The Cook Book

Recipe for the week of May 13 - 19

Larger than Life Metastable Waffles

We continue a series of snapshots, begun last week, and the week before, illustrating self-organization at various parameter locations in the Larger than Life (LtL) phase diagram. This week's soup is a large 'waffle' for the range 7 Box rule with birth interval [75,170] and survival interval [100,200], after 100 updates, with all 0's boundary conditions. The initial configuration was a central lattice circle of occupied sites with radius 10, together with four occupied quarter circles of radius 25 in the corners. Our 'level-sets' color scheme paints dead cells the very dark red of a hot electric fry pan, wheras the birth times of live cells are rendered in assorted shades of golden brown and maple syrup.

Starting from a 50-50 random mix of live and dead cells, the dynamics produce a familiar nucleation scenario - most of the space relaxes to all 0's, but isolated pockets of activity form seeds for spreading colonies of 1's. Early on we observe an assortment of waffles, most imperfect, but some highly patterned and nearly symmetric. Later the imperfections take over and cover the space with another variety of pseduo-random 'seething gurp' somewhat reminiscent of that seen in the pretzel regime.

Intrigued by the delicate transient droplet patterns, we segue into 'engineering' mode in search of the perfect waffle. Kellie Evans discovered that for the LtL rule under discussion, if we start from a radius 10 lattice circle, then the waffle appears to grow 'perfectly' for more than 70 updates before beginning to unravel. The thumbnail image above shows that evolution about half way along, while the central configuration of our soup shows the rectilinear waffle pattern being overrun by gurp 30 updates after the transition begins.

Going way out on a limb, we conjecture that in this region of phase space, very carefully chosen large-range LtL rules, started from very carefully chosen initial states, can grow perfect waffles of arbitrarily large size in comparison with the range. This suggests the possibility of an unstable invariant waffle-tiling of the plane for the limiting Euclidean LtL dynamics. One of these weeks we'll be serving up quite a breakfast if this idea pans out. Meanwhile, a new scenario for non-monotone growth is evident. Namely, the system spreads for some time according a metastable local pattern before an entirely different stable pattern emerges. An interesting, somewhat ill-posed question in this connection is whether the limiting shapes for the metastable and stable growth might differ.

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