Strobing rule

A rule (i.e. a Life-like cellular automaton) is said to be strobing iff it contains B0 (dead cells get born if they have zero live neighbors) but not S8 (live cells do not survive if they have eight live neighbors). In strobing rules, the entire universe will "light up" (get born) and die again in successive alternate generations, creating a visual "strobing" effect.[note 1]

Equivalent strobing rules of self-complementary rules

Self-complementary outer-totalistic rules (i.e. outer-totalistic Life-like cellular automata that are invariant under black/white reversal) necessarily include precisely one of B0 and S8. Applying the transformation below to a self-complementary rule with S8 but not B0 yields a rule with B0 but not S8, i.e. a strobing rule in the above sense. This rule is called the equivalent strobing rule of the original rule.

Applying the "strobing rule" transformation a second time yields the original rule again, so each (non-strobing) self-complementary rule and its equivalent strobing rule form a pair. As there are exactly 29 = 512 self-complementary rules, and each equivalent strobing rule is itself self-complementary, there are exactly 28 = 256 such pairs.

Patterns evolve identically in the strobing and non-strobing versions of a self-complementary rule except that in the strobing version, every other generation is inverted.

Determining the equivalent strobing rule of a self-complementary rule

The strobing version of a given self-complementary rule is determined the same way as its black/white reversal, except that the rule's original B and S conditions are not negated.

That is to say, to determine the strobing version of a given rule:

  1. Subtract each condition in B and S from 8, yielding B′ and S′.
  2. The equivalent strobing rule of B/S is S′/B′.

For self-complementary rules, this can be reduced to the following, easier method:

  1. Negate each condition in B and S, yielding B′ and S′.
  2. The equivalent strobing rule of B/S is B′/S′.

Using the latter algorithm, it is easy to see that the rule integers of a rule and its strobing equivalent's always add up to 262143 (218 - 1), which is also the amount of possible range-1 outer-totalistic rules (minus one).

Example

Applying the first method to the rule B356/S014678, we have:

  1. B = 356; S = 014678
  2. B′ = 235; S′ = 012478

Or, using the second method:

  1. B = 356; S = 014678
  2. B′ = 012478; S′ = 235

Therefore, the equivalent strobing rule of B356/S014678 is B012478/S235.

Notes
  1. This effect may not actually be rendered by software supporting such rules, e.g. Golly.
gollark: Maybe not ARMv4 though.
gollark: Lua runs on ARM.
gollark: Indeed.
gollark: <@178948413851697152> https://en.m.wikipedia.org/wiki/AMULET_microprocessor
gollark: I think the trendy thing is just aggressive power/clock gating now.
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