**NB:** A

PDF
version of this announcement (suitable for posting) is also available.

## Rigorous Results for Digital Snowflake Growth

### David Griffeath

University of Wisconsin -- Madison

###
Thursday, January 13, 2005

L02 Carson Hall, 4 pm

Tea 3:30 pm, Math Lounge

In Honor of Laurie Snell's 80th Birthday

**Abstract: ** Cellular automata (CA) models for the evolution
of snowflakes were introduced by Norman Packard in 1984, and have been
popularized by Stephen Wolfram ever since. These are very simple
two-dimensional algorithms with qualitative features of real
snowflakes, but lacking many of the more subtle relevant effects of
diffusive aggregation, temperature fluctuation, etc. Nevertheless,
they are beautiful mathematical structures, exterior boundaries of
which converge to the fractal known as the (von) Koch Snowflake that
was introduced in 1904. For the past 20 years Wolfram has presented
these automata as examples of complex dynamics generated by simple
local interactions, with the suggestion that they are beyond the scope
of rigorous mathematical methods.

In fact, the 16 Packard
Snowflakes on the hexagonal lattice can be analyzed by traditional
mathematical methods, albeit with invaluable help from computer
visualization. For instance, we prove that each process has an
asymptotic density as it fills the plane, independent of the finite
initial seed, and we are able to compute the density exactly in 8 of
the 16 cases.
E.g., for the rule
featured in the September 1984 *Scientific American*, Steven
Levy's 1992 book *Artificial Life*, and Wolfram's postcards and
various other publications, the density is 5/6. In the remaining 8
cases we give a convincing explanation why the density cannot be
evaluated exactly, but show how (in principle) to derive bounds of
arbitrary accuracy, and we determine the density to within .0001 for
the most basic snowflake.

This talk will be accessible to graduate and advanced undergraduate students.