Black/white reversal (or complement, or on/off reversal) can refer to two related concepts:
- 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 called an anti-pattern; for instance, the black/white reversal of a glider (in an appropriate rule) is referred to as an anti-glider.
- The black/white reversal of a rule is a transformation of a rule in such a way that the black/white reversal of any pattern (in the previous sense) will behave the same way under the new rule as the unreversed pattern did under the original rule.
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:
- Negate the rule's B and S conditions, yielding B′ and S′.
- Subtract each condition in B′ and S′ from 8, yielding B″ and S″.
- The black/white reversal of B/S is S″/B″.
For example, using the rule B36/S125:
- B = 36; S = 125
- B′ = 0124578; S′ = 034678
- B″ = 0134678; S″ = 012458
Therefore, the black/white reversal of B36/S125 is B012458/S0134678.
- B = 2-a, S=12
- B′ = 012a345678; S′ = 0345678 (note how B2-a, upon negation, becomes B2a)
- B″ = 0123456a78; S′ = 0123458
Therefore, the black/white reversal of B2-a/S12 is B0123458/S0123456a78.
- Gamedziner (November 20, 2017). Re: Miscellaneous Discoveries in Other Cellular Automata (discussion thread) at the ConwayLife.com forums