OK, made a picture of this, if anyone other than me is curious — animated gif up to generation 41, at which point the blob has diameter 385 and 389906 grains of sand. It does indeed look quite similar to the version where all the grains of sand are dropped in the center. https://drive.google.com/file/d/1VB2pfvesiLtASbrCpruDp54bs1Tri-eQ/view --Michael On Sat, Jan 2, 2021 at 9:45 AM Michael Kleber <michael.kleber@gmail.com> wrote:
If you want a square-grid automaton whose growth pattern looks like a circle, then check out Jim Propp's "rotor-router" model, a de-randomized version in the same family as chip-firing and sandpile models. (My Mathematical Intelligencer piece on this in 2005 had some good pictures — arXiv link https://arxiv.org/abs/math/0501497, with pics like https://arxiv.org/PS_cache/math/ps/0501/0501497v1.3million-bb.png for rotor-router and https://arxiv.org/PS_cache/math/ps/0501/0501497v1.gsp3million-bb.png for the greedy sandpile — thought there are probably better visualizations generated in the 15 years since!)
The way these are often described doesn't have exponential growth, because they involve taking a snapshot after every chip/particle/etc gets added, so the area is forced to increase linearly with time. But the notion of time is pretty flexible, especially because these models are commutative — there's no ambiguity even if you add many chips "all at once".
A natural variation of sandpiles that *does* grow exponentially is: * At time 0, place a single grain of sand at the origin. * At time i, add a grain of sand to *every* cell that already has one or more grains in it. Then at each stage, apply the usual "greedy sandpile" redistribution mechanism: every cell that has >4 grains of sand topples, giving one grain to each of its neighbors; repeat until quiescence.
Is this a novel variation? Not clear to me how similar the result will be. It isn't going to look exactly the same, of course, since for the above pictures we're adding all the grains of sand at the origin, rather than distributed over the whole circular region.
--Michael
On Fri, Jan 1, 2021 at 9:39 PM Brad Klee <bradklee@gmail.com> wrote:
https://groups.google.com/g/alt.religion.buddhism.nichiren/c/-2v4wQvVd6g
The short of it is:
One point goes to Wolfram & Co. for Code 746, one of the more memorable results of New Kind Of Science:
https://demonstrations.wolfram.com/AlmostCircularCellularAutomaton/
If a stolon network grows at a constant radial rate, then Code 746 is okay, but from other Japanese research we know that congregations involve multiple individuals:
Little is known about how many clones exist in a population and how large an area does a single clone occupy.
That is an interesting question. Is anyone in the United States working on this at all ? ? ? ?
Happy New Year,
--Brad _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush.
-- Forewarned is worth an octopus in the bush.