Packard Snowflakes on the von Neumann Neighborhood
Charles D.Brummitt, Hannah Delventhal and Michael Retzlaff
In 1984, Packard [1] introduced simple planar cellular automata to emulate the growth of snow crystals. These Packard Snowflakes have since been popularized by S. Wolfram and others, most recently in [2]. The present paper provides a rigorous analysis of the simplest examples: those with nearest neighbor interaction on the two-dimensional integers. In each case we determine the asymptotic density with which the spreading crystal fills the plane. For the basic Exactly 1rule started from a singleton, we establish alternate representations of the final state as a uniform tag system and as a substitution system.