The Cook Book

Recipe for the week of June 3 - 9

Level Sets of Bootstrap Percolation

Here in the Kitchen we began displaying successive CA updates by means of a large periodic palette in order to enhance the visualization of nonlinear growth models. Most recently, such level sets have been used to capture Larger than Life self-organization in four different regions of phase space, by painting empty cells with a fixed background color while coding occupied cells with the cyclically colored time of last birth. Experimenting, one finds that the most striking images result from systems which relax to a final fixed configuration over a long period of time, thereby producing a rich 'fossil record.' Even if the final state is 'all 1's,' a decidedly dreary configuration per se, the time trace can be visually beautiful and mathematically illuminating.

In this spirit we have begun revisiting some stock soups, to check out their level set representations. This week's offering shows a variant of one of our favorite rules, the bootstrap percolation CA that graces the Kitchen's title screen. As explained on the Particle Postcards page, that graphic superimposes many final states with varying parameters p for range 1 Diamond Threshold Growth with threshold 2. Our new soup shows the level sets of a single range 1 Box Threshold Growth with threshold 4 (and free boundary) started from an initial density p = .1 of occupied cells.

Of all the soups in our archive, this image most resembles last summer's Nucleation of Threshold Growth. Whereas that process was supercritical, leading to well-delineated concentric rings that approximate the rule's limiting polygonal shape, bootstrap percolation occurs in critical cases. The name refers to the fact that droplets can only spread indefinitely if they get a boost from external occupied cells. Consequently the level sets are much more intriguing, especially insofar as they capture interaction between nucleating droplets. An unfortunate consequence of increased complexity is the unusually large file size, by Kitchen standards, of the gif (239K). We hope those with modem connections will find it worth the download.

Next week we'll revisit another staple recipe for a level-set rendition that has given rise to some new and interesting mathematical results.

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