Black/white reversal

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.[1]
  • 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:

  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.

gollark: I certainly haven't gotten that.
gollark: x² such that x in (A union C) and x < 8.
gollark: I should do that some year.
gollark: Or Haskell, where your code is just a mapping from inputs to outputs which the compiler does ??? to.
gollark: As opposed to, say, SQL's declarative style, where you just describe what information you want and the database works out how to get it.

References
  1. Gamedziner (November 20, 2017). Re: Miscellaneous Discoveries in Other Cellular Automata (discussion thread) at the ConwayLife.com forums
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