Black/white reversal

Black/white reversal (or complement, or on/off reversal) can refer to two related concepts.

For two-state patterns, the black/white reversal of a pattern is the result of toggling the state of each cell in the universe: bringing dead cells to life, and killing live cells. The black/white reversal of a pattern is sometimes described by prefixing the name or description of the pattern with 'anti-'; for instance, the black/white reversal of a glider (in an appropriate rule) is referred to as an anti-glider.

For two-state rules, the black/white reversal of a rule is a transformation of the original rule, that behaves in "essentially the same way", except that the roles of "off" and "on" states are exchanged. Specifically, evolving a pattern P under the black/white reversal of a rule R is equivalent to first taking the black/white reversal of the pattern P, then evolving it (for the same number of ticks) under the original rule R, and finally taking the black/white reversal of the resulting evolved pattern.

Each rule has precisely one black/white reversal; if this is the same as the rule itself, the rule is said to be self-complementary. Such rules necessarily include precisely one of B0 or S8; in the latter case, there exists an equivalent strobing rule, such as B01245/S0125 for Day & Night.

Determining the black/white reversal of a rule

To determine the black/white reversal of a given rule:

1. Negate the rule's B and S conditions, yielding B′ and S′.
2. Subtract each condition in B′ and S′ from 8, yielding B″ and S″.
3. The black/white reversal of B/S is S″/B″.

For example, using the rule B36/S125:

1. B = 36; S = 125
2. B′ = 0124578; S′ = 034678
3. B″ = 0134678; S″ = 012458

Therefore, the black/white reversal of B36/S125 is B012458/S0134678.

Non-totalistic rules

The black/white reversal of a non-totalistic rule can be computed in the same manner as above if every neighborhood configuration is considered and negated individually.

For example, in Hensel notation, the black/white reversal of the rule B2-a/S12 is:

1. B = 2-a, S = 12
2. B′ = 012a345678; S′ = 0345678 (note how B2-a, upon negation, becomes B2a)
3. B″ = 0123456a78; S′ = 0123458

Therefore, the black/white reversal of B2-a/S12 is B0123458/S0123456a78.

Note, however, that B4 and S4 conditions are not only subtracted from 8 (which would map them to themselves), but must be considered on a per-letter basis. After the negation, elements of the pairs (4c,4e), (4i,4t), (4n,4r), (4y,4j), and (4q,4w) swap with their partners, but 4k, 4a, and 4z are conserved.