The Cook Book

Recipe for the week of August 7 - 13

The Geometry of Critical Minimal Droplets (revisited)

Recall that the week of March 20 featured a close-to-critical Threshold Growth CA started from a square. We mentioned that such a construction gave upper bounds on the thresholds for which droplets of size s can nucleate in a threshold s rule. At that time we also promised to return with a corresponding lower bound graphic.

This week's soup shows an architecture for growth of droplets of size s in a threshold s rule that we believe to be quite close to optimal. The model has a range 100 Box neighborhood and threshold 16,399. The maroon colored region with 4 points and an octagonal hole was the initial state. That configuration, designed by Kellie Evans on the basis of similar architectures with smaller range, has 16,399 cells. After the fashion of our March soup, subsequent updates are coded as successive colors from a broad palette. The growth finally reaches the tips of the initial state at time 611, and is shown here at time 623. As an indication of the delicacy of the construction, we know of no configuration of 16,400 cells that grows for threshold 16,400.

The interactive visualization features of our WinCA software (available in the Kitchen Sink) were crucial to the design of this droplet. The third beta version of our program, to be released next month, will include a hole.xpt demo experiment that grows the time 623 state. The mathematics behind our lower bound will appear in a forthcoming research paper.

Take me higher...
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