Packard Snowflakes on the von Neumann Neighborhood
Charles D.Brummitt, Hannah Delventhal and Michael Retzlaff
In 1984, Packard  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 . 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.