- 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.