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

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. 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. 2000 Mathematics Subject Classification. Primary 37B15. Secondary 68Q80, 11B05, 60K05.