ConwayLife.com

Has anybody tried making a computer program that plays game of life in a hyperbolic plane?

grisha5 wrote:Has anybody tried making a computer program that plays game of life in a hyperbolic plane?

Yes, here's some links:

http://dmishin.blogspot.com/2011/10/hyp ... ation.html
http://dmishin.blogspot.com/2014/10/cel ... ic-54.html

Thank you!

grisha5 wrote:Thank you!

*tips hat*

How about just using pentagon tiling for cellular automaton?

simsim314 wrote:How about just using pentagon tiling for cellular automaton? [image]

I recognize the first fourteen pentagon tilings, from Tilings and Patterns and mathpuzzle.com for example. But where does that multicolored fifteenth tiling come from? ... Ah, found it.

As far as running cellular automata on these things -- well, they're not vertex-transitive, so it's hard to get reliable emergent patterns going. Different cells often have different numbers of neighbors... It will certainly be possible to find or design CAs that permit gliders on these tilings, but they may have to be kind of custom-built for each tiling.

Basically it looks like CA patterns on these tilings will be significantly more complex and harder to understand on average, than CAs on regular grids. That said, if someone wants to spend half a century or so investigating the details of a particular rule on one of these tilings, with the same level of attention as Conway's Life has gotten, then I'm sure all kinds of interesting structures and patterns would emerge...!

-- Seems to me that CAs on pentagons has come up before somewhere, but I can't find the link. There's Calcyman's glider on Penrose tilings, of course, and papers like this one, and there are mentions of pentagons in A New Kind of Science and in an article by Carter Bays.

Ready can simulate CAs on arbitrary polygonal meshes, right? That's probably what I'm thinking of. Unfortunately I have yet to own a computer good enough to run Ready...!

dvgrn wrote:Seems to me that CAs on pentagons has come up before somewhere, but I can't find the link.

I don't know if this is what you're looking for, but I ran into this recently.

I can't help but wonder if there's a hyperbolic tiling that gives more interesting results. In particular, what effect does the number of edges of a cell have on the general behavior of outer-totalistic CAs on that geometry? What about the number of cells meeting at a vertex? I can't help but notice that this sort of investigation as to these influences might help us better understand why 3D life-like cellular automata seem to lack complexity, and how we might be able to make simple 3+ dimensional CAs that behave in interesting and complex ways.

Since I'm currently already working on a project using hyperbolic geometry, when I'm done with my current project and have some spare time I'll try and do some investigating with this.

I posted work on various tilings awhile back, including the then-14 types of pentagonal tilings, and the various uniform and dual-uniform tilings, in these two threads:

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viewtopic.php?f=11&t=1124&p=8223