Self-complementary

An (outer-totalistic) rule is said to be self-complementary if it remains unchanged under black/white reversal, i.e. if a pattern composed of live cells in a dead universe behaves the same as an equivalent pattern composed of dead cells in a live universe.

A self-complementary rule's birth conditions completely determine its survival conditions, and vice versa, but no other constraints exist; as such there are 2⁹ = 512 distinct self-complementary outer-totalistic rules.

Every self-complementary rule contains precisely one of B0 and S8. In the former case, the rule is strobing; in the latter case, there exists an equivalent strobing rule. Non-strobing and strobing versions of the same self-complementary rule behave identically, except that in the strobing version, patterns are inverted in every other generation, so of the 512 distinct self-complementary outer-totalistic rules, only 256 exhibit fundamentally different behavior.

Examples

The most well-known and well-investigated self-complementary rule is Day & Night (B3678/S34678); its equivalent strobing version is B01245/S0125.