[slowenough]

No I don't mean the automata rules, I mean the calculation describing how the cells move when you drag them around.

[roywiggins]

It depends on what model they're using. If they're just using the Poincare disk (as is displayed) then translations/rotations/reflections- isometries- of the hyperbolic plane are modeled by the Mobius transformations that map the unit circle (the edge) to itself. Geometrically speaking, Mobius transformations in the plane are 1) ordinary reflection, translation, and scaling, and 2) circle inversions, and form a group. Translations of the poincare disk are compositions of two circle inversions that end up mapping the center of the circle to some other point inside the disk, and map the disk to itself.

https://mathoverflow.net/questions/313671/classification-of-...

This doesn't really get into transformations, but here's one explication of a hyperbolic tiling I wrote:

http://roy.red/folding-tilings-.html

[pharrington]

Projection of the cells to the hyperbolic disk is in https://github.com/dmishin/hyperbolic-ca-simulator/blob/mast...

[seiferteric]

I can't say for sure, but I think it is using a conformal mapping.