Checkerboard dual

The checkerboard dual of an isotropic non-totalistic cellular automaton is another such automaton such that any finite arrangement of live cells in an empty universe in the original rule will behave like finite perturbations in a checkerboard universe in the dual.

That is to say, for a finite pattern P, a rule R and its checkerboard dual C(R):

R(P) = C(P ⊕ checkerboard) ⊕ checkerboard

where ⊕ denotes the XOR operation.

Checkerboard duality is similar to strobing duality, but applies to isotropic rules; the checkerboard equivalent of an outer-totalistic rule (such as Life) is not generally outer-totalistic. The checkerboard dual of an isotropic non-totalistic rule is also isotropic non-totalistic if and only if the original rule is self-complementary.

The checkerboard dual of Day & Night is B1e2cn3acjkr4cny5einqy6ei7c/S01e2-ei3acjkr4-ejr5einqy6-cn7c8^[1].

Finding the checkerboard dual of a given rule

Let the 'B/S conditions' of a rule be the set of 3-by-3 boxes of cells that, after iterating by one generation, result in a live centre cell. Clearly, a rule is determined uniquely by its B/S conditions. Black/white symmetric rules have exactly 256 B/S conditions (out of a possible 512).

The correct sister rule is obtained by XORing each B/S condition with a 5-cell X and the output with 1, or each condition with a 4-cell O and preserving the output, analogous to how the strobing dual is obtained by XORing each B/S condition with a 9-cell solid box.