of Indra in which the whole Universe was mirrored in each pearl in a net of pearls. Practically impossible to represent by hand, this idea barely existed outside the imagination, until the 1980s when the authors embarked on the first computer investigation of Mein's vision. In this extraordinary book they explore the path from some basic mathematical ideas to the simple algorithms that create delicate fractal filigrees, most appearing in print for the first time. Step-by-step instructions for writing computer programs allow beginners to generate the images.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 2002

Количество страниц: 396

Добавлена в каталог: 11.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\infty$      see infinity
$\mu$-parameter      259 287
$\mu$-value      302
3-manifold      387
3-sphere      387
Ahlfors, Lars      384 386
Aigand diagiam      41
Apollonian gasket      196—208 228 231 245 252 259 265 274 292 325 330 341 362 368
Apollonian packing      see Apollonian gasket
Apollonius      194 199
Arctangent      42
Area-filling curves      329
Associative law      22
Asymptotic self-similarity      317
Automaton      356
Automorphic function      374 375
Avatamsaka Sutra      xvii
Axioms      377
Beltrami, Eugenio      379
Bers, Lipman      280 291 312 313 384
BFS      see breadth-first search
Bilateral symmetry      6
Binary notation      128
Bolyai, Janos      xv
Boundary tracing      301—307
Breadth-first search      113—117
Butterfly effect      133
Cannon, James      331 356
Cantor set      136
Cantor, Georg      3 129 155 312 328 329
Cardano's Formula      38
Cardano, Girolamo      37
Cayley graph      105
Cayley map      89
Cayley, Arthur      105
Chaotic      xvii 133 141 153
Commutative law      48
Commutator      167 169 181 185 265
Commutator, trace      193
Commuting maps      15 169
complex numbers      36—61
Complex numbers, absolute value      43
Complex numbers, argument      43
Complex numbers, imaginary part      39
Complex numbers, modulus      43
Complex numbers, polar representation      43 45
Complex numbers, programming      46—48
Complex numbers, purely imaginary      39
Complex numbers, real part      39
Complex plane      41
Complex square roots      60
composition      14—17 23 33 61
Concatenation      278 283 325
conjugate      173
Conjugate, complex      39 44
Conjugate, group      173—178 195 266
Conjugation      26—28
Conjugation of maps      27 51
Conjugation, method of      64
Continued fraction      212 294 308 316 349
Convex hull boundary      390
Core chain      271
Countable set      129 155
Cubic equation      37 38 304
Cusp      273
Cusp group      see group cusp
Cyclic permutation, order      143
Decimals      125—129 154
Decimals, repeating      126 155 291
Depth-first search      113 122 141—150
Determinant      70
DFS      see depth-first search
Dictionary ordering      116
DIMENSION      328 see
Diophantine equation      192
Discrete group      see group discrete
Elliptic functions      374
Elliptic map      see map elliptic
Epstein, D.B.A.      356 358
Erlanger Programm      xvi 1 71
Euclidean algorithm      213 222 306—307
Exponential function      51—54
Exponential shrinking      103 118 170
Extended real line      86
Farey addition      212 276 316
Farey fractions      223 301 306
Farey John      212
Farey neighbours      212 223 278
Farey sequence      155 301 306
Farey tessellation      285
Fibonacci numbers      316
Figure of Eight Knot      388
Fixed point      51 78
Fixed point, attracting      51
Fixed point, equation      78 155
Fixed point, neutral      82
Fixed point, repelling      51
Ford circles      221
Fractal      138—141
Fractal dimension      140 253 254 see
Fractal dust      121
Fricke identities      189
Fricke, Robert, xvi      3 161
Fuchs, Immanuel      179 376
Fundamental region      35
Fundamental theorem of algebra      46 287
Galois, Evariste      20
Gasket      see Apollonian Gasket
Gauss, Carl Friedrich      209
Gaussian integers      59
Generators      see group generators
Genus of a surface      383
Geodesies      390
Geometrization conjecture      389
Golden mean      316
Grandma      85 92 227—230 264 293
Group      3 17—21 23 51
Group, $\theta$-Schottky      118
Group, automatic      356
Group, classical Schottky      102
Group, cusp      248 266 268—309
Group, degenerate      312 369
Group, discrete      239
Group, double cusp      204 248 268 275 291—301
Group, doubly-degenerate      312 339
Group, free      98 132 356
Group, Fuchsian      161 178 180
Group, Fuchsian Schottky      118 177
Group, full modular      210
Group, general Schottky      102 274
Group, generators      20 232
Group, kissing Schottky      158 170—172
Group, Kleinian      353 360
Group, Maskit      259 260 291—296
Group, maximally cusped      204
Group, modular      206—218 223 224 374
Group, more than 2 generators      365—370
Group, non-discrete      232
Group, non-free      353—364
Group, once-punctured torus      190 228
Group, parabolic commutator      228
Group, parameterized family      171
Group, quasifuchsian      161
Group, Riley      258
Group, Schottky      98—103 157
Group, single cusp      248
Group, singly-degenerate      312
Group, totally degenerate      312
Group, two-generator, Grandma's      265
Group, unit circle      89
Group, upper half plane      86
Hausdorff dimension      136 139 155 256—259 309 313
Hokusai, Great Wave      371
Holt, Derek      356 358 360 381
Hyperbolic 3-space      386 388
Hyperbolic functions      47
Hyperbolic geometry      314 333 373 376
Hyperbolic map      see map hyperbolic
Hyperbolic plane      379 381
Hyperbolic space      164
Hyperbolization theorem      389
Ideal triangle      196 199 221
Identity transformation      19 93 98 354
IFS      see random search
Image      10
Imaginary axis      41
incircle      197 199
Independence of starting point      133 155
Indra's net      ii xvii
infinity      55—57
Invariant      15 98 101 164 373 374
Inverse, in group      23
inversion      54—56
Iowa      4
Irrational number      34 50 53 214 310 314 333 350
Iteration      5 51 153 303
Jergensen's doubly-degenenerate group      331
Jergensen's point      319
Jergensen's singly-degenerate group      314
Jergensen, Troels      256 312 333 388
Jordan curve      226
Jordan, Camille      20
Julia set      291
KBMAG      358 360 381
Kissing chain of circles      157
Klein, Felix      vii xvi 1 2 20 38 62 71 100 105 161 190 373 374 377 380
Kleinian group      see group Kleinian
Knot complements      388
Kra, Irwin      367 372
Lie, Sophus      20
Limit, algebraic      343
Limit, geometric      343
Limit, point      121—125
Limit, set      100 121
Lindenmaier or L-system      329
Loxodrome      64
Loxodromic map      see map loxodromic
Mandelbrot set      viii 215 291
Mandelbrot, Benoit      viii 121 139 152 253 329
MAP      10 12
Map, affine      51 60
Map, angle preserving      77
Map, classification      81
Map, conformal      383 384
Map, conjugating      27 64
Map, elliptic      79
Map, extremal quasiconformal      384
Map, hyperbolic      79 87
Map, inverse      12 61
Map, linear      33
Map, loxodromic      79
Map, Markov      338
Map, Moebius      62—95
Map, nearly parabolic      243 252
Map, non-linear      66
Map, orientation preserving      77
Map, pairing      91 96 97
Map, parabolic      79
Map, quasiconformal      384
Map, stretch      89
Map, trace      84
Mapping      10
Markov chain      154
Markov conjecture      191 283 352
Markov identity      189
Maskit's slice      287—292
Matrix algebra      72—74
Matrix inverse      73
McMullen, Curt      x xi 139 140 156 197 256 291 317 321 324 333 345 349
Minsky, Yair      291 324 391
Modular function      374
Modular group      see group modular
Modular necklace      214
Modular, arithmetic      144
Moduli      383
Moebius map      see map Moebius
Moebius, August      xvi 71
Motions      4
Multiplier      79
Necklace conditions      168 361
Necks, problem of      184 248—252
Neighbour fractions      see Farey neighbours
Newton solver      302—307 309
Newton's method      287 302
nodes      103
Non-discrete group      see group non-discrete
Non-Euclidean geometry      373 377 see geometry
Non-Euclidean space      164 see space
Normalization      227
Orbit      18 21 50 51
Ordinary set      109 161
Orientation, of generators      297
Parabolic map      see map parabolic
Parallel postulate      378
Parameters      111 171 227
Peano curves      329
Period      126
Plotting limit points      132—135
Poincare conjecture      389
Poincare disk      380
Poincare, Henri      100 376
Point at infinity      see infinity
Polar coordinates      42—45
Polynomial equation of degree n      46
Prefix      126
Preserves angles and circles      58
Programming      28—33
Programming, automatic groups      358—360
Programming, boundary tracing      304—307
Programming, depth-first search      145—150
Programming, drawing curves      182—187
Programming, Euclid's algorithm      307
Programming, Moebius maps      76 91
Programming, Newton's method      302—307
Programming, Schottky groups      111—117
Programming, sealing gaps      215—221
Programming, special words algorithm      217—221 260—264
Pseudo-code      30
Puncture      189
Quadratic equation      37
Quadratic form      209
Quasicircle      161
Quasifuchsian      see group quasifuchsian
RADIANS      43
Random search      113 152—154
Real axis      41
Real trace ray      296 390
Recipes, Grandma's      227—233
Recipes, Grandma's Special Four-Alarm      260—261 362
Recipes, Jergensen's      256—257
Recipes, Maskit's      259—260
Recipes, Moebius maps      85—92
Recipes, Riley's      258—259
Recipes, Schottky groups      117—119
Recursion      150
Reflection      6
Regular set      see ordinary set
Renormalization      322
Repetend      126
Richardson, Lewis Fry      253
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