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.

Generalizations

Self-complementary rules exhibiting complex behavior, spaceships, still lifes, oscillators.replicators, rake and guns are known in to exist in higher range HROT rules. Somewhat similar behavior has been also discovered in isotropic non-totalistic self-complementary 1D cellular automata with higher ranges and custom neighborhoods.

It is also possible to define generalizations of self-complementary rules with more than 2 cyclical states, e.g. a 3-state rule in which every pattern behaves exactly the same, if every cell of state 1 is changed to state 2, state 2 to state 3 and state 3 back to state 1.

External links

Self-complementary at Adam P. Goucher's Catagolue