User:Apple Bottom/Incubator/Almost oscillator

From LifeWiki
Jump to navigation Jump to search

An almost-oscillator (or almost oscillator, or local oscillator) is a pattern such that every cell in the universe is periodic.

For finitely-supported patterns, this condition is equivalent to the condition that there exists a period P such that the pattern returns to its original state after P generations, i. e. that it is an oscillator in the usual sense. In general, however, the first condition is strictly weaker; there are almost-oscillators that are not oscillators by the usual definition. A (necessarily infinitely-supported) almost-oscillator for which no such P exists is said to have period infinity.

If patterns with infinite population are allowed, the disjoint union of oscillators of periods 1, 2, 3, 4, … is a trivial example of an almost-oscillator that is not an oscillator. It is also possible to construct a period-infinity almost-oscillator which has finite population in each generation; this is left as an exercise to the reader.

External links