Recipe for the week of February 13

The Cook Book

Recipe for the week of February 13 - 19

A Way-Cool CyberFlake

CA enthusiasts have been growing virtual snowflakes for many years. N. Packard was an early pioneer; more recently such patterns have been featured in a series of articles in Nature by Nowak and May. On the mathematical side growth models have played a key role in the development of spatial complex systems theory, dating back to seminal contributions of Hammersley in the 50's, Eden in the 60's, and Richardson in the '70's. An extensive collection of rigorous results, for both random and deterministic dynamics, has been developed over the past twenty years. The theory of nucleation, aggregation, shape, interfaces, and droplet interaction for basic growth models sheds new light on many areas of applied science.

This week's soup depicts the elaborate evolution of a very simple CA growth model that we learned from Janko Gravner. An empty cell becomes occupied if there are exactly 3 occupied cells in its Moore (range 1 Box) neighborhood, and once a cell is occupied it remains so forever. Thus this rule is Conway's Game of Life without death. Of course one expects the occupied region to grow over time, but how does it evolve from simple initial seeds? Our graphic shows the configuration after 1200 updates when one starts from a lattice disc of radius 50 (the yellow orb in the middle). Note the central vertical and horizontal 'ladders' that evolve by a kind of weaving pattern that seems to outrun the more complex growth elsewhere, and therefore should be expected to continue indefinitely. We have no idea, though, whether the crystal as a whole continues to grow forever, or whether it grinds to a halt except for these spokes.

The actual image for our soup is 1024 by 768 pixels, but has been reduced to the standard format for our Kitchen. Those fortunate enough to have large desktops should check out the original bigflake.gif. That graphic captures the beautifully intricate swirls within the main body of the cyberflake. They are quite reminiscent of the ice formations on my bedroom window, here in the dead of a Wisconsin winter.