From: Marc LeBrun <mlb@well.com> Date: 5/4/20, 7:04 PM

Anyway, it seems a finite square board with edges identified would be logically equivalent to an infinite quilt of square patches in a regular board, each of which are initialized with a suitably rotated/reflected copy of the initial configuration.

Yes, as long as with Langton's ant you have synchronized ants running in all the patches.

Indeed this avoids local snarling, but curling up a CA generally mostly seems to introduce additional "global" phenomena that depend sensitively only on arbitrary details, such as the exact diameter of the universe along each axis.

Yes, that's been my holy grail... well semi-holy. Right now it occurs to me you can twist the Klein bottle by any number of cells.  Same with a torus.  That is, shift the top left-right relative to the bottom. I guess that corresponds to brick-laying patterns. Wait, does this mean one can do those tilings with two sizes of squares? Marc, you may have widened one of the cracks in my brain.

...

Does this mean that a doubly-twisted "purse" board will behave the same as an untwisted "torus" board?

Duh, sorry, obviously only if the initial configuration was symmetrical...

And not Langton's Ant.  Or else two or four ants always meeting and doing some kind of do-se-do whenever they go to corners, meh. That reminds me of an experiment I've never done: put mirrors in a dish of water so that waves in the water pass straight through the mirrors. I wonder whether you could set up the tiles systematically out of sync, say, by some number of clock cycles for each tile-sized step to the North.  I guess that creates a problem setting up the initial state...  --Steve