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Symmetric rules

Whenever the lattice underlying a cellular automaton is symmetric, there are plausible reasons to work with rules which have the same symmetry; Moore neighborhoods have square symmetry on account of the lattice, with additional symmetries due to complementation which are sometimes taken into account.

Although there are eight symmetry operations for a square, not all symmetry classes have eight members; for example there is only one member of the class containing the zero neighborhood. The symmetries preserve distance, so that the central cell always stands by itself; nor are diagonal neighbors ever mixed with lateral neighbors. All told, there are 102 symmetry classes, 51 each for the two states of the center cell.

It would be an option to leave out reflections, to consider only rotational symmetries of the neighborhoods. This might split some of the symmetry classes, but hardly all of them; choosing this degree of specialization assumes that the handedness of the rules have importance.

Harold V. McIntosh