This article is about the general concept. For the 12-cell oscillator, see Phoenix 1.
A phoenix (plural phoenices or phoenixes) is a pattern all of whose cells die in every generation, but that never dies as a whole. A pattern that is a phoenix is also said to be phoenician.
An inherent property of phoenices is that the heat equals twice the average population.
Every finite phoenix oscillator has period 2. Infinite phoenix oscillators (agars or wicks) are known for periods 2, 4, 8, and 6n (for any positive integer n).[2][3] Many of these are members of a higher XOR-based family of effectively arbitrarily high periods or phoenix signal loops. In January 2000, Stephen Silver showed that a period-3 oscillator in Life cannot be a phoenix,[1] in September 2019, Alex Greason showed that there are no period-5 phoenix oscillators in Life,[4] and in December 2023Keith Amling showed that there are no phoenix oscillators of periods 7, 9, or 11 in Life[5][6] (barring potential unknown bugs). In January 2024, Adam P. Goucher proved that there are no finite phoenices in Life with any period other than 2, along with the corollary that all phoenices (finite or infinite) must have constant population.[7]
As with any general pattern, there are innumerable instances of phoenices, some of which are striking enough to be shown on their own individual pages. Many are based on simple avatars such as the ones shown in the figure, which means that there would be considerable redundancy in exhibiting more than a few prototypes of any class of phoenices. The first pair (croaker and flutter respectively) were among the first discovered, along with the realization that they could be strung out quite arbitrarily into long filaments and even into closed loops.
Dominoes, either vertical or horizontal, can be stacked and even staggered slightly, as long as they are parallel and spaced by a single width. Unlike having monominoes and dominoes alternating, the chains are not flexible enough to create elaborate patterns.
On the other hand, monominoes alone can also be used to create diagonal phoenix chains. Finite chains of this type quickly disintegrate, but in a departure from strict phoenicity the ends of the chains can sometimes be anchored. Such is the origin of the barberpole family.
The final example is a section of a phoenix agar incorporating avatars of the preceding styles. Note that while the blue lattice generates the red lattice, the red lattice does not regenerate the blue lattice; rather each generation is translated to the northeast by a single Life cell. The larger unit cells of the agar are 4 × 6, whose least common multiple (and hence the period of the agar as an oscillator) is 12.
