The checkerboard dual of an isotropic non-totalistic Life-like 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 ⊕ checkerboard) = S(P) ⊕ checkerboard
where ⊕ denotes the XOR operation.
Checkerboard duality is similar to strobing duality, but applies to isotropic rules; the checkerboard equivalent of a semi-totalistic rule (such as Life) is not generally semi-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.
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 such that, after iterating by one generation, result in a live centre cell. Clearly, a rule is determined uniquely by its B/S conditions. These black/white symmetric rules have exactly 256 B/S conditions (out of a possible 512).
The correct sister rule (in your case) is obtained by XORing each B/S condition with a 5-cell X, analogous to how the strobing dual is obtained by XORing each B/S condition with a 9-cell solid box.
- AforAmpere (August 4, 2017). Re: Day & Night (B3678/S34678) (discussion thread) at the ConwayLife.com forums
- Adam P. Goucher (January 16, 2018). Re: Thread for basic questions (discussion thread) at the ConwayLife.com forums
- Aidan F. Pierce (January 16, 2018). Re: Day & Night (B3678/S34678) (discussion thread) at the ConwayLife.com forums