consider a variation of Conway's Game of Life in which the neighbors of a cell are the 8 cells that are a knight's move away: knight Life. this repairs a geometric non-uniformity of the Moore neighborhood which has some neighbors at distance sqrt(2) but others at distance 1.
similarly, consider 3-4-5 Life: 12 neighbors 5 units away: 4 neighbors orthogonally and 8 more by the 3-4-5 Pythagorean right triangle.
easiest proof that any cell can affect any other cell in 3-4-5 Life:
(0,0) + (5,0) = (5,0)
(5,0) - (4,3) = (1,-3)
(1,-3) + (0,5) = (1,2)
(1,2) is a knight's move. knights can get to any square (knight's tour, etc).
we can also synthesize a ferz move:
(0,0) + (-3,4) = (-3,4)
(-3,4) + (4,-3) = (1,1)
a knight's move and a ferz move reaches an adjacent square: (1,2) - (1,1) = (0,1).
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