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Euclidean tilings such as {5/2,10} have defined vertex figures, but do not tile periodically and end up creating an infinite density mess everywhere. Nonetheless, if we ignore this complication, can we define a separate, abstract notion of "distance" between two given facets, unrelated to Euclidean physical proximity, in the sense that facet B can only be reached from facet A by crossing two dihedral edges from two attached facets?

Indeed, can a cellular automaton be run on grids such as {5/2,10}?

muzik wrote: March 16th, 2025, 5:44 pm Euclidean tilings such as {5/2,10} have defined vertex figures, but do not tile periodically and end up creating an infinite density mess everywhere. Nonetheless, if we ignore this complication, can we define a separate, abstract notion of "distance" between two given facets, unrelated to Euclidean physical proximity, in the sense that facet B can only be reached from facet A by crossing two dihedral edges from two attached facets?

Indeed, can a cellular automaton be run on grids such as {5/2,10}?

A cellular automaton can be run on anything that can be "honestly" represented as a graph (in the sense of graph theory). The "cells" become vertices of the graph; when a "cell" A is a neighbour of a "cell" B that is represented by a directed edge from A to B ("the current state of A can affect the state of B in the next generation").

So your question probably resolves to whether or not you can represent the situation as a graph, without losing something important that you would want to preserve?

While not strictly necessary, a large part of what makes many CA interesting is the ability to look and see how the reactions proceed. If it ends up looking like "infinite density mess everywhere" when displayed, people might have difficulty actually exploring it. (Sounds like cellular automata on regular tilings of a hyperbolic plane would be very intuitive and easy to explore in comparison; a conformal projection (such as the conformal disk model or the upper half-plane model) helps to see symmetries and what is going on.)

confocaloid wrote: March 16th, 2025, 6:18 pma large part of what makes many CA interesting is the ability to look and see how the reactions proceed. If it ends up looking like "infinite density mess everywhere" when displayed, people might have difficulty actually exploring it.

Indeed this is something I want to account for - if only cells with a certain "distance" to a given cell of interest would be rendered, then we'd be able to see intelligible subsets of this grid.

Perhaps there are other questions worth asking:
- If infinite cells can fit into a finite bounded region, is exponential growth possible?
- Could we also run these on spherical tilings such as {7/3,3}, as these too would have infinite facets?

muzik wrote: March 16th, 2025, 7:04 pm [...] - Could we also run these on spherical tilings such as {7/3,3}, as these too would have infinite facets?

Before trying to think too hard about {7/3, 3}, I personally would be interested in seeing what is possible on the 20 faces of the icosahedron. It looks like there may be some interesting questions waiting to be answered. See the post viewtopic.php?p=207140#p207140 in the forum thread viewtopic.php?f=11&t=594 "Penrose tilings, hyperbolic tesselations"

(In the event that two-state cellular automata on the icosahedron happen to be "boring", one can always increase the number of cellstates)

muzik wrote: March 16th, 2025, 7:04 pm would have infinite facets?

they don't

muzik wrote: November 3rd, 2017, 4:45 pm Nothing came of the OEIS, but punching "11 101 15641" in Google including the quotation marks led to a German paper which seems to give multiple more numbers which look like they might fit this sequence.

11, 101, 15641, 3512981, 6655541, 20769311, 26919791, 41487071, 71541641, 160471601, 189425981

This now finally appears to have actually been submitted: https://oeis.org/A382970

Edit: Nevermind. It’s not.

Former post:
I read through the posts, and I think this thread is probably going to get the Conwaylife Skyscraper treatment and get locked under the guise of being a Random Posts thread in disguise.

Also, the locking is probably inevitable now that I pointed this out.

Looking at pages 14 and 15 (using the default settings for page numbering), it looks fine to me.

hotdogPi wrote: September 26th, 2025, 8:34 am Looking at pages 14 and 15 (using the default settings for page numbering), it looks fine to me.

Thank you. I was worrying that I just opened a Pandora’s box by saying that, but turns out not. Whew. I said that when I was half-tired this morning without thinking straight. I rethought it out and decided that the thread is more or less innocent.