Bibliography

1. Books

[1] Abraham, R. H., and Shaw, Ch. D., Dynamics, The Geometry of Behavior, Part I-N, Aerial Press, Santa Cruz.

[2] Avnir, D., (ed.) The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, Polymers, Wiley, Chichester, 1989.

[3] Banchoff, T. F., Beyond the Third Dimension, Scientific American Library, 1990.

[4] Barnsley, M., Fractals Everywhere, Academic Press, 1988.

[5] Becker K.-H. and Dorfter, M., Computergraphische Experimente mit Pascal, Vieweg, Braunschweig, 1986.

[6] Beckmann, P., A History of Pi, Second Edition, The Golem Press, Boulder, 1971.

[7] Bondarenko, B., Generalized Pascal Triangles and Pyramids, Their Fractals, Graphs and Applications, Tashkent, Fan, 1990, in Russian.

[8] Borwein, J. M., and Borwein, P. B., Pi and the AGM- A Study in Analytic Number Theory, Wiley, New York, 1987.

[9] Briggs, J., and Peat, F. D., Turbulent Mirror, Harper & Row, New York, 1989.

[10] Cherbit, G. (ed.), Fractals, Non-integral Dimensions and Applications, John Wiley & Sons, Chichester, 1991.

[11] Campbell, D., and Rose, H. (eds.), Order in Chaos, North-Holland, Amsterdam, 1983.

[12] Collet, P., and Eckmann, J.-P., Iterated Maps on the Interval as Dynamical Systems, Birkhauser, Boston, 1980.

[13] Devaney, R. L., An Introduction to Chaotic Dynamical Systems, Second Edition, Addison-Wesley, Redwood City, 1989.

[14] Devaney, R. L., Chaos, Fractals, and Dynamics, Addison-Wesley, Menlo Park, 1990.

[15] Durham, T., Computing Horizons, Addison-Wesley, Wokingham, 1988.

[16] Edgar, G., Measures, Topology and Fractal Geometry, Springer-Verlag, New York, 1990.

[ 17] Engelking, R., Dimension Theory, North Holland, 1978.

[18] Escher, M. C., The World of M. C. Escher, H. N. Abrams, New York, 1971.

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

434 Bibliography

[20] Falconer, K.,Fractal Geometry, Mathematical Foundations and Applications, Wiley, New York, 1990.

[21] Family, F., and Landau, D. P. (eds.), Aggregation and Gelation, North-Holland, Ams­terdam, 1984.

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

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

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

[25] Gleick, J., Chaos, Making a New Science, Vlking, New York, 1987.

[26] Golub, G. H., and Loan, C. F. van, Matrix Computations, Second Edition, Johns Hopkins, Baltimore, 1989.

[27] Guyon, E., Stanley, H. E., (eds.), Fractal Forms, Elsevier/North-Holland and Palais de Ia ~couverte, 1991.

[28] Haken, H., Advanced Synergetics, Springer-Verlag, Heidelberg, 1983.

[29] Haldane, J. B. S., On Being the Right Size, 1928.

[30] Hall, R., Illumination and Color in Computer Generated lmagery,l, Springer-Verlag, New York, 1988.

[31] Hausdorff, F., Grundzuge der Mengenlehre, Verlag von Veit & Comp., 1914.

[32] Knuth, D. E., The Art of Computer Programming, Volume 2, Seminumerical Algorithms, Addison-Wesley, Reading, Massachusetts.

[33] Kuratowski, C., Topologie II, PWN, 1961. [34] Lauwerier, H., Fractals, Aramith Uitgevers, Amsterdam, 1987.

[35] Lehmer, D. H., Proc. 2nd Symposium on Large Scale Digital Calculating Machinery, Harvard University Press, Cambridge, 1951.

[36] Lindenmayer, A., and Rozenberg, G., (eds.), Automata, Languages, Development, North-Holland, 1975.

[37] Mandelbrot, B. B., Fractals: Form, Chance, and Dimension, W. H. Freeman and Co., San Francisco, 1977.

[38] Mandelbrot, B. B., The Fractal Geometry of Nature, W. H. Freeman and Co., New York, 1982.

[39] McGuire, M., An Eye for Fractals, Addison-Wesley, Redwood City, 1991.

[40] Menger, K., Dimensionstheorie, Leipzig, 1928.

[41] Mey, J. de, Bomen van Pythagoras, Aramith Uitgevers, Amsterdam, 1985.

[42] Moon, F.C., Chaotic Vibrations, John Wiley & Sons, New York, 1987.

[43] Parchomenko, A. S., Was ist eine Kurve, VEB Verlag, 1957.

[44] Peitgen, H.-0. and Richter, P. H., The Beauty of Fractals, Springer-Verlag, Berlin, 1986.

[45] Peitgen, H.-0., and JUrgens, H., Fraktale: Gezahmtes Chaos, Carl Friedrich von Siemens Stiftung, Miinchen, 1990.

[46] Peitgen, H.-0., and Saupe, D., (eds.), The Science of Fractal/mages, Springer-Verlag, New York, 1988.

Bibliography 435

[47] Prigogine, 1., and Stenger, 1., Order out of Chaos, Bantam Books, New York, 1984.

[ 48] Press, W. H., Flannery, B. P., Teuk:olsky, S. A. and Vetterling, W. T., Numerical Recipes, Cambridge University Press, Cambridge, 1986.

[49] Prusinkiewicz, P., and Lindenmayer, A., The Algorithmic Beauty of Plants, Springer­Verlag, New York, 1990.

[50] Rasband, S. N., Chaotic Dynamics of Nonlinear Systems, John Wiley & Sons, New York, 1990.

[51] Richardson, L. F., Weather Prediction by Nwnerical Process, Dover, New York, 1965.

[52] Ruelle, D., Chaotic Evolution and Strange Attractors, Cambridge University Press, Cambridge, 1989.

[53] Sagan, C., Contact, Pocket Books, Simon & Schuster, New York, 1985.

[54] SchrOder, M., Fractals, Chaos, Power Laws, Freeman, New York, 1991.

[55] Schuster, H. G., Deterministic Chaos, Physik-Verlag, Weinheim and VCR Publishers, New York, 1984.

[56] Stauffer, D., Introduction to Percolation Theory, Taylor & Francis, London, 1985.

[57] Stauffer, D., and Stanley, H. E., From Newton to Mandelbrot, Springer-Verlag, New York, 1989.

[58] Stewart, 1., Does God Play Dice, Penguin Books, 1989.

[59] Stewart, 1., Game, Set, & Math, Basil Blackwell, 1989.

[60] Thompson, D' Arcy, On Growth an Form, New Edition, Cambridge University Press, 1942.

[61] Vicsek, T., Fractal Growth Phenomena, World Scientific, London, 1989.

[62] Wade, N., The Art and Science of Visuall/lusions, Routledge & Kegan Paul, London, 1982.

[63] Wall, C. R., Selected Topics in Elementary Number Theory, University of South Car­olina Press, Columbia, 197 4.

[64] Weizenbaum, J., Computer Power and Human Reason, Penguin, 1984.

[65] Wolfram, S. , Farmer, J. D., and Toffoli, T., (eds.) Cellular Automata: Proceedings of an Interdisciplinary Workshop, in: Physica lOD, 1 and 2 (1984).

2. General Articles

[66] Barnsley, M. F. , Fractal Modelling of Real World Images, in: The Science of Fractal Images, H.-0. Peitgen and D. Saupe (eds.), Springer-Verlag, New York, 1988.

[67] Davis, C., and Knuth, D. E., Number Representations and Dragon Curves, Journal of Recreational Mathematics 3 (1970) 66-81 and 133-149.

[68] Douady, A., Julia sets and the Mandelbrot set, in: The Beauty of Fractals, H.-0. Peitgen, P. Richter, Springer-Verlag, Heidelberg, 1986.

[69] Dewdney, A. K., Computer Recreations: A computer microscope zooms in for a look at the most complex object in mathematics, Scientific American (August 1985) 16-25.

436 Bibliography

[70] Dewdney, A. K., Computer Recreations: Beauty and profundity: the Mandelbrot set and a flock of its cousins called Julia sets, Scientific American (November 1987) 140--144.

[71] Dyson, F., Characterizing Irregularity, Science 200 (1978) 677-()78.

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

[73] Hofstadter, D. R., Strange attractors : Mathematical patterns delicately poised between order and chaos, Scientific American 245 (May 1982) 16-29.

[74] Mandelbrot, B. B., How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science 155 (1967) 636-()38.

[75] Peitgen, H.-0. and Richter, P. H., Die unendliche Reise, Geo 6 (Juni 1984) 100-124.

[76] Peitgen, H.-0., Haeseler, F. v., and Saupe, D., Cayley's problem and Julia sets, Math­ematical Intelligencer 6.2 (1984) 11-20.

[77] Peitgen, H.-0., Jiirgens, H., and Saupe, D., The language of fractals, Scientific Amer­ican 263, 2 (August 1990) 40-47.

[78] Peitgen, H.-0. , Jiirgens, H., Saupe, D., and Zahlten, C., Fractals - An Animated Discussion, Video film, Freeman 1990. Also appeared in German as Fraktale in Filmen und Gespriichen, Spektrum der Wissenschaften Videothek, Heidelberg, 1990.

[79] Ruelle, D., Strange Attractors Math. Intelligencer 2 (1980) 126-137.

[80] Stewart, 1., Order within the chaos game? Dynamics Newsletter 3, no. 2, 3, May 1989, 4-9.

[81] Voss, R., Fractals in Nature, in: The Science of Fractal Images, H.-0. Peitgen and D. Saupe (eds.), Springer-Verlag, New York, 1988.

3. Research Articles

[82] Abraham, R., Simulation of cascades by video feedback, in: "Structural Stability, the Theory of Catastrophes, and Applications in the Sciences", P. Hilton (ed.), Lecture Notes in Mathematics vol. 525, 1976, 10--14, Springer-Verlag, Berlin.

[83] Bak, P., The devil's staircase, Phys. Today 39 (1986) 38-45.

[84] Bandt, Ch., Self-similar sets I. Topological Markov chains and mixed self-similar sets, Math. Nachr. 142 (1989) 107-123.

[85] Bandt, Ch., Self-similar sets lll. Construction with sofic systems, Monatsh. Math. 108 (1989) 89-102.

[86] Barnsley, M. F. and Demko, S., Iterated function systems and the global construction of fractals, The Proceedings of the Royal Society of London A399 (1985) 243-275

[87] Barnsley, M. F., Ervin, V., Hardin, D., and Lancaster, J., Solution of an inverse problem for fractals and other sets, Proceedings of the National Academy of Sciences 83 (1986) 1975-1977.

[88] Barnsley, M. F., Elton, J. H., and Hardin, D. P., Recurrent iterated function systems, Constructive Approximation 5 (1989) 3-31.

Bibliography 437

[89] Bedford, T., Dynamics and dimension for fractal recurrent sets, J. London Math. Soc. 33 (1986) 89-100.

[90] Berger, M., Encoding images through transition probablities, Math. Comp. Modelling 11 (1988) 575-577.

[91] Berger, M., Images generated by orbits of2D-Markoc chains, Chance 2 (1989) 18-28.

[92] Berry, M. V., Regular and irregular motion, in: Jorna S. (ed.), Topics in Nonlinear Dynamics, Amer. lnst of Phys. Conf. Proceed. 46 (1978) 16-120.

[93] Blanchard, P., Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. 11 (1984) 85-141.

[94] Borwein, J. M., Borwein, P. B., and Bailey, D. H., Ramanujan, modular equations, and approximations to 1r, or how to compute one billion digits of 1r, American Mathematical Monthly 96 (1989) 201-219.

[95] Brent, R. P., Fast multiple-precision evaluation of elementary functions, Journal Assoc. Comput Mach. 23 (1976) 242-251.

[96] Brolin, H., Invariant sets under iteration of rational functions, Arkiv f. Mat. 6 (1965) 103-144.

[97] Cantor, G., Uber unendliche, lineare Punktmannigfaltigkeiten V, Mathematische An­nalen 21 (1883) 545-591.

[98] Carpenter, L., Computer rendering of fractal curves and surfaces, Computer Graphics (1980) 109ff.

[99] Cayley, A., The Newton-Fourier Imaginary Problem, American Journal of Mathematics 2 (1897) p. 97.

[100] Cremer, H., Uber die Iteration rationaler Funktionen, Jahresberichte der Deutschen Mathematischen Vereinigung 33 (1925) 185-210.

[101] Crutchfield, J., Space-time dynamics in video feedback, Physica 10D (1984) 229-245. [102] Dekking, F. M., Recurrent Sets, Advances in Mathematics 44, 1 (1982) 78-104.

[103] Douady, A., Hubbard, J. H., Iteration des polynomes quadratiques complexes, CRAS Paris 294 (1982) 123-126.

[104] Dubuc, S., and Elqortobi, A., Approximations of fractal sets, Journal of Computational and Applied Mathematics 29 (1990) 79-89.

[105] Elton, J., An ergodic theorem for iterated maps, Journal of Ergodic Theory and Dy­namical Systems 7 (1987) 481-488.

[106] Fatou, P., Sur les equations fonctionelles, Bull. Soc. Math. Fr. 47 (1919) 161-271, 48 (1920) 33-94, 208-314.

[107] Farmer, J. D., Ott, E., and Yorke, J. A., The dimension of chaotic attractors, Physica 7D (1983) 153-180.

[108] Feigenbaum, M., Universality in complex discrete dynamical systems, in: Los Alamos Theoretical Division Annual Report (1977) 98-102.

[109] Feigenbaum, M., Universal behavior in nonlinear systems in: Campbell, D., and Rose, H.,(eds.) Order in Chaos North-Holland, Amsterdam, 1983., pp.16-39.

[110] Fournier, A., Fussell, D. and Carpenter, L., Computer rendering of stochastic models, Comm. of the ACM 25 (1982) 371-384.

438 Bibliography

[111] Goodman, G. S., A probabilist looks at the chaos game, in: in: FRACfAL 90 -Proceedings of the }St IFIP Conference on Fractals, Lisbon, June 6-8, 1990 (H.-0. Peitgen, J. M. Henriques, L. F. Penedo, eds.), Elsevier, Amsterdam, 1991.

[112] GroSman, S., and Thomae, S., Invariant distributions and stationary correlation func­tions of one-dimensional discrete processes Zeitschrift fiir Naturforschg. 32 (1977) 1353-1363.

[113] Haeseler, F. v., Peitgen, H.-0., and Skordev, G., Pascal's triangle, dynamical systems and attractors, to appear.

[114] Hart, J. C., and DeFanti, T., Efficient Anti-aliased Rendering of 3D-Linear Fractals, Computer Graphics 25, 4 (1991) 91-HJO.

[115] Hentschel, H. G. E. and Procaccia, 1., The infinite number of generalized dimensions of fractals and strange attractors, Physica 8D (1983) 435-444.

[116] Hepting, D., Prusinkiewicz, P., and Saupe, D., Rendering methods for iterated function systems, in: FRACfAL 90 - Proceedings of the }St IFIP Conference on Fractals, Lisbon, June 6-8, 1990 (H.-0. Peitgen, J. M. Henriques, L. F. Penedo, eds.), Elsevier, Amsterdam, 1991.

[ 117] Hilbert, D., Uber die stetige Abbildung einer Unie auf ein Fliichenstii.ck, Mathematische Annalen 38 (1891) 459-460.

[118] Holte, J., A recurrence relation approach to fractal dimension in Pascal's triangle, ICM-90.

[119] Hutchinson, J., Fractals and self-similarity, Indiana University Journal of Mathematics 30 (1981) 713-747.

[120] Jacquin, A. E., Image coding based on a fractal theory of iterated contractive image transformations, to appear in: IEEE Transactions on Signal Processing, March 1992.

[121] Julia, G., Sur /'iteration des fonctions rationnelles, Journal de Math. Pure et Appl. 8 (1918) 47-245.

[122] Julia, G., Memoire sur /'iteration des fonctions rationnelles Journal de Math. Pure et Appl. 8 (1918) 47-245.

[123] Kawaguchi, Y., A morphological study of the form of nature, Computer Graphics 16,3 (1982).

[124] Koch, H. von, Sur une courbe continue sans tangente, obtenue par une construction geometrique elementaire, Arkiv for Matematik 1 (1904) 681-704.

[125] Koch, H. von, Une methode geometrique elementaire pour l' etude de certaines ques­tions de la theorie des courbes planes, Acta Mathematica 30 (1906) 145-174.

[126] Kummer, E. E., Uber Ergiinzungssiitze zu den allgemeinen Reziprozitiitsgesetzen, Jour­nal fiir die reine und angewandte Mathematik 44 (1852) 93-146.

[127] Li, T. Y. and Yorke, J. A., Period 3 Implies Chaos, American Mathematical Monthly 82 (1975) 985-992.

[128] Lindenmayer, A., Mathematical models for cellular interaction in development, Parts I and II, Journal of Theoretical Biology 18 (1968) 280-315.

[129] Lorenz, E. N., Deterministic non-periodic flow, J. Atmos. Sci. 20 (1963) 130-141.

[130] Lovejoy, S. and Mandelbrot, B. B., Fractal properties of rain, and a fractal model, Tellus 37A (1985) 209-232.

Bibliography 439

[131] Mandelbrot, B. B., Fractal aspects of the iteration of z ...... ..\z(1 - z) for complex ..\ and z, Annals NY Acad. Sciences 357 (1980) 249--259.

[132] Mandelbrot, B. B., Comment on computer rendering of fractal stochastic models, Comm. of the ACM 25,8 (1982) 581-583.

[133] Mandelbrot, B. B., Self-a/fine fractals and fractal dimension, Physica Scripta 32 (1985) 257-260.

[134] Mandelbrot, B. B., On the dynamics of iterated maps V.· conjecture that the boundary of theM-set has fractal dimension equal to 2, in: Chaos, Fractals and Dynamics, Fischer and Smith (eds.), Marcel Dekker, 1985.

[135] Mandelbrot, B. B., An introduction to multifractal distribution functions, in: Fluctua­tions and Pattern Formation, H. E. Stanley and N. Ostrowsky (eds.), Kluwer Academic, Donlrecht, 1988.

[136] Mandelbrot, B. B. and Ness, J. W. van, Fractional Brownian motion, fractional noises and applications, SIAM Review 10,4 (1968) 422-437.

[137] Mauldin, R. D., and Williams, S. C., Hausdorff dimension in graph directed construc­tions, Trans. Amer. Math. Soc. 309 (1988) 811-829.

[138] Matsushita, M., Experimental Observation of Aggregations, in: The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, Polymers, D. Avnir (ed.), Wiley, Chichester 1989.

[139] May, R. M., Simple mathematical models with very complicated dynamics, Nature 261 (1976) 459--467.

[140] Menger, K., Allgemeine Raume und charakteristische Raume, Zweite Mitteilung: "fiber umfassenste n-dimensionale Mengen" Proc. Acad. Amsterdam 29 (1926) 1125-1128.

[141] Mitchison, G. J., and Wtlcox, M., Rule governing cell division in Anabaena, Nature 239 (1972) 110-111.

[142] Musgrave, K., Kolb, C., and Mace, R., The synthesis and the rendering of eroded fractal terrain, Computer Graphics 24 (1988).

[143] Peano, G., Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen 36 (1890) 157-160.

[144] Pietronero, L., Evertz, C., and Siebesma, A. P., Fractal and multifractal structures in kinetic critical phenomena, in: Stochastic Processes in Physics and Engineering, S. Albeverio, P. Blanchard, M. Hazewinkel, L. Streit (eds.), D. Reidel Publishing Company (1988) 253-278. (1988) 405-409.

[145] Prusinkiewicz, P., Graphical applications of L-systems, Proc. of Graphics Interface 1986- Vision Interface (1986) 247-253.

[146] Prusinkiewicz, P. and Hanan, J., Applications of L-systems to computer imagery, in: "Graph Grammars and their Application to Computer Science; Third International Workshop", H. Ehrig, M. Nagl, A. Rosenfeld and G. Rozenberg (eds.), (Springer­Verlag, New York, 1988).

[147] Prusinkiewicz, P., and Hammel, M., Automata, languages and iterated function systems Images in: Lecture Notes for the SIGGRAPH '91 course ''Fractal Modeling in 3D Computer Graphics and Imagery".

440 Bibliography

[148] Reuter, L. Hodges, Rendering and magnification of fractals using iterated function systems, Ph. D. thesis, School of Mathematics, Georgia Institute of Technology (1987).

[149] Richardson, R. L., The problem of contiguity: an appendix of statistics of deadly qua"els, General Systems Yearbook 6 (1961) 139-187.

[150] Ruelle, F., and Takens, F., On the nature of turbulence, CommMath. Phys. 20 (1971) 167-192, 23 (1971) 343-344.

[151] Salamin, E., Computation of 1r Using Arithmetic-Geometric Mean, Mathematics of Computation 30, 135 (1976) 565-570.

[152] Semetz, M., Gelleri, B., and Hofman, F., The Organism as a Bioreactor, Interpretation of the Reduction Law of Metabolism interms of Heterogeneous Catalysis and Fractal Structure, Journal Theoretical Biology 117 (1985) 209-230.

[153] Sierpinski, W., C. R. Acad. Paris 160 (1915) 302.

[154] Sierpinski, W., Sur une courbe cantorienne qui content une image biunivoquet et con­tinue detoute courbe donnee, C. R. Acad. Paris 162 (1916) 629-632.

[155] Shanks, D., and Wrench, J. W. Jr., Calculation of 1r to 100,000 Decimals, Mathematics of Computation 16, 77 (1962) 76-99.

[156] Sarkovski, A. N., Coexistence of cycles of continuous maps on the line, Ukr. Mat. J. 16 (1964) 61-71 (in Russian).

[157] Shishikura, M., The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, SUNY Stony Brook, Institute for Mathematical Sciences, Preprint #199In.

[158] Shonkwiller, R., An image algorithm/or computing the Hausdorff distance efficiently in linear time, Info. Proc. Lett 30 (1989) 87-89.

[159] Smith, A. R., Plants, fractals, andformallanguages, Computer Graphics 18,3 (1984) 1-10.

[160] Stanley, H. E., and Meakin, P., Multifractal phenomena in physics and chemistry, Nature 335.

[161] Stefan, P., A theorem of Sarkovski on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977) 237-248.

[162] Stevens, R. J., Lebar, A. F., and Preston, F. H., Manipulation and presentation of multidimensional image data using the Peano scan, IEEE Transactions on Pattern Analysis and Machine Intelligence 5 (1983) 520-526.

[163] Sullivan, D., Quasiconformal homeomorphisms and dynamics I, Ann. Math. 122 (1985) 401-418.

[164] Sved, M., and Pitman, J., Divisibility of binomial coefficients by prime powers, a geometrical approach, Ars Combinatoria 26A (1988) 197-222.

[165] Tan Lei, Similarity between the Mandelbrot set and Julia sets, Report Nr 211, Institut fiir Dynamische Systeme, Universitat Bremen, June 1989, and, Common. Math. Phys. 134 (1990) 587-617.

[166] Velho, L., and Miranda Gomes, J de, Digital halftoning with space-filling curves, Computer Graphics 25,4 (1991) 81-90.

[167] Voss, R. F., Random fractal forgeries, in : Fundamental Algorithms for Computer Graphics, R. A. Earnshaw (ed.), (Springer-Verlag, Berlin, 1985) 805-835.

Bibliography 441

[168] Voss, R. F., and Tomkiewicz, M., Computer Simulation of Dendritic Electrodeposition, Journal Electrochemical Society 132, 2 (1985) 371-375.

[169] Vrscay, E. R., Iterated function systems: Theory, applications and the inverse problem, in: Proceedings of the NATO Advanced Study Institute on Fractal Geometry, July 1989. Kluwer Academic Publishers, 1991.

[170] Williams, R. F., Compositions of contractions, Bol.Soc. Brasil. Mat. 2 (1971) 55-59.

[171] Wilson, S., Cellular automata can generate fractals, Discrete Appl. Math. 8 (1984) 91-99.

[172] Witten, I. H., and Neal, M., Using Peano curves for bilevel display of continuous tone images, IEEE Computer Graphics and Applications, May 1982, 47-52.

[173] Witten, T. A., and Sander, L. M., Phys. Rev. Lett. 47 (1981) 1400-1403 and Phys. Rev. B27 (1983) 5686-5697.

Index

.A-map, 14 J.t-map, 14 c-collar, 289 M-set, 14

Abraham, Ralph, 22 adaptive cut algorithm, 367 addresses, 332

addressing scheme, 332 for IFS Attractors, 337 for Cantor set, 85, 336 for Sierpinski gasket, 93, 332 language of, 335 space of, 336 three-digit, 332 of a point, 335 of a subtriangle, 334

aggregation, 399 of a zinc ion, 401

Alexandroff, Pawel Sergejewitsch, 122 algorithm, 39, 61 allometric growth, 159 ammonite, 157 approximations, 168

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

Archimedes, 6, 171, 211 arctangent series, 179 Aristotle, 142 arithmetic series, 213 arithmetic triangle, 96 Astronomica Nova, 47 attractor, 258, 365

covering of the, 365 for the dynamical system, 279 problem of approximating, 365

totally disconnected, 338 attractorlets, 352, 368 Augustus de Morgan, 7 Avnir, D., 399

Banach, Stefan, 259, 284 Bamsley's fern, 276

transformations, 277 Bamsley, Michael F., 41, 255, 298, 320,

352 BASIC, 206 BASIC program, 149

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

Beckmann, Petr, 178 Berger, Marc A., 255 Bernoulli, Daniel, 282 Bernoulli, Jacob, 212 binary extension, 116 blueprint, 263 body, 159

height, 159 weight, 237

Borel measure, 354 Borwein, Jonathon M., 175, 180 Borwein, Peter B., 175, 180 Bourbaki, 10 Bouyer, Martine, 179 box self-similarity, 199 box-counting dimension, 229, 240, 241 Brahe, Tycho, 45 brain function anomalies, 62 branching order, 132 Branner, Bodil, 15 Brent, R. P., 180

Index

broccoli, 153 Brooks, R., 10, 15 Brouwer, Luitzen Egbertus Jan, 122, 123 Brown, Robert, 319 Brownian motion, 400, 406

fractional, 420 one-dimensional, 417

Brownian Skyline(), 431 Buffon, L. Comte de, 347 butterfly effect, 49

calculator, 58 camera, 24 Cantor, Georg, 75, 79, 122, 123, 193 Cantor brush, 133 Cantor maze, 268 Cantor set, 75, 79, 191, 271, 366

addresses for the, 85 construction, 80 program, 251

Cantor Set and Devil's Staircase(), 252 capacity dimension, 229 Casio fx-70000, 56 Cauchy sequence, 286 cauliflower, 77, 120, 153, 161, 255 Cech, Eduard, 122 central limit theorem, 408 Ceulen, Ludolph von, 176 chaos, 54, 62, 64, 69, 89 chaos game, 41, 43, 320, 323, 329, 331,

352, 365 analysis of the, 330 density of points, 348 game point, 320 statistics of the, 353 with equal probabilities, 339

Chaos Game(), 376 characteristic equation, 197 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 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 compass dimension, 229, 235, 237 compass settings, 218, 227 complete metric space, 286, 289 complex plane, 139 complexity, 20, 45 complexity of nature, 152

degree of, 229 computer, 3 computer graphics, 13

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

factor, 288 mapping principle, 309 ratio, 342

control parameter, 69 control unit, 21, 23 convergence, 286

test, 116 correlation, 423 correlation length, 391, 392 cover, order of, 126 covering dimension, 124 Cremer, Hubert, 139 Crutchfield, James P., 24

443

444

curdling, 249 curves, 129, 235

non planar, 129 planar, 129 self-similar, 235

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

Dase, Johann Martin Zacharias, 178 decimal, 330 decimal MRCM, 330

numbers, 78 system, 330

decoding, 281, 326 images, 326 method, 281

Democritus, 6 dendritic structures, 399 Descartes, 5 deterministic, 40, 54, 60

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

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

diffusion limited aggregation, 20, 402 mathematical model, 404

digits, 34 dimension, 121, 122, 123, 229

covering, 123 fractal, 221 Hausdodl, 123

displacement, 406 mean square, 406 proportional, 407

distance,284,289,294

between two images, 289 between two sets, 289 Hau~di,284, 289 in the plane, 294 of points, 294

distribution, 80 bell-shaped, 407 Gaussian, 407

DLA, 402 Douady, Adrien, 14 dragon,267 dual representations, 195 dust, 85 dynamic law, 21 dynamic process, 286 dynamical system, 259

attractor for the, 279 dynamical systems theory, 259, 287 dynamics of an iterator, 71

Eadem Mutata Resurgo, 212 electrochemical deposition, 399

mathematical modeling, 400 ENIAC, 57, 356 error propagation, 49, 64 escape set, 90, 141 escaping sequence, 89 Euclid, 7 Euclidean dimension, 229 Eudoxus, 6 Euler, Leonhard, 175, 187, 282 expansion, 83, 182

binary, 83 continued fraction, 183 decimal, 83 dual, 195 triadic, 84

experimental mathematics, 2

feedback, 21, 66 clock, 23 cycle, 24 experiment, 22 loop, 257, 287 machine, 21, 37

feedback system, 35, 40, 42, 89

Index

Index

geometric, 214 class of, 287 quadratic, 139

Feigenbaum, Mitchel, 62 Feller, William, 13 Fermat, Pierre de, 96 Fermi, Enrico, 12 fern, 306

non-self-similar, 309 Fibonacci, Leonardo, 78 Fibonacci, 35

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

fibrillation of the heart, 62 final curve, 111 fixed, 258 fixed point equation, 186

of the IFS, 279 forest fires, 387

simulation, 389 Fortune Wheel Reduction Copy Machine,

'324, 345 Fourier, Jean Baptiste Joseph de, 282 Fourier, 181 ·

a.nalysis, 282 series, 282 Transfprmaion techniques, 181

fractal branching structures, 379 fractal dimension, 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

Galilei, Galileo, 12, 156 Galle, Johann G., 45

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

construction process, 165 geometry, 8 Giant sequoias, 157 Gleick, James, 49 Goethe, Johann, Wolfgang von, 3 golden mean, 36, 171, 184

445

continued fraction expansion, 184 graphical iteration, 360, 396 Graphical Iteration(), 72 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 growth, 221 growth law, 159, 221 growth rate, 51

allometric, 223 cubic, 224 proportional, 159

Guilloud, Jean, 179

Hadamard, Jacques Salomon, 138 half-tone image, 353 Hausdorff dimension, 229 Hausdorff distance, 167, 284, 289 Hausdorff, Felix, 75, 122, 123, 259, 284 head size, 159 Herschel, Friedrich W., 45 Hewitt, E., 10 hexagonal web, 102 Hilbert curve, 75 Hilbert, David, 109, 122, 123 Hints for PC Users, 72, 150 homeomotphism, 121

446

HP 285,58 Hubbard, John H., 14 human brain, 279

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

325 contractivity, 291

Hutchinson, J., 190, 255, 259, 284

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

fractal dimension for the attractors, 293

hierarchical, 306, 312, 361 image, 257

attractor image, 328 coding, 43, 354 color, 354 compression, 279 decoding, 326 encoding, 282 example of a coding, 278 final, 258 half-tone, 353 initial, 263 leaf, 298 perception, 279 target, 298

imitations of coastlines, 428 incommensurability, 142, 182 information dimension, 229 initiator, 104 input, 21, 33, 37, 41 input unit, 21, 23 interest rate, 51, 52 intersection, 122 invariance, 89 invariance property, 193 inverse problem, 281, 298, 354 isometric growth, 159 iterated function system, 256 iteration, 22, 50, 58, 65, 66

graphical, 67 iterator, 21, 44, 65

Index

Journal of Experimental Mathematics, 1 Julia set, 139 Julia, Gaston, 10, 75, 138

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 Koch curve, 103, 128, 227

construction, 105 Koch's Original Construction, 103 Length of, 106 self-similarity dimension, 232

Koch Curve(), 207 Koch island, 165, 227

area of the, 167 Koch, Helge von, 103, 161, 169 Krantz, S., 15 Kummer, Ernst Eduard, 149, 273

L-system, 19 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 law of gravity, 47 leaf, 144, 299

spiraling, 144 Lebesgue, Henri L., 122, 124 Leibniz, Gottfried Wilhelm, 105, 175 lens system, 28, 31 Liber Abaci, 35 limit, 152

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

Index

boundary of the, 169 Lindemann, F., 178 Lindenmayer, Aristid, 147 linear congruential method, 362 linear mapping, 261 log-log diagram, 220

of the Koch curve, 227 for the coast of Britain, 220 main idea, 221

logistic equation, 17, 50, 53, 56, 356 Lorenz experiment, 56 Lorenz, Edward N., 49, 54, 62 Lucas, Edouard, 3

Machin, John, 176,179 Magnus, Wtlhelm, 13 Mandelbrojt, Szolem, 138 Mandelbrot set, 14 Mandelbrot, Benoit B., 103, 209, 420 Manhattan Project, 57 mapping ratio, 24 mappings, 261

affine linear, 262 linear, 261

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 M., 17, 50, 62 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, J. P., 15 meter, 78 meter stick, 331

method of least squares, 221 metric, 294

compact, 126 complete, 286 choice of the, 294 Euclidian, 285 Manhattan, 286 maximum, 285 space, 126, 285 suitable, 297 topology, 284

middle-square generator, 363 mixing, 67 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 MRCM, 28, 256, 259

adaptive iteration, 367 blueprint of, 278, 298 decimal, 330 lens systems, 309 limit image, 309 mathematical description, 309 networked, 306, 361 with three lens system, 257

MRCM Iteration(), 316 multifractals, 13, 244

447

multiple reduction copy machine, 28, 40, 43

Mumford, David, 1

natural flakes, 104 NCfM, 30 Neumann, John von, 12, 356, 363 Newton's method, 34, 186 Newton, Sir Isaac, 105 nonlinear effects, 31 nonlinear mathematics, 3

448

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

one-step machines, 33 organism, 109

fractal nature of, 237 output, 21, 33, 37, 41 output unit, 21, 23 overlap, 295 overlapping attractors, 371

parabola, 66 parameter, 22, 33 Pascal triangle, 96

a color coding, 148 Pascal, Blaise, 102, 273 Peano curve, 109, 245

construction, 110 space-filling, 114

Peano, Giuseppe, 109, 122, 123, 249 Peitgen, Heinz-Otto, 15 percolation, 380, 383

cluster, 393 models, 380 threshold, 387, 394

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 planets, 45, 48 plant, 17 Plath, Peter, 401 Plato, 6 Platonic solids, 45 Plutarch, 7 Poincare, Henri, 4, 62, 122 point set topology, 167 point sets, 167

Index

Pontrjagin, Lew Semjenowitsch, 122 population dynamics, 62, 35, 50 Portugal, 209, 225 power law, 221

behavior, 226 Principia Mathematica, 151 prisoner set, 141 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, 37, 41, 50 prognun,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, 19 pseudo-random, 356 Pythagoras of Samos, 142 Pythagorean tree, 143

quadratic, 50, 61, 69 dynamic law, 61

quadratic iterator, 44 quadratic law, 61

Rabbit problem, 38 Ramanujan, Srinivasa, 175 random function, 419

rescaled, 419 random, 381

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

Index

randomness, 40, 319, 381 reduction, 262 reduction factor, 28, 32, 231, 262 reflection, 262 renormalization, 393

technique, 393 Richter, Peter H., 15 romanesco, 153, 161 rotation, 262 Ruelle, David, 14 Rutherford, William, 178

Sagan, Carl, 181 Salamin, Eugene, 180 Santillana, G. de, 7 scaling factor, 154, 231, 332 self-affine, 161, 304 self-intersection, 109, 242 self-similar, 304

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

self-similarity, 111, 153, 230 of the Koch curve, 232 operational characterization of strict,

204 statistical, 161

self-similarity dimension, 229, 233 sensitive dependence on initial conditions,

56 series, 164

arithmetic, 213 geometric, 164

set, 81 countable, 81

set theory, 79 Shanks, Daniel, 179 shift, 88

binary, 117 Shishikura, M., 16 Sierpinski carpet, 95, 135, 274 Sierpinski fern, 313 Sierpinski gasket, 30, 42, 75, 193, 272

binary characterization, 195 perfect, 29

program, 148 variation,266

449

Sierpinski Gasket by Binary Addresses(), 150

Skewed Sierpinski Gasket(), 150 Sierpinski triangle, 93 Sierpinski, Waclaw, 75, 91, 193 similar, 28 similarity, 154 similarity transformation, 28, 154, 230 similitude, 28, 32 singularity, 389 slope, 228 snowflake curve, 103 software, 48 space-filling, 109

curves, 109 structures, 109

Spain, 209, 225 spectral characterization, 426 spiders, 128

monster, 133 order of, 132

Spira Mirabilis, 212 spirals, 211

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

square, encoding o~ 270 square root, 32, 34, 185

approximation of, 185 of two, 185

square root spiral, 142 stability, 30 stable, 65, 66, 69 staircase, 246

boundary of the, 247 statistical tests, 363 statistics of the chaos game, 353 Stewart, Ian, 356 stock market, 4

450

Stone, Marshall, 10 strange attractor, 275 Strassnitzky, L. K. Schulz von, 178 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 Sumerian, 33 super object, 128, 129 super-cluster, 394 super-site, 393

Tan Lei, 16 tent transformation, 38 Thomae, Stefan, 62 three-body problem, 48 Tombaugh, Clyde W., 45 topological dimension, 229 topological invariance, 123 Topology, 121 touching points, 335 traditionalists, 16 transformation, 31, 32, 64, 123, 154

affine, 31, 248, 323 affine linear, 260 Cantor's, 123 for the Barnsley fern, 277 invariance, 187 linear, 31 nonlinear, 141 renormalization, 396 similarity, 154, 187, 230, 260, 323

tree, 78, 269 decimal number, 78 Pythagorean, 143

triadic numbers, 81, 87 triangle, encoding of, 270 triangular construction, 424 triangular lattice, 386 turbulence, 62 twig, 269 twin christmas tree, 267 two-step method, 34, 37

Index

Ulam, Stanislaw Marcin, 12, 57, 356 uniform distribution, 409 universal object, 130 universality, 128

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

unstable, 66, 69 Urysohn, Pawel Samuilowitsch, 122 Utah, 226 Utopia, 8

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

setup, 22 Vieta, Fran~ois, 17 4 viscous fingering, 405 visualization, 4 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 Wilson, Ken G., 398 Witters, E., 6 worst case scenario, 280 Wrench, John W., Jr., 179

Zu Chong-Zhi, 173