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Experimental Mathematics Volume 15, 2006 - Issue 4
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Original Article

Modeling Snow Crystal Growth I: Rigorous Results for Packard's Digital Snowflakes

Janko Gravner Mathematics Department, University of California, Davis, CA 95616
&
David Griffeath Department of Mathematics, University of Wisconsin, Madison, WI 53706
Pages 421-444 | Published online: 30 Jan 2011

  • J. A. Adam. "Flowers of Ice—Beauty, Symmetry, and Complexity," a review of The Snowflake: Winter's Secret Beauty. Notices Amer. Math. Soc. 52:4 (2005), 402–416.
  • J.-P. Allouche and J. Shallit. Automatic Sequences: Theory, Applications, Generalizations. Cambridge: Cambridge University Press, 2003.
  • R. J. Baxter. Exactly Solved Models in Statistical Mechanics. New York: Academic Press, 1982.
  • W. A. Bentley and W. J. Humphreys. Snow Crystals. New York: McGraw-Hill, 1931 (reprinted New York: Dover, 1962).
  • E. R. Berlekamp, J. H. Conway, R. K. Guy. Winning Ways for Your Mathematical Plays, second edition, vol. 4. Wellesley, MA: A K Peters 2004.
  • A. Cobham. "Uniform Tag Sequences." Math. Systems Theory 6 (1972), 164–192.
  • M. Cook. "Universality in Elementary Cellular Automata. Complex Systems 15 (2005), 1–40.
  • R. Descartes. Les Météores, 1637, edited by Adam et Tannery. Paris, Vrin, t. IV, 1965.
  • R. Dixon. Mathographics. New York: Dover, 1991.
  • W. Feller. An Introduction to Probability Theory and Its Applications, vol. 1. New York: Wiley, 1968.
  • M. Gardner. "Mathematical games." Scientific American 133 (December 1976), 124–128.
  • J. Gleick. Chaos: Making a New Science. New York: Penguin, 1987.
  • J. Gravner and D. Griffeath. "Cellular Automaton Growth on ℤ2: Theorems, Examples and Problems." Advances in Applied Mathematics 21 (1998), 241–304.
  • J. Gravner and D. Griffeath. "Modeling Snow Crystal Growth II." In preparation.
  • J. Gravner and D. Hickerson. "Asymptotic Density of an Automatic Sequence Is Uniform." In preparation.
  • D. Griffeath. "The Primordial Soup Kitchen." Available online (http://psoup.math.wisc.edu/extras/1ormore/1ormore.html), 2006. http://psoup.math.wisc.edu/extras/1ormore/1ormore.html
  • D. Griffeath and D. Hickerson. "A Two-Dimensional Cellular Automaton with Irrational Density." In New Constructions in Cellular Automata, edited by D. Griffeath and C. Moore. Oxford: Oxford Univ. Press, 2003.
  • D. Hickerson. Personal communication, 1999.
  • R. Hooke. Micrographia, 1665. New York: Dover, 2003.
  • J. Kepler. Strena Seu de Nive Sexangula, 1611; The Six-Cornered Snowflake, translated by Colin Hardie. Oxford: Clarendon Press, 1966.
  • B. W. King. "Snowflake Curves." Math. Teacher 57 (1964), 219–222.
  • H. von Koch. "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire." Arkiv för Mathematik, Astronomi och Fysik 1 (1904), 681–702.
  • S. Levy, Artificial Life: The Quest for a New Creation. Pantheon Books, 1992.
  • K. Libbrecht. "The Physics of Snow Crystals." Reports on Progress in Physics 68 (2005), 855-895.
  • K. Libbrecht. "Snowflakes and Snow Crystals." Available online (http://www.its.caltech.edu/\char126atomic/snowcrystals/), 2006. http://www.its.caltech.edu/char126atomic/snowcrystals/
  • K. Libbrecht and P. Rasmussen. The Snowflake: Winter's Secret Beauty. Osceola, WI: Voyageur Press, 2003.
  • O. Magnus. De Variis Figuris Nivium, 1555.
  • R. D. Mauldin, S. C. Williams. "Hausdorff Dimension in Graph Directed Constructions. Trans. Amer. Math. Soc. 309 (1988), 811–829.
  • P. Meakin. Fractals, Scaling and Growth Far from Equilibrium. Cambrige: Cambridge University Press, 1998.
  • U. Nakaya. Snow Crystals: Natural and Artificial. Cambridge: Harvard University Press, 1954.
  • N. H. Packard. "Lattice Models for Solidification and Aggregation." Preprint, Institute for Advanced Study, 1984 (reprinted in Theory and Application of Cellular Automata, edited by S.Wolfram, pp. 305–310. River Edge, NJ: World Scientific, 1986.
  • A. Pimpinelli and J. Villain. Physics of Crystal Growth. Cambridge: Cambridge University Press, 1999.
  • S. J. Willson. "Cellular Automata Can Generate Fractals." Discrete Appl. Math. 8 (1984), 91–99.
  • M. Wójtowicz. "Mirek's Cellebration: A 1D and 2D Cellular Automata explorer." Available online (http://www.mirwoj.opus.chelm.pl/ca/), 2005. http://www.mirwoj.opus.chelm.pl/ca/
  • S. Wolfram. "Computer Software in Science and Mathematics." Scientific American 251 (September 1984), 188–203.
  • S. Wolfram. "Computation Theory of Cellular Automata." Comm. Math. Phys. 96 (1984), 15–57.
  • S. Wolfram. A New Kind of Science. Wolfram Media, 2002.

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