- 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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