Experimental Mathematics

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

Janko Gravner and David Griffeath

Full-text: Open access

Abstract

Digital snowflakes are solidifying cellular automata on the triangular lattice with the property that a site having exactly one occupied neighbor always becomes occupied at the next time step. We demonstrate that each such rule fills the lattice with an asymptotic density that is independent of the initial finite set. There are some cases in which this density can be computed exactly, and others in which it can only be approximated. We also characterize when the final occupied set comes within a uniformly bounded distance of every lattice point. Other issues addressed include macroscopic dynamics and exact solvability.

Article information

Source
Experiment. Math. Volume 15, Issue 4 (2006), 421-444.

Dates
First available in Project Euclid: 5 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.em/1175789778

Mathematical Reviews number (MathSciNet)
MR2293594

Zentralblatt MATH identifier
1122.37057

Subjects
Primary: 37B15: Cellular automata [See also 68Q80]
Secondary: 68Q80: Cellular automata [See also 37B15] 11B05: Density, gaps, topology 60K05: Renewal theory

Keywords
Asymptotic density cellular automaton exact solvability growth model macroscopic dynamics thickness

Citation

Gravner, Janko; Griffeath, David. Modeling Snow Crystal Growth I: Rigorous Results for Packard's Digital Snowflakes. Experiment. Math. 15 (2006), no. 4, 421--444. http://projecteuclid.org/euclid.em/1175789778.


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