Phoenix

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 spaceship cannot be a phoenix in Conway's Game of Life, and in fact every finite phoenix eventually evolves into an oscillator.[1] The first discovered phoenix, phoenix 1, was a period-2 oscillator with 12 cells.

This article is about the general concept. For the 12-cell oscillator, see Phoenix 1.

An inherent property of phoenices is that its heat equals its population.

Every known finite phoenix oscillator has period 2. Infinite phoenix oscillators are known for periods 2, 4, 6, 8 and 12.[2] In January 2000, Stephen Silver showed that a period 3 oscillator can not be a phoenix;[1] and in September 2019, Alex Greason showed that there are no period 5 phoenix oscillators.[3] The situation for other periods is unknown.

some avatars

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 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; such an arrangement adorned an issue of Scientific American soon after the original presentation of Life.

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 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.

Also see
  • List of phoenices
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References
  1. Dave Greene (December 7, 2018). "Re: Thread for basic questions". Retrieved on December 7, 2018.
  2. Alex Greason (November 4, 2019). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  3. Alex Greason (September 29, 2019). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
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