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[28] Falconer, K., The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1985.

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[31] Family, E, Vicsek, T. (eds.), Dynamics of Fractal Suifaces, World Scientific, Singapore, 1991.

[32] Feder, J., Fractals, Plenum Press, New York 1988.

[33] Fleischmann, M., Tildesley, D. J., Ball, R. c., Fractals in the Natural Sciences, Prince­ton University Press, Princeton, 1989.

[34] Garfunkel, S., (Project Director), Steen, L. A. (Coordinating Editor) For All Practical Purposes, Second Edition, W. H. Freeman and Co., New York, 1988.

[35] GEO Wissen - Chaos und KreativiUit, Gruner + Jahr, Hamburg, 1990.

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2. General Articles

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[101] Douady, A., Julia sets and the Mandelbrot set, in: The Beauty of Fractals, H.-O. Peitgen, P. H. Richter, Springer-Verlag, 1986.

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[103] Gilbert, W. J., Fractal geometry derived from complex bases, Math. Intelligencer 4 (1982) 78-86.

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Index

Bold entries refer to this volume, and the entries in regular type indicate page numbers from Part One of Fractals for the Classroom.

M -set, 14 "-collar, 289

Abraham, Ralph, 22 absolute value, 360 adaptive cut algorithm, 367 addresses, 332, 335

addressing scheme, 332 dynamics, 233 for IFS Attractors, 337 for Sierpinski gasket, 29, 93, 332 for the Cantor set, 85, 336 language of, 335 of period-doubling branches, 232 space of, 336 three-digit, 332

aggregation, 399 of a zinc ion, 401

Alexandroff, Pawel Sergejewitsch, 122 alga, 13 algorithm, 39, 61 allometric growth, 159 ammonite, 157 anabaena catenula, 13, 15 angle

periodic, 389 pre-periodic, 389

approximations, 168 finite stage, 168 pixel, 199 quality of the, 298

Archimedes, 6, 171, 211 arctangent series, 179 area reduction, 282 Aristotle, 142 arithmetic precision, 192 arithmetic series, 213 arithmetic triangle, 96

Astronomica Nova, 47 attractive, 203, 399 attractor, 258, 365, 374, 397

coexisting, 338 covering of the, 365 derivative criterion, 399 for the dynamical system, 279 problem of approximating, 365 totally disconnected, 338

Attractor (BASIC program), 349 attractorlets, 352, 368 Augustus de Morgan, 7 Avnir, D., 399 axiom, 14, 18, 46

Banach, Stefan, 259, 284 band merging, 240 band splitting, 240, 249 Barnsley's fern, 276

transformations, 277 Barnsley, Michael F., 41, 255, 298, 320, 352 BASIC, 70, 206

DIM statement, 318 LINE,71 PSET, 71 SCREEN,72

BASIC program, 149 basin boundary, 339 basin of attraction, 281, 338, 341, 357, 373,

429 basin of infinity, 377 Beckmann,Petr, 178 Benard cells, 316 Benedicks, Michael, 294 Berger, Marc A., 255 Bernoulli, Daniel, 282 Bernoulli, Jacob, 212 bifurcation, 196, 214, 228, 254

490

calculation, 215 period-doubling, 221

bifurcation diagram, 216 binomial coefficients, 69, 81

divisibility properties, 81, 85, 96, 100 Birkhoff, George David, 295 bisection, 464 blinkers, 76 blueprint, 263 body, 159

height, 159 mass, 237

Bohr, Niels, 1 Boll, Dave, 431 Bondarenko, Boris A., 67 Borel measure, 354 Borwein, Jonathon M., 175, 180 Borwein, Peter B., 175, 180 boundary crisis, 264 Bourbaki, 10 Bouyer, Martine, 179 box self-similarity, 199 box-counting dimension, 229, 240, 241, 338 Brahe, Tycho, 45 brain function anomalies, 62 branch point, 449 branching, 57 branching order, 132 Branner, Bodil, 15 Brent, R. P., 180 broccoli, 153 Brooks, R., 10, 15 Brouwer, Luitzen Egbertus Jan, 122, 123 Brown, Robert, 319 Brownian motion, 319, 400, 406

fractional, 420 one-dimensional, 417

Brownian Skyline (BASIC program), 431 Buffon, L. Comte de, 347 bush,58 butterfly effect, 49

calculator, 58, 185 camera, 24 cancellation of significant digits, 369 Cantor brush, 133 Cantor maze, 268 Cantor set, 21, 39, 75, 79, 191, 234, 271, 286,

322,366,406 addresses for the, 85 construction, 80 dynamics on, 231 program, 251

Index

Cantor Set and Devil's Staircase (BASIC pro-gram),252

Cantor set of sheets, 311 Cantor, Georg, 75, 79, 122, 123, 193, 443 capacity dimension, 229 Caratheodory, Constantin, 444, 446 Carleson, Lennart, 294 carry, 89, 97, 98, 108, 111 Cartesian coordinates, 360 Casio fx-7000G, 56 catalytic oxidation, 111 Cauchy sequence, 286 cauliflower, 77, 120, 153, 161, 255 Cayley, Sir Arthur, 356, 357 tech, Eduard, 122 cell division, 13 cellular automata, 71, 72, 113

linear, 82 Cellular Automata (BASIC program), 114 central limit theorem, 408 Ceulen, Ludolph von, 176 Chaitin, Gregory J., 90 chaos, 54, 62, 64, 69, 89, 136, 148, 152, 195,

237,249 acoustic, 335 icons of, 274 routes to, 195, 253

chaos game, 41, 43, 320, 323, 329, 331, 352, 365,397,400

analysis of the, 330 density of points, 348 game point, 320 statistics of the, 353 with equal probabilities, 339

Chaos Game (BASIC program), 376 chaos machine, 151 chaotic transients, 262 characteristic equation, 197 Charkovsky sequence, 251 Charkovsky, Alexander N., 251 chemical reaction, 271 circle, encoding of, 270 classical fractals, 139 climate irregularities, 62 cloud,240,429 cluster, 385

correlation length, 391 dendritic, 405 incipient percolation, 391 maximal size, 389 of galaxies, 380 percolation, 391

coast of Britain, 225

Index

box-counting dimension, 243 complexity of the, 225 length of the, 225

code, 61 collage, 298

design of the, 302 fern, 298 leaf, 299 mapping, 265 optimization, 302 quality of the, 302

Collatz, Lothar, 39 color image, 354 comb, 445 compass dimension, 229, 235, 237 compass settings, 218, 227 complete metric space, 286, 289 complex argument, 361 complex conjugate, 364 complex division, 364 complex number, 359 complex plane, 139 complex square root, 367, 397, 413 complexity, 20, 45

degree of, 229 complexity of nature, 152 composition, 221 computer, 3 computer arithmetic, 145 computer graphics, 3, 13

the role of, 4 computer hardware, 181 computer languages, 70 concept of limits, 152 connected, 387 continued fraction expansions, 182 continuity, 180 contraction, 260

factor, 288 mapping principle, 284, 287, 309 ratio, 342

control parameter, 69 control unit, 21, 23 convection, 315, 317, 319 convergence, 286

test, 116 Conway, John Horton, 74 correlation, 423 correlation length, 391, 392 Coulomb's law, 385 Courant, Richard, 300 cover dimension, 124 cover, order of, 126

Cremer, Hubert, 139 crisis, 260, 325 critical exponent, 262 critical line, 247, 248 critical orbit, 407 critical point, 407 critical value, 246, 407, 421 critical value lines, 246 Crutchfield, James P., 24 curdling, 249 curves, 129, 235

non planar, 129 parametrized, 28 planar, 129 self-similar, 235 space-filling, 30

Cusanus, Nicolaus, 173 cycle, 40, 42, 67, 167

periodic, 141

Dase, Johann Martin Zacharias, 178 decay rate, 140 decimal, 330

numbers, 78 system, 330

decimal MRCM, 330 decoding, 281, 326

images, 326 method, 281

Democritus, 6 dendrite, 448, 462 dendritic structures, 399 dense, 168 derivative, 298 derivative criterion, 399, 431, 437, 440 Descartes, 5 deterministic, 40, 54, 60

feedback process, 54 fractals, 321 iterated function system, 325 rendering of the attractor, 365 shape, 321 strictly, 324

Devaney, Robert L., 148 devil's staircase, 245, 251

area of the, 246 boundary curve, 249 program, 251

dialects of fractal geometry, 256 die, 41, 320

biased, 346, 350 ordinary, 320 perfect, 339

491

492

differential equation, 295, 298, 314, 317, 340, 346

numerical methods, 301 system, 302

diffusion limited aggregation, 20, 402 mathematical model, 404

digits, 34 dimension, 121-123, 229

covering, 123 fractal, 104, 221 Hausdorff, 123 self-similarity, 104

displacement, 406 mean square, 406 proportional, 407

distance, 284, 289, 294 between two images, 289 between two sets, 289 Hausdorf~ 284, 289 in the plane, 294 of points, 294

distribution, 80 bell-shaped, 407 Gaussian, 407

divisibility properties, 70, 82, 85, 113 divisibility set, 86 DLA,402 DNA,9 Douady, Adrien, 14,351,352,384,445 dough, 166 dragon, 32, 33, 43, 267 dust, 85 dynamic law, 21 dynamic process, 286 dynamical system, 259, 272

attractor for the, 279 conservative, 270 dissipative, 270 linear, 204

dynamical systems theory, 259, 287 dynamics of an iterator, 71 Dynkin, Evgeni B., 67

Eadem Mutata Resurgo, 212 Edgar, Gerald, 434 eigenvalue, 239, 292 electrochemical deposition, 399

mathematical modeling, 400 electrostatic field, 384 encirclement, 378, 392, 403

algorithm, 382 energy, 269 ENIAC, 57, 356

equipotential, 385, 394, 467 ergodic, 170 ergodicity, 138 erosion model, 11 error amplification, 368 error development, 191 error propagation, 49, 64, 123, 126 escape set, 90, 141,373,375 escape time, 262 escaping sequence, 89 Escher, Mauritz c., 69 Ettingshausen, Andreas von, 92 Euclid,7 Euclidean dimension, 229 Eudoxus,6 Euler step, 301 Euler, Leonhard, 175, 187,282 expansion, 83, 182

Index

binary, 83, 116, 164, 195, 171, 172, 185 binary coded decimal, 185 continued fraction, 183 decimal, 83,89 dual, 195 p-adic,89 triadic, 84

experimental mathematics, 2

factorial, 69 factorization, 83, 87, 90 Faraday, Michael, 334 Fatou, Pierre, 356 feedback, 21, 66

clock,23 cycle, 24 experiment, 22 loop, 257, 287 machine, 21, 37

feedback system, 35, 40, 42, 89 geometric, 214

feedback systems class of, 287 quadratic, 139 sub-class of, 287

Feigenbaum constant, 224, 230, 293, 324 Feigenbaum diagram, 197,265,312 Feigenbaum point, 193, 200, 222, 231, 234,

241 Feigenbaum scenario, 292, 335 Feigenbaum (BASIC program), 266 Feigenbaum, Mitchell, 62, 197,200, 224 Feller, William, 13 Fermat, Pierre de, 96 Fermi, Enrico, 3, 12

Index

fern, 255, 306 non-self-similar, 309

Fibonacci, 35 -Association, 36 -Quarterly, 36 generator, 363 generator formula, 364 numbers, 36 sequence, 35, 171

Fibonacci, Leonardo, 78 fibrillation of the heart, 62 field line, 384, 387, 444,445

angle, 388 figure-eight, 410 final curve, 111 final state, 120, 197, 340 final state diagram, 197, 265 final state sensitivity, 338 fixed point, 203, 210, 221, 254, 353, 435

(un)stable, 203 attractive, 203, 429, 437 indifferent, 429, 440 of the IFS, 279 parabolic, 440 repelling, 399, 413, 437 stability, 204, 205 super attractive, 207, 429 unstable, 214

fixed point equation, 186 floating point arithmetic, 146 folded band, 307 forest fires, 387

simulation, 389 Fortune Wheel Reduction Copy Machine, 324,

345 Fourier, 181

analysis, 282 series, 282 Transformaion techniques, 181

Fourier, Jean Baptiste Joseph de, 282 fractal branching structures, 379 fractal dimension, 83, 221, 229

prescribed, 380 universal, 392

fractal geometry, 28, 69 fractal surface construction, 424 fractal surfaces, 238 fractals, 41, 89

classical, 75 construction of basic, 165 gallery of historical, 147

FRCM,324 friction, 269

Friedrichs, Kurt Otto, 300

Galilei, Galileo, 12, 156 Galle, Johann G., 45 Game of Life, 74

majority rule, 77 one-out-of-eight rule, 76 parity rule, 78

game point, 42, 320 Gauss, Carl Friedrich, 45, 81, 175 Gaussian distribution, 407 Gaussian random numbers, 408 generator, 104, 235 generic parabola, 158, 245 geometric feedback system, 214 geometric intuition, 9 geometric series, 164, 224, 246

construction process, 165 geometry, 8 Giant sequoias, 157 Gleick, James, 49 glider, 75, 76 Goethe, Johann Wolfgang von, 3 golden mean, 36, 171, 184,441,443,456

continued fraction expansion, 184 Golub, Jerry, 118

493

graphical iteration, 120, 193, 204, 360, 396 backward, 403

Graphical Iteration (BASIC program), 72 grass, 58 Great Britain, 218 Greco-Roman technology, 7 Greek Golden Age, 5 Gregory series, 177 Gregory, James, 175, 176 grids, 199

choice of, 205 GroBmann, Siegfried, 62, 242 growth, 221

allometric, 223 cubic, 224 proportional, 159

growth law, 159, 221 growth rate, 51 Guckenheimer, John, 272 Guilloud, Jean, 179 guns, 76

Hadamard, Jacques Salomon, 138 half-tone image, 353 Hamilton, William R., 411 Hanan, James, 20 Hao, Bai-Lin, 273

494

Hausdorff dimension, 229 Hausdorff distance, 167, 284, 289 Hausdorff, Felix, 75, 122, 123, 229, 230, 259,

284 head size, 159 Henon attractor, 275, 278, 284

dimension, 287 invariance, 278, 279

Henon transformation, 274 decomposition, 275 Feigenbaum diagram, 292 fixed points, 291 inverse, 286

Herman, M. R., 442 Herschel, Friedrich W., 45 Heun's method, 348 Hewitt, E., 10 hexagonal web, 102 Hilbert curve, 47, 51, 75 Hilbert, David, 31, 46, 75, 109, 122, 123 Hints for PC Users, 72, 150 Hirsch, Morris w., 117 histogram, 139,244

spike, 217, 247 Holmes, Philip, 272 Ho1zfuss, Joachim, 335 homeomorphism, 121 homoclinic point, 259 HP 285,58 Hubbard, John H., 14,351,352,384,445 human brain, 279

encoding schemes, 279 Hurewicz, Witold, 122 Hurst exponent, 420 Hurst, H. E., 420 Hutchinson equation, 98 Hutchinson operator, 40, 101, 191, 264, 291,

325 contractivity, 291

Hutchinson, J., 190, 255, 259, 284 hydrodynantics, 337 Henon, Michel, 274

ice cristals, 270 iconoclasts, 8 IFS, 98, 100,256,325,352

fractal dimension for the attractors, 293 hierarchical, 51, 105, 108,306, 312, 361

image, 257 attractor image, 328 code for the, 43, 354 color, 354 compression, 279

encoding, 282 example of a coding, 278 final, 258 half-tone, 353 initial, 263 leaf,298 perception, 279 scanner, 298 target, 298 the problem of decoding, 326

imitations of coastlines, 428 incommensurability, 142, 182 indifferent, 399 information dimension, 229 initial value, 298 initial value problem, 298 initiator, 104 injective, 333 input, 21, 33, 37, 41 input unit, 21, 23 interest rate, 51, 52 intermittency, 253, 259, 264, 325

scaling law, 260 intersection, 122 invariance, 89 invariance property, 193 invariant set, 402 inverse problem, 281, 298, 354 irreducible, 84 isometric growth, 159 iterated function system, 256

Index

iteration, 22, 50, 58, 65, 66, 159, 162, 166 graphical, 67

iterator, 21, 44, 65

Julia set, 139, 354, 357, 373, 374, 397, 402, 413,449

by chaos game, 397 invariance, 400 iterated function system, 402 quatemion,411 self-similarity, 400 structural dichotomy, 417

Julia, Gaston, 10, 75, 138, 355, 357 JuliaSets (BASIC program), 414

Kadanoff, Leo P., 398 Kahane, I. P., 13 Kepler's model of the solar system, 45 Kepler, Johannes, 45, 47 kidney, 109, 238 Klein, Felix, 5 Kleinian groups, 13

Index

kneading, 148, 166 cut-and-paste, 157 stretch-and-fold, 157, 159,274,307 substitution property, 156

Koch curve, 23, 38, 65, 75, 103, 128,227 construction, 105 Koch's original construction, 103 length of, 106 random, 381, 60 self-similarity dimension, 232

Koch Curve (BASIC program), 207 Koch island, 44, 165,227

area of the, 167 random, 381, 60,

Koch, Helge von, 75, 103, 161, 169 Kolmogorow, Andrej N., 117 Kummer criterion, 90, 92, 97, 98 Kummer, Ernst Eduard, 86, 88, 149,273

L-system, 11, 17, 19, 34, 38, 62 extensions, 61 parametric, 61 stochastic, 59

L-Systems (BASIC program), 63 labeling, 85 Lagrange, Joseph Louis, 282 Lange, Ehler, 356 language, 167, 256 Laplace equation, 405 Laplace, Pierre Simon, 347 Laplacian fractals, 405 laser instabilities, 62 Lauterborn, Werner, 335 law of gravity, 47 leaf, 144,299

spiraling, 144 Lebesgue, Henri L., 122, 124 Legendre's identity, 92 Legendre, Adrien Marie, 88, 91, 92 Leibniz, Gottfried Wilhelm, 21, 105, 175 lens system, 28, 31 level set, 425 Lewy, Hans, 300 Li, Tien-Yien, 271 Libchaber, Albert, 118 Liber Abaci, 35 Lichtenberger, Ralph, 412 limit, 152 limit object

self-similarity property, 199 limit objects, 164 limit structure, 169

boundary of the, 169

495

Lindemann, E, 178 Lindenmayer, Aristid, 9, 11, 12, 20, 61, 147 linear congruential method, 362 linear mapping, 261 Liouville monster, 444 Liouville number, 443 Liouville, Joseph, 443 Ljapunov exponent, 127, 129, 136 Ljapunov, Alexander Michailowitsch, 127 locally connected, 444, 446 logllog diagram, 220, 221

of the Koch curve, 227 for the coast of Britain, 220

logistic equation, 17, 50, 53, 56, 295, 356 look-up table, 73, 81, 113 Lorenz attractor, 271, 315, 316, 319, 348

dimension, 322 model,321 reconstruction, 333, 334

Lorenz experiment, 56 Lorenz map, 310, 320 Lorenz system, 315, 317, 350

crisis, 326 intermittency, 325 Lorenz map, 321, 322 periodic solution, 323 physical model, 315 streamlines, 319 temperature profile, 319

Lorenz, Edward N., 49, 54, 62, 123, 191,295, 315, 317

1963 paper, 272 Lozi attractor, 289 Lozi, Rene, 288 Lucas' criterion, 90 Lucas, Edouard, 3, 88, 90, 95

Machin, John, 176, 179 Magnus, Wilhelm, 13 Mandelbrojt, Szolem, 138 Mandelbrot set, 14,415,417,467

algorithm, 423, 467 atom, 438 buds, 427, 438 central piece, 429 dimension, 425 encirclement, 419, 422, 467 equipotentials, 425 field lines, 425 level set, 425 potential function, 425 secondary, 448, 463 self-similarity, 448

496

Mandelbrot Set Pixel Game (BASIC program), 471

Mandelbrot Test (BASIC program), 471 Mandelbrot (BASIC program), 470 Mandelbrot, Benoit Bo, 14, 75, 103, 138, 209,

415,417,420 Manhattan Project, 57 mapping ratio, 24 mappings, 261

affine linear, 262 linear, 261

Margolus, Norman, 77 Markov operator, 354 Mars, 47 mass, 249 mathematics, 151

applied,2 in school, 315 new objectivity, 151 new orientation, 11 without theorems, 6

Matsushita, Mitsugu, 399 May, Robert Mo, 17, 50, 62 Mane, Ricardo, 333 mean square displacement, 406 memory, 37, 41 memory effects, 27 Menger sponge, 124

construction of, 124 Menger, Karl, 122, 124, 131 Mephistopheles, 3 metabolic rate, 237 Metelski, Jo P., 15 meter, 78 meter stick, 331 method of least squares, 221 metric, 294

choice of the, 294 Euclidian, 285 Manhattan, 286 maximum, 285 suitable, 297 topology, 284

metric space, 126, 284, 285 compact, 126 complete, 286

middle-square generator, 363 Misiurewicz point, 460 Misiurewicz, Michal, 288, 460 Mitchison, Go Jo, 13 mixing, 67, 133, 169, 174 mod-p condition, 91, 96, 104 mod-p criterion, 111

modulo, 81, 82, 85 modulus, 360 monitor, 24 monitor-inside-a-monitor, 24 monster, 115

fractal, 255 of mathematics, 9, 75

monster spider, 133 Monte Carlo methods, 347 Montel, Paul, 12 moon, 48,91 mountains, 19

Index

MRCM, 11,21, 28,40,43,45,256,259,397 adaptive iteration, 367 blueprint of, 278, 298 decimal, 330 lens systems, 309 limit image, 309 mathematical description, 309 networked, 51, 306, 361

MRCM Iteration (BASIC program), 316 Mullin, Tom, 337 multifractals, 13, 244 Multiple Reduction Copy Machine, see MRCM multiplier, 459 Mumford, David, 1

natural flakes, 104 NCTM,30 Neumann, John von, 3, 12, 72, 356, 363 Newton's method, 34, 186,357 Newton, Sir Isaac, 21, 47, 105 noise, 166 nonlinear effects, 31 nonlinear mathematics, 3 Norton, Vo Alan, 412

object, 155 one-dimensional, 128 scaled-up, 155

one-step machines, 33 open set, 181 orbit, 119

backward, 286, 404 ergodic, 138 periodic, 119, 141, 146,215

organs, 109 fractal nature of, 237

oscillator, 314 output, 21, 33, 37, 41 output unit, 21, 23 overlap, 295 overlapping attractors, 371

Index

parabola, 66, 221, 310, 409 parameter, 22, 33 parametrization, 430 Pascal triangle, 67, 80, 96

a color coding, 68, 95, 148 coordinate systems, 86

Pascal, Blaise, 96, 102, 273 pattern formation, 83 Peano curve, 30, 41, 53, 75, 109, 245

construction, 110 S-shaped, 50 space-filling, 114

Peano, Giuseppe, 30, 75, 109, 122, 123, 249 pendulum over magnets, 339, 343, 357 percolation, 380, 383

cluster, 393 models, 380 threshold, 387, 394

period, 198 period-doubling, 198, 202, 221, 223, 230, 231,

249,291,292,323,335,448 periodic orbit, 215, 240, 435

derivative criterion, 437 periodic point, 136, 167, 171, 173, 180, 185,

252 periodic trajectory, 323 periodic window, 247, 324, 448 Peyriere, J., 13 phase transition, 391 phylotaxis, 305, 306 pi, 171,179,347

approximations of, 176, 179 Cusanus' method, 173 Ludolph's number, 176 Machin's formula, 178 Rutherford's calculation, 178

Pisa,35 Pisano, Leonardo, 35 pixel approximation, 199 pixel game, 353, 358 planets, 45, 48 plant, 17 Plath, Peter, 401 Plato, 6 Platonic solids, 45 Plutarch, 7 Poincare map, 313 Poincare, Henri, 4, 62, 122, 117, 195,259,313 point at infinity, 365, 374 point charge, 385 point of no return, 376 point set topology, 167 point sets, 167

polar coordinate, 361, 385 poly-line, 204 polynomial, 80, 82 Pontrjagin, Lew Semjenowitsch, 122 population dynamics, 35, 50, 62 Portugal, 209, 225 potential energy, 385 potential function, 385, 396, 425 power law, 221

behavior, 226 preimage, 203, 212, 217, 379, 397 Principia Mathematica, 151

497

prisoner set, 141,373,375,378,384,386,403, 409,418

(dis )connected, 417 connected, 409

probabilities, 327 badly defined, 358 choice of, 327 for the chaos game, 373 heuristic methods for choosing, 352

probability theory, 96 problem, 302

optimization, 302 traveling salesman, 302

processing unit, 23, 41, 50 production rules, 14 program, 70, 148, 206, 251, 315, 375, 430

chaos game for the fern, 375 graphical iteration, 70 iterating the MRCM, 315 random midpoint displacement, 430

proportio divina, 36 Prusinkiewicz, Przemyslaw, 12, 19,20,61 pseudo-random, 356 Pythagoras of Samos, 142 Pythagorean tree, 143

quadratic, 50, 61, 69 dynamic law, 61

quadratic equation, 212, 370 quadratic iterator, 44, 133, 148, 191, 196, 265,

289,308,403 equiValence to tent transformation, 177 generalization to two dimensions, 274 in low precision, 145

quadratic law, 61 quatemions, 411

rabbit, 400, 412 rabbit problem, 38 Ramanujan, Srinivasa, 175 random, 381

498

fractals, 381 Koch curve, 381 Koch snowflake, 381 midpoint displacement, 412 number generator, 56, 346 process, 321 successive additions, 426

random function, 419 rescaled, 419

randomness, 40, 319, 381 Rayleigh, Lord, 335 reaction rate, 111 real number, 359 reconstruction,272,328,331

acoustic chaos, 337 reduction, 262 reduction factor, 28, 32, 231, 262 reflection, 262 renormalization, 393

technique, 393 repeller, 203, 214, 397

derivative criterion, 399 repelling, 399 rescaling, 238 rest point, 203 return map, 308 Richter, Peter H., 15 Riemannian sphere, 365 romanesco, 153, 161 rose garden, 25 rotation, 262, 456 Rozenberg, Grzegorz, 9 rubber band, 151 Ruelle, David, 14, 118, 271 Rutherford, William, 178 Rossler attractor, 305, 306, 348

paper model, 308 reconstruction, 331, 332

Rossler system, 304, 314 Feigenbaum diagram, 312 return map, 309

Rossler, Otto. E., 304

saddle point, 255, 259 saddle-node bifurcation, 255 Sagan, Carl, 181 Salarnin, Eugene, 180 Saltzman, B., 318 Santillana, G. de, 7 saw-tooth transformation, 155, 171 scaling factor, 154, 231, 332 Schwarzian derivative, 228 self-affine, 161, 163, 248, 304

self-intersection, 109, 242 self-similar, 88, 304

at a point, 163 perfect, 89 statistically, 420 strictly, 163, 304

Index

self-similarity, 111, 153, 230, 231, 248, 400, 448

asymptotic, 459, 466 at a point, 453 at Feigenbaum point, 236 of the Feigenbaum diagram, 198 operational characterization of strict, 204 statistical, 161

self-similarity dimension, 229, 233 of the Koch curve, 232

sensitive dependence on initial conditions, 56, 122, 150, 166, 173, 177, 282, 320, 389

sensitivity, 122, 137, 144, 193 sensitivity constant, 166 series, 164

arithmetic, 213 geometric, 164

set, 81 countable, 81

set theory, 79 shadowing lemma, 186 Shanks, Daniel, 179 Shaw, Robert, 134 shift, 88

binary, 117 shift on two symbols, 166 shift operator, 164 shift transformation, 170, 184, 388 Shishikura, M., 16, 425 Siegel disk, 441, 445 Siegel, Carl Ludwig, 441 Sierpinski arrowhead, 25, 28, 40 Sierpinski carpet, 95, 135, 274 Sierpinski fern, 313 Sierpinski gasket, 24, 26, 28, 30, 42, 68, 75,

93,97,106,193,272 binary characterization, 98, 195 perfect, 29 program, 148 variation, 266

Sierpinski Gasket by Binary Addresses (BASIC program), 150

Skewed Sierpinski Gasket (BASIC pro­gram), 150

Sierpinski, Waclaw, 75, 91, 193 similar, 28

Index

similarity, 154 similarity transformation, 28, 154, 230 similitude, 28, 32 singularity, 389 slope, 228 Smale, Stephen, 117,259,272 Smith, Alvy Ray, 20 snowflake curve, 103 software, 48 space-filling, 31, 109 Spain, 209, 225 Sparrow, Colin, 324 spectral characterization, 426 spiders, 128

monster, 133 order of, 132

Spira Mirabilis, 212 spirals, 211, 462, 466

Archimedian, 211 golden, 217, 456 length of, 211, 217 logarithmic, 157, 211 polygonal, 214 smooth,216 square root, 142

square root, 32, 34, 185 approximation of, 185 complex, 413 of two, 185

square root spiral, 142 square, encoding of, 270 stability, 30, 117, 121,221,259 stability condition, 300 stable, 65, 66, 69 staircase, 246

boundary of the, 247 star ships, 76 statistical tests, 363 statistics of the chaos game, 353 Steen, Lynn Arthur, 67 stereographic projection, 366 Stewart, H. B., 67, 314 Stewart, Ian, 356 Stirling, James, 70 stock market, 4 Stone, Marshall, 10 strange attractor, 270, 275, 310

characterization, 287 coexistence, 293 reconstruction, 328

Strassnitzky, L. K. Schulz von, 178 stream function, 317 Stroemgren, Elis, 48

structures, 231 basic,306 branching, 379 complexity of, 20 dendritic, 399 in space, 240 in the plane, 240 mass of the, 249 natural, 77 random fractal dendritic, 380 self-similar, 231 space-filling, 109 tree-like, 380

Sucker, Britta, 356 Sullivan, Dennis, 442 Sumerian, 33

499

super attractive, 207-209, 221, 427, 435, 437 super object, 128, 129 super-cluster, 394 super-site, 393 survivors, 134 Swinney, Harry, 118

Takens, Floris, 118, 271, 333 Tan Lei, 16, 452 tangent bifurcation, 254, 255, 291 target set, 392, 420 temperature, 317 tent transformation, 38, 155, 172, 321

binary representation, 172 equivalence to quadratic iterator, 177

Thomae, Stefan, 62, 242 Thompson, J. M. T., 11, 314 three-body problem, 48 threshold radius, 378 time delay, 331 time profile, 209, 213, 218 time series, 119, 191, 196 Time Series (BASIC program), 192 Toffoli, Tommaso, 77 Tombaugh, Clyde w., 45 topological dimension, 229 topological invariance, 123 topology, 121 totally disconnected, 387 touching point, 439 touching points, 335 traditionalists, 16 trajectory

periodic, 311 transcendental, 443 transformation, 31, 32, 64, 123, 154, 154

affine, 31, 248, 323

500

affine linear, 260 Cantor's, 123 for the Barnsley fern, 277 invariance, 187 linear, 31 nonlinear, 141, 397 renormalization, 396 shift, 164, 171 similarity, 154, 187, 230, 260, 323

transitivity, 148 transversal, 468 trapezoidal method, 301 trapping region, 280 tree, 62, 78, 269

decimal number, 78 Pythagorean, 143

triadic numbers, 81, 87 triangle, encoding of, 270 triangular construction, 424 triangular lattice, 386 triangulation, 467 truncation, 145 turbulence, 62

acoustic, 334 turtle graphics, 34, 62

step length, 39 turtle state, 36

stacking of, 57 twig, 54, 269 twin christmas tree, 267 two-step method, 34, 37

Ulam, Stanislaw Marcin, 12,57,72, 356 uncertainty exponent, 339 uniform distribution, 346, 409 unit disk, 386 universal object, 130 universality, 128, 196,200,228,230,238

of the Menger sponge, 131 of the Sierpinski carpet, 128

unstable, 66, 69, 203 Urysohn, Pawel Samuilowitsch, 122 Uspenski, Wladimir A., 67 Utah, 226 Utopia, 8

vascular branching, 238 vectors, 37 Verhulst, Pierre F., 50, 52, 53 vessel systems, 109 vibrating plate, 334 video feedback, 22

setup, 22

Vieta's law, 438 Vieta, Fran~ois, 174 viscous fingering, 405 visualization, 4, 19, 34 Voss, Richard F., 20 Voyager II, 62

Wallis, J. R., 13 Wallis, John, 175 weather model, 49 weather prediction, 54, 69 Weierstrass, Karl, 105 wheel of fortune, 41 Wilcox, Michael, 13 Wilson, Ken G., 398 wire, 385 Witten, E., 6 Wolfram, Stephen, 67, 72 worst case scenario, 280 Wrench, John W., Jr., 179

Yorke, James A., 134,271

Zu Chong-Zhi, 173 Zuse, Konrad, 72

Index