Difference between revisions of "Phoenix"

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(Removing the less detailed statement about fixed population, leaving the one I just added that specifies "(finite or infinite)"... I hadn't noticed that the rest of that was duplicated elsewhere.)
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|caption      = p4 phoenix wick stabilised by phoenixes, although the fence posts have stators.
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Revision as of 01:32, 14 February 2024

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.

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

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 2023 Keith 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]

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 (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.

Gallery

x = 76, y = 53, rule = B3/S23 4bo18bo19bo19bo$2bobo18bobo17bobo17bobo$6bo14bo19bo5bo13bo$2o24b2o19bo bo16b2o$6b2o12b2o18b2o18b2o$bo25bo22b2o15bo$3bobo16bo19bobo17bobo4bo$ 3bo24b2o14bo6bo12bo4bobo$22b2o22bo19bo$28bo23b2o18b2o$24bobo19b2o18b2o $26bo25bo19bo$48bobo17bobo$50bo19bo3$7bo16bo25bo21bo$7bobo12bobo23bobo 19bobo$3bobo20bo19bo5bo15bo5bo$3bo6b2o8b2o22bobo19bobo$bo24b2o24b2o10b o9b2o$11bo9bo20b2o18bobo$2o26bo22bo19bobo$12b2o8b2o18bo17b2o9bo$2bo25b 2o20b2o15bobo$12bo9bo17b2o19bo5bo$2b2o6bo16bo19bobo13bobo$8bobo9b2o19b o5bo15bo$4bobo19b2o15bobo$6bo14bo21bo$23bobo$23bo6$7bo15bo7bo11bo7bo$ 7bobo13bobo5bobo9bobo5bobo$3bobo5bo9bo5bobo11bo5bobo$3bo7bobo13bo6b2o 11bo6b2o$bo18b2o18b2o$14b2o18bo19bo$2o20bo19bo$14bo17b2o18b2o$2bobo7bo 9b2o18b2o$4bo5bobo19bo20bo$6bobo15bo16bo$8bo21b2o22b2o$24b2o14b2o6bo$ 30bo15bobo5bo$26bobo13bobo5bobo$28bo15bo7bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 8 WIDTH 700 HEIGHT 500 AUTOSTART GPS 2 LOOP 2 ]]
All p2 phoenices in a 16 × 16 box[8]
(click above to open LifeViewer)
RLE: here Plaintext: here
x = 58, y = 95, rule = B3/S23 3bo$3bobo$bo5bo$7bobo9bo7bo$2o15bobo5bobo$10b2o9bobo5bo$2bo12b2o6bo$10b o18b2o$2b2o12bo$8bobo17bo$4bobo10b2o$6bo3bobo14b2o$14bobo2bobo$10bo3b o6bo7bo$8bobo12bo$29b2o$6b2o3b2o10b2o$28bo$6bo5bo9bo$4bo15bobo4b2o$2b obo8b2o$18b2o3b2o4bo$2o11bo$18bo4bo5b2o$bo9b2o$3bo12b2o3b2o5bo$3bobo6b o$7bo9bo3bo5b2o$7bobo2bobo4bo$19bo6bo$10b2o2bob2o$25b2o$12bobobo$24bo $14bo$23b2o$15b2o$22bo$16bo$21b2o$17b2o$23bo$18bo$20bobob2o$20bo$26b2o 2$28b2o2$30b2o$37bo$32b2obobo$39bo$34bo$39b2o$35b2o$41bo$35bo$41b2o$33b 2o$43bo$33bo$41bobobo$31b2o$40b2obo2b2o$31bo6bo$38bo4bobo2bobo$29b2o5b o3bo9bo$45bo6bobo$29bo5b2o3b2o12bo$45b2o9bo$27b2o5bo4bo$44bo11b2o$28b o4b2o3b2o$43b2o8bobo$29b2o4bobo15bo$35bo9bo5bo$29bo$33b2o10b2o3b2o$27b 2o$34bo12bobo$28bo7bo6bo3bo$36bobo2bobo$29b2o14bobo3bo$39b2o10bobo$29b o17bobo$41bo12b2o$27b2o18bo$34bo6b2o12bo$28bo5bobo9b2o$30bobo5bobo15b 2o$30bo7bo9bobo$50bo5bo$52bobo$54bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 5 WIDTH 700 HEIGHT 500 AUTOSTART GPS 2 LOOP 2 ]]
Connected phoenix parts
(click above to open LifeViewer)
RLE: here Plaintext: here
x = 206, y = 75, rule = B3/S23 11bo27bo19bo27bo19bo27bo19bo27bo$11bobo23bobo19bobo23bobo19bobo23bobo 19bobo23bobo$9bo31bo15bo31bo15bo31bo15bo31bo$14b2o19b2o25b2o19b2o25b2o 19b2o25b2o19b2o$8b2o31b2o13b2o31b2o13b2o31b2o13b2o31b2o$15bo19bo27bo 19bo27bo19bo27bo19bo$7bo9bo15bo9bo11bo9bo15bo9bo11bo9bo15bo9bo11bo9bo 15bo9bo$17bobo11bobo31bobo11bobo31bobo11bobo31bobo11bobo$6b2o35b2o9b2o 35b2o9b2o35b2o9b2o35b2o$20b2o7b2o37b2o7b2o37b2o7b2o37b2o7b2o$6bobo33bo 13bo33bo13bo33bo13bo33bo7b2o$8bobo10bo7bo10bobo13bobo10bo7bo10bobo13bo bo10bo7bo10bobo13bobo10bo7bo10bobo6bo2bo$4bobo186bobo$4b2o5b2o9b2o3b2o 9b2o3b2o9b2o3b2o9b2o3b2o9b2o3b2o9b2o3b2o9b2o3b2o9b2o3b2o9b2o3b2o9b2o3b 2o9b2o3b2o5bob2o$7bo187b3o$12bo9bo5bo9bo5bo9bo5bo9bo5bo9bo5bo9bo5bo9bo 5bo9bo5bo9bo5bo9bo5bo9bo5bo7b2o$6b2o8bobobo11bobobo11bobobo11bobobo11b obobo11bobobo11bobobo11bobobo11bobobo11bobobo11bobobo11bo$12bobobo5bo 5bobobo5bo5bobobo5bo5bobobo5bo5bobobo5bo5bobobo5bo5bobobo5bo5bobobo5bo 5bobobo5bo5bobobo5bo5bobobo5bo5bobobobo$8bo5bo5bobobo5bo5bobobo5bo5bob obo5bo5bobobo5bo5bobobo5bo5bobobo5bo5bobobo5bo5bobobo5bo5bobobo5bo5bob obo5bo5bobobo5bo$8bobobo11bobobo11bobobo11bobobo11bobobo11bobobo11bobo bo11bobobo11bobobo11bobobo11bobobo11bobobo6b2o$14bo5bo9bo5bo9bo5bo9bo 5bo9bo5bo9bo5bo9bo5bo9bo5bo9bo5bo9bo5bo9bo5bo9bo$4b3o188bo$4b3obo5b2o 3b2o9b2o3b2o9b2o3b2o9b2o3b2o9b2o3b2o9b2o3b2o9b2o3b2o9b2o3b2o9b2o3b2o9b 2o3b2o9b2o3b2o9b2o5b2o$5bobobo186bobo$6bo2bo6bobo10bo7bo10bobo13bobo 10bo7bo10bobo13bobo10bo7bo10bobo13bobo10bo7bo10bobo$7b2o7bo33bo13bo33b o13bo33bo13bo33bobo$28b2o7b2o37b2o7b2o37b2o7b2o37b2o7b2o$14b2o35b2o9b 2o35b2o9b2o35b2o9b2o35b2o$25bobo11bobo31bobo11bobo31bobo11bobo31bobo 11bobo$15bo9bo15bo9bo11bo9bo15bo9bo11bo9bo15bo9bo11bo9bo15bo9bo$23bo 19bo27bo19bo27bo19bo27bo19bo$16b2o31b2o13b2o31b2o13b2o31b2o13b2o31b2o$ 22b2o19b2o25b2o19b2o25b2o19b2o25b2o19b2o$17bo31bo15bo31bo15bo31bo15bo 31bo$19bobo23bobo19bobo23bobo19bobo23bobo19bobo23bobo$19bo27bo19bo27bo 19bo27bo19bo27bo10$22bo15bo79bo15bo$22bobo11bobo79bobo11bobo$10bo9bo 19bo9bo16bo7bo5bo7bo16bo9bo19bo9bo16bo7bo5bo7bo$8bobo14b2o7b2o14bobo 14bobo5bobobobo5bobo14bobo14b2o7b2o14bobo14bobo5bobobobo5bobo$12bobo4b 2o19b2o4bobo16bo5bobo9bobo5bo16bobo4b2o19b2o4bobo16bo5bobo9bobo5bo$6b 2o6bo11bo7bo11bo6b2o16bo6bo6bo16b2o6bo11bo7bo11bo6b2o16bo6bo6bo$16bobo 23bobo19b2o10bo3bo10b2o19bobo23bobo19b2o10bo3bo10b2o$7bo10bo8b2o3b2o8b o10bo20bobo3bobo20bo10bo8b2o3b2o8bo10bo20bobo3bobo21b2o$63bo29bo65bo 29bo10bo$8b2o9b2o4b2obobobob2o4b2o9b2o8bobo8b2o9b2o8bobo8b2o9b2o4b2obo bobob2o4b2o9b2o8bobo8b2o9b2o8bobo6bobobo$197bobo2b3o$9bo9bo6bob2ob2obo 6bo9bo7b2o3b2o7bo9bo7b2o3b2o7bo9bo6bob2ob2obo6bo9bo7b2o3b2o7bo9bo7b2o 3b2o3bobobo3bo$11bobobo27bobobo27bobobo27bobobo27bobobo27bobobo22bobob 2obo$9bo5bobobo7bo5bo7bo5bobobo7bo5bo7bo5bobobo7bo5bo7bo5bobobo7bo5bo 7bo5bobobo7bo5bo7bo5bobobo7bo5bo6bo3bo$5bobobobo5bo5bo11bobobobobo5bo 5bo11bobobobobo5bo5bo11bobobobobo5bo5bo11bobobobobo5bo5bo11bobobobobo 5bo5bo11bo3b2obo$bobo3bo11bobobobobo5bo5bo11bobobobobo5bo5bo11bobobobo bo5bo5bo11bobobobobo5bo5bo11bobobobobo5bo5bo11bobobobobo5bo3b2obobo$o 2bo7bo5bo7bo5bobobo7bo5bo7bo5bobobo7bo5bo7bo5bobobo7bo5bo7bo5bobobo7bo 5bo7bo5bobobo7bo5bo7bo5bobobo5b3o$obo24bobobo27bobobo27bobobo27bobobo 27bobobo27bobobo9bobo$b3o7b2o3b2o7bo9bo7b2o3b2o7bo9bo6bob2ob2obo6bo9bo 7b2o3b2o7bo9bo7b2o3b2o7bo9bo6bob2ob2obo6bo9bo6b2o$3b2o$3bo9bobo8b2o9b 2o8bobo8b2o9b2o4b2obobobob2o4b2o9b2o8bobo8b2o9b2o8bobo8b2o9b2o4b2obobo bob2o4b2o9b2o$4b3o8bo29bo65bo29bo$6bo19bobo3bobo20bo10bo8b2o3b2o8bo10b o20bobo3bobo20bo10bo8b2o3b2o8bo10bo$16b2o10bo3bo10b2o19bobo23bobo19b2o 10bo3bo10b2o19bobo23bobo$23bo6bo6bo16b2o6bo11bo7bo11bo6b2o16bo6bo6bo 16b2o6bo11bo7bo11bo6b2o$17bo5bobo9bobo5bo16bobo4b2o19b2o4bobo16bo5bobo 9bobo5bo16bobo4b2o19b2o4bobo$19bobo5bobobobo5bobo14bobo14b2o7b2o14bobo 14bobo5bobobobo5bobo14bobo14b2o7b2o14bobo$19bo7bo5bo7bo16bo9bo19bo9bo 16bo7bo5bo7bo16bo9bo19bo9bo$70bobo11bobo79bobo11bobo$70bo15bo79bo15bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ZOOM 5 WIDTH 1100 HEIGHT 500 AUTOSTART GPS 2 LOOP 2 ]]
p4 phoenix wick stabilised by phoenixes, although the fence posts have stators.
(click above to open LifeViewer)


See also

References

  1. 1.0 1.1 Dave Greene (December 7, 2018). Re: Thread for basic questions (discussion thread) at the ConwayLife.com forums
  2. Alex Greason (November 4, 2019). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  3. wwei47 (January 12, 2024). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  4. Alex Greason (September 29, 2019). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  5. Keith Amling (December 10, 2023). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  6. Keith Amling (December 10, 2023). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
  7. "Every finite phoenix has period 2". Complex Projective 4-Space. Retrieved on January 20, 2024.
  8. wwei23 (November 17, 2021). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums

External links