The Cook Book
Recipe for the week of July 24 - 30
Asymptotic Polygonal Shapes for Threshold Growth
- We're back after an extended summer vacation. This week's soup shows
that limiting polygonal shapes for various Threshold Growth CA rules.
Recall that, under these dynamics, an empty sites becomes occupied iff
it sees at least theta previously occupied states within its neighborhood.
When the threshold theta is sufficiently small and the initial seed
sufficiently large, the occupied region spreads out at a linear rate with
an asymptotic shape. As Janko Gravner and I show in our new paper,
First Passage Times for Discrete Threshold Growth Dynamics, these
shapes are always polygons. To download a postscript version of our
manuscript, with 3 hi-res pictures, click here.
- Pictured are the 10 'supercritical' shapes for a range 2 box neighborhood.
This image was generated by running the dynamics from the small initial seed
shown in the middle of the figure. Clearly, smaller polygons correspond to
larger thresholds. The most interesting observation is that the shapes for
theta = 7 and theta = 8 appear to both equal a diamond of the same size. This
can be confirmed by an explicit computation based on the underlying mathematical
- Further recent advances in our understanding of Threshold Growth
dynamics will be presented over the next few weeks.