References

[1] Abelson, H., and A. A. diSessa, 1981, Turtle Geometry: The Computer as a Medium for Exploring Mathematics (MIT Press, Cambridge).

[2] Aggarwal, S., 1973, ``Local and global Garden of Eden theorems,'' University of Michigan technical report No. 147.

[3] Aladyev, V., 1974, ``Survey of research in the theory of homogeneous structures and their applications,'' Math. Biosci. 22, 121.

[4] Aladyev, V., 1976, ``The behavioural properties of homogeneous structures,'' Math. Biosci. 29, 99.

[5] Alekseev, V. M. and M. V. Yakobson, 1981, ``Symbolic dynamics and hyperbolic dynamic systems,'' Phys. Rep. 75, 287.

[6] Amoroso, S., and G. Cooper, 1971, ``Tessellation structures of reproduction of arbitrary patterns,'' J. Comput Syst. Sci. 5, 455.

[7] Apostol, T. M., 1976, Introduction to Analytic Number Theory (Springer, Berlin).

[8] ApSimon, H. G., 1970a, ``Periodic forests whose largest clearings are of size 3,'' Philos. Trans. R. Soc. London, Ser. A 266, 113.

[9] ApSimon, H. G., 1970b, ``Periodic forests whose largest clearings are of size n 4,'' Proc. R. Soc. London, Ser. A 319, 399.

[10] Arbib, M. A., 1969, Theories of Abstract Automata (Prentice-Hall, Englewood Cliffs).

[11] Atrubin, A. J., 1965, ``A one-dimensional real-time iterative multiplier,'' IEEE Trans. Comput. EC-14, 394.

[12] Baer, R. M., and H. M. Martinez, 1974, ``Automata and biology,'' Ann. Rev. Biophys. 3, 255.

[13] Banks, E. R., 1971, ``Information processing and transmission in cellular automata,'' MIT Project MAC report No. TR-81.

[14] Barricelli, N. A., 1972, ``Numerical testing of evolution theories,'' J. Statist. Comput. Simul. 1, 97.

[15] Berlekamp, E. R., 1968, Algebraic Coding Theory (McGraw-Hill, New York).

[16] Berlekamp, E. R., J. H. Conway, and R. K. Guy, 1982, Winning Ways for Your Mathematical Plays (Academic, New York), Vol. 2, Chap. 25.

[17] Buckingham, D. J., 1978, ``Some facts of life,'' Byte 3, 54.

[18] Burks, A. W., 1970, Essays on Cellular Automata (University of Illinois, Urbana).

[19] Burks, A. W., 1973, ``Cellular Automata and Natural Systems,'' Proceedings of the 5th Congress of the Deutsche Gessellschaft für Kybernetik, Nuremberg.

[20] Butler, J. T., and S. C. Ntafos, 1977, ``The vector string descriptor as a tool in the analysis of cellular automata systems,'' Math. Biosci. 35, 55.

[21] Codd, E. F., 1968, Cellular Automata (Academic, New York).

[22] Cole, S. N., 1969, ``Real-time computation by n-dimensional iterative arrays of finite-state machines,'' IEEE Trans. Comput. C-18, 349.

[23] Conway, J. H., 1970, unpublished.

[24] Deutsch, E. S., 1972, ``Thinning algorithms on rectangular, hexagonal and triangular arrays,'' Commun. ACM 15, 827.

[25] Farmer, J. D., 1982a, ``Dimension, fractal measures, and chaotic dynamics,'' in Evolution of Order and Chaos in Physics, Chemistry and Biology, edited by H. Haken (Springer, Berlin).

[26] Farmer, J. D., 1982b, ``Information dimension and the probabilistic structure of chaos,'' Z. Naturforsch. 37a, 1304.

[27] Fine, N. J., 1947, ``Binomial coefficients modulo a prime,'' Am. Math. Mon. 54, 589.

[28] Fischer, P. C., 1965, ``Generation of primes by a one-dimensional real-time iterative array,'' J. ACM 12, 388.

[29] Flanigan, L. K., 1965, ``An experimental study of electrical conduction in the mammalian atrioventricular node,'' Ph.D. thesis (University of Michigan).

[30] Fredkin, E., 1981, unpublished, and PERQ computer demonstration (Three Rivers Computer Corp.).

[31] Gach, P., G. L. Kurdyumov, and L. A. Levin, 1978, ``One-dimensional uniform arrays that wash out finite islands,'' Probl. Peredachi. Info., 14, 92.

[32] Gardner, M., 1971, ``Mathematical games,'' Sci. Amer. 224, February, 112; March, 106; April, 114.

[33] Gardner, M., 1972, ``Mathematical games,'' Sci. Amer. 226, January, 104.

[34] Geffen, Y., A. Aharony, B. B. Mandelbrot, and S. Kirkpatrick, 1981, ``Solvable fractal family, and its possible relation to the backbone at percolation,'' Phys. Rev. Lett. 47, 1771.

[35] Gerola, H., and P. Seiden, 1978, ``Stochastic star formation and spiral structure of galaxies,'' Astrophys. J. 223, 129.

[36] Glaisher, J. W. L., 1899, ``On the residue of a binomial-theorem coefficient with respect to a Prime Modulus,'' Q. J. Math. 30, 150.

[37] Golomb, S. W., 1967, Shift Register Sequences (Holden-Day, San Francisco).

[38] Grassberger, P., 1982, ``A new mechanism for deterministic diffusion,'' Wuppertal preprint WU B 82-18.

[39] Greenberg, J. M., B. D. Hassard, and S. P. Hastings, 1978, ``Pattern formation and periodic structures in systems modelled by reaction-diffusion equations,'' Bull. Am. Math. Soc. 84, 1296.

[40] Griffeath, D., 1970, Additive and Cancellative Interacting Particle Systems (Springer, Berlin).

[41] Haken, H., 1975, ``Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems,'' Rev. Mod. Phys. 47, 67.

[42] Haken, H., 1978, Synergetics, 2nd ed. (Springer, Berlin).

[43] Haken, H., 1979, Pattern Formation by Dynamic Systems and Pattern Recognition (Springer, Berlin).

[44] Haken, H., 1981, Chaos and Order in Nature (Springer, Berlin).

[45] Hardouin-Duparc, J., 1974, ``Paradis terrestre dans l'automate cellulaire de conway,'' R.A.I.R.O. 8 R-3, 63.

[46] Hardy, G. H., and E. M. Wright, 1979, An Introduction to the Theory of Numbers, 5th ed. (Oxford University Press, Oxford).

[47] Harris, B., 1960, ``Probability distributions related to random mappings,'' Ann. Math. Stat. 31, 1045.

[48] Harvey, J. A., E. W. Kolb, and S. Wolfram, 1982, unpublished.

[49] Herman, G. T., 1969, ``Computing ability of a developmental model for filamentous organisms,'' J. Theor. Biol. 25, 421.

[50] Honsberger, R., 1976, ``Three surprises from combinatorics and number theory,'' in Mathematical Gems II, Dolciani Math. Expositions (Mathematical Association of America, Oberlin), p. 1.

[51] Hoogland, A., et al., 1982, ``A special-purpose processor for the Monte Carlo simulation of Ising spin systems,'' Delft preprint.

[52] Hopcroft, J. E., and J. D. Ullman, 1979, Introduction to Automata Theory, Languages and Computation (Addison-Wesley, Reading).

[53] Kauffman, S. A., 1969, ``Metabolic stability and epigenesis in randomly constructed genetic nets,'' J. Theor. Biol. 22, 437.

[54] Kimball, S. H., et al., 1958, ``Odd binomial coefficients,'' Am. Math. Mon. 65, 368.

[55] Kitagawa, T., 1974, ``Cell space approaches in biomathematics,'' Math. Biosci. 19, 27.

[56] Knuth, D. E., 1973, Fundamental Algorithms (Addison-Wesley, Reading).

[57] Knuth, D. E., 1981, Seminumerical Algorithms, 2nd ed. (Addison-Wesley, Reading).

[58] Kosaraju, S. R., 1974, ``On some open problems in the theory of cellular automata,'' IEEE Trans. Comput. C-23, 561.

[59] Landauer, R., 1979, ``The role of fluctuations in multistable systems and in the transition to multistability,'' Ann. N.Y. Acad. Sci. 316, 433.

[60] Langer, J. S., 1980, ``Instabilities and pattern formation in crystal growth,'' Rev. Mod. Phys. 52, 1.

[61] Levy, Y. E., 1982, ``Some remarks about computer studies of dynamical systems,'' Phys. Lett. A 88, 1.

[62] Lifshitz, E. M., and L. P. Pitaevskii, 1981, Physical Kinetics (Pergamon, New York).

[63] Lindenmayer, A., 1968, ``Mathematical models for cellular interactions in development,'' J. Theoret. Biol. 18, 280.

[64] MacWilliams, F. J., and N. J. A. Sloane, Theory of Error-Correcting Codes (North-Holland, Amsterdam).

[65] Mandelbrot, B., 1977, Fractals: Form, Chance and Dimension (Freeman, San Francisco).

[66] Mandelbrot, B., 1982, The Fractal Geometry of Nature (Freeman, San Francisco).

[67] Manna, Z., 1974, Mathematical Theory of Computation (McGraw-Hill, New York).

[68] Manning, F. B., 1977, ``An approach to highly integrated, computer-maintained cellular arrays,'' IEEE Trans. Comput. C-26, 536.

[69] Martin, O., A. Odlyzko, and S. Wolfram, 1983, ``Algebraic properties of cellular automata,'' Bell Laboratories report (January 1983).

[70] McIlroy, M. D., 1974, ``The numbers of 1's in binary integers: bounds and extremal properties,'' SIAM J. Comput. 3, 255.

[71] Miller, J. C. P., 1970, ``Periodic forests of stunted trees,'' Philos. Trans. R. Soc. London, Ser. A 266, 63.

[72] Miller, J. C. P., 1980, ``Periodic forests of stunted trees,'' Philos. Trans. R. Soc. London Ser. A 293, 48.

[73] Minsky, M. L., 1967, Computation: Finite and Infinite Machines (Prentice-Hall, Englewood Cliffs).

[74] Moore, E. F., 1962, ``Machine Models of Self-Reproduction,'' Proceedings of a Symposium on Applied Mathematics 14, 17, reprinted in Essays on Cellular Automata, edited by A. W. Burks (University of Illinois, Urbana, 1970), p. 187.

[75] Nicolis, G., and Prigogine, I., 1977, Self-Organization in Nonequilibrium Systems, (Wiley, New York).

[76] Nicolis, G., G. Dewel, and J. W. Turner, editors, 1981, Order and Fluctuations in Equilibrium and Nonequilibrium Statistical Mechanics, Proceedings of the XVIIth International Solvay Conference on Physics (Wiley, New York).

[77] Nishio, H., 1981, ``Real time sorting of binary numbers by 1-dimensional cellular automata,'' Kyoto University report.

[78] Ott, E., 1981, ``Strange attractors and chaotic motions of dynamical systems,'' Rev. Mod. Phys. 53, 655.

[79] Pearson, R., J. Richardson, and D. Toussaint, 1981, ``A special purpose machine for Monte-Carlo simulation,'' Santa Barbara preprint NSF-ITP-81-139.

[80] Peterson, W. W., and E. J. Weldon, Error-Correcting Codes, 2nd ed. (MIT Press, Cambridge).

[81] Preston, K., M. J. B. Duff, S. Levialdi, Ph. E. Norgren, and J.-I. Toriwaki, 1979, ``Basics of Cellular Logic with Some Applications in Medical Image Processing,'' Proc. IEEE 67, 826.

[82] Prigogine, I., 1980, From Being to Becoming (Freeman, San Francisco).

[83] Renyi, A., 1970, Probability Theory (North-Holland, Amsterdam).

[84] Roberts, J. B., 1957, ``On binomial coefficient residues,'' Can. J. Math. 9, 363.

[85] Rosen, R., 1981, ``Pattern generation in networks,'' Prog. Theor. Biol. 6, 161.

[86] Rosenfeld, A., 1979, Picture Languages (Academic, New York).

[87] Schewe, P. F., editor, 1981, ``Galaxies, the Game of Life, and percolation,'' in Physics News, Amer. Inst. Phys. Pub R-302, 61.

[88] Schulman, L. S., and P. E. Seiden, 1978, ``Statistical mechanics of a dynamical system based on Conway's game of life,'' J. Stat. Phys. 19, 293.

[89] Shannon, C. E., 1951, ``Prediction and entropy of printed English,'' Bell Syst. Tech. J., 30, 50.

[90] Sierpinski, W., 1916, ``Sur une courbe dont tout point est un point de ramification,'' Pr. Mat.-Fiz. 27, 77; Oeuvres Choisis, (Panstowe Wydawnictwo Naukowe, Warsaw) Vol. II, p. 99.

[91] Smith, A. R., 1971, ``Simple computation-universal cellular spaces,'' J. ACM 18, 339.

[92] Sternberg, S. R., 1980, ``Language and architecture for parallel image processing,'' in Pattern Recognition in Practice, edited by E. S. Gelesma and L. N. Kanal (North-Holland, Amsterdam), p. 35.

[93] Stevens, P. S., 1974, Patterns in Nature (Little, Brown, Boston).

[94] Stolarsky, K. B., 1977, ``Power and exponential sums of digital sums related to binomial coefficient parity,'' SIAM J. Appl. Math. 32, 717.

[95] Sutton, C., 1981, ``Forests and numbers and thinking backwards,'' New Sci. 90, 209.

[96] Thom, R., 1975, Structural Stability and Morphogenesis (Benjamin, New York).

[97] Thompson, D'A. W., 1961, On Growth and Form, abridged ed. edited by J. T. Bonner (Cambridge University, Cambridge, England).

[98] Toffoli, T., 1977a, ``Computation and construction universality of reversible cellular automata,'' J. Comput. Sys. Sci. 15, 213.

[99] Toffoli, T., 1977b, ``Cellular automata mechanics,'' Ph.D. thesis, Logic of Computers Group, University of Michigan.

[100] Toffoli, T., 1980, ``Reversible computing,'' MIT report MIT/LCS/TM-151.

[101] Toffoli, T., 1983, ``Squareland: a hardware cellular automaton simulator,'' MIT LCS preprint, in preparation.

[102] Turing, A. M., 1936, ``On computable numbers, with an application to the Entscheidungsproblem,'' Proc. London Math. Soc. Ser. 2 42, 230; 43, 544E, reprinted in The Undecidable, edited by M. Davis (1965; Hewlett, New York), p. 115.

[103] Turing, A. M., 1952, ``The chemical basis of morphogenesis,'' Philos. Trans. R. Soc. London, Ser. B 237, 37.

[104] Ulam, S., 1974, ``Some ideas and prospects in biomathematics,'' Ann. Rev. Bio., 255.

[105] von Neumann, J., 1963, ``The general and logical theory of automata,'' in J. von Neumann, Collected Works, edited by A. H. Taub, 5, 288.

[106] von Neumann, J., 1966, Theory of Self-Reproducing Automata, edited by A. W. Burks (University of Illinois, Urbana).

[107] Wainwright, R. T., 1971--73, Lifeline, 1--11.

[108] Wainwright, R. T., 1974, ``Life is Universal!,'' Proceedings of the Winter Simulation Conference, Washington, D.C., ACM, p. 448.

[109] Waksman, A., 1969, ``A model of replication,'' J. ACM 16, 178.

[110] Willson, S., 1982, ``Cellular automata can generate fractals,'' Iowa State University, Department of Mathematics, preprint.

[111] Witten, T. A., and L. M. Sander, 1981, ``Diffusion-limited aggregation, a kinetic critical phenomenon,'' Phys. Rev. Lett. 47, 1400.

[112] Wolfram, S., et al., 1981, ``SMP Handbook,'' Caltech.

[113] Wolfram, S., 1982a, ``Cellular automata as simple self-organizing systems,'' Caltech preprint CALT-68-938 (submitted to Nature).

[114] Wolfram, S., 1982b, ``Geometry of binomial coefficients,'' to be published in Am. Math. Monthly.

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