Cellular Automata in Hyperbolic Space

[CARuler]

In hyperbolic space, it is possible for a moving replicator to replicate endlessly without interfering with itself.

yes, in fact there are now 2 examples of this behaviour
example1:

Code: Select all

B2aho/S1ho2a X X0a X1a

example2:

Code: Select all

B2ampx3m/S0a3c X3a X0 X1a

===============================================================================
edit:
i don't think all of these work as symmetries

Code: Select all

permute (OT)
rotate37 (minimum symmetry)
rotate37reflectx (reflect on hexagons only) might not work (MNW)
rotate37reflectp (reflect on heptagons only) MNW
rotate37reflect (isotropic)
rotate67 (psuedo symmetry)
rotate67reflectx MNW
rotate67reflectp MNW
rotate67reflect

[Rhombicubocta]

tile5-4.png (27.15 KiB) Viewed 65 times

tile5-4.png (27.15 KiB) Viewed 65 times

Here is a proposal for the naming of directions in hyperbolic grids.

Let M be the number of sides of the polygons tiling the hyperbolic plane.
The idea is labeling cells via N-ary tree staring from the origin where N is chose so that at least 1/2M of the hyperbolic plane.
In the picture M is 5. This means 1/10 of the plane must be labeled to label all directions.
So we use a base-3 tree because it is the smallest that covers at least 1/10 of the hyperbolic plane.
After each branch the labeling of the next branches is oriented based on the rotated based on the turns taken on N-nary branching tree.
(This in addition to the holonomy.)
For example the sequence (...,0,2,2,2) is the same positionas (...) because that is same going around vertex of the grid. However the orientation of the next branch of the tree would be different. For example (0,2,2,2,2) is the same position as (0) and again with a different orientation of the further branches on the tree. A 0 and than twenty 2s would give you (0) in the same orientation.