Still life

A still life (or stable pattern) is a pattern that does not change from one generation to the next, and thus may be thought of as an oscillator with period 1. Still lifes are sometimes assumed to be finite and non-empty. The two main subgroups of still lifes are strict still lifes and pseudo still lifes. In some contexts, the term "still life" may refer to stable objects rather than stable patterns in general, or strict still lifes rather than stable objects in general.

Classification of still lifes (stable patterns). Click to enlarge.

Strict still lifes

A strict still life is a still life that is either connected (i.e., has only one island), or is such that removing one or more its islands destroys the stability of the pattern. For example, beehive with tail is a strict still life because it is connected, and table on table is a strict still life because neither of the tables are stable by themselves.

Beehive is a strict still life because it is connected.
Beehive with tail is a strict still life because it is connected, even though it contains a smaller still life.
Table on table is a strict still life because neither table is stable without the other.

Pseudo still lifes

A pseudo still life consists of two or more islands which can be partitioned (either individually or as sets) into non-interacting subpatterns which are by themselves each still lifes. Furthermore, there must be at least one dead cell that has more than three alive neighbours in the overall pattern but has less than three alive neighbours in the subpatterns. This final restriction removes patterns such as bakery, blockade and fleet from consideration, as the islands are not "almost touching".

Note that a pattern may have multiple disconnected components and still be a strict (as opposed to pseudo) still life if the disconnected components are dependent on each other for stability (for example, table on table above). Some pseudo still lifes have also been found by Gabriel Nivasch that can be partitioned into a minimum of three and four stable subpatterns, respectively, as in the second and third images below.[1] The stable subpatterns themselves may be either strict or pseudo still lifes. It is not possible to construct a pseudo still life that can be partitioned into a minimum of greater than four stable subpatterns because of the Four Color Theorem.[1]

Bi-block is a pseudo still life because each block is stable by itself.
The triple pseudo still life, which can be partitioned into three to five independent stable subpatterns, but not two. RLE: here
The quad pseudo still life, which can be partitioned into four still lifes, but not two or three. RLE: here

It has been shown that it is possible to determine whether a still life pattern is a strict still life or a pseudo still life in polynomial time by searching for cycles in an associated skew-symmetric graph.[2][3]

Constellations

A (stable) constellation is a still life that is composed of two or more non-interacting objects. This contrasts with pseudo and quasi still lifes, in which the objects in question must interact. Compare for instance the bi-block and blockade:

Bi-block is a pseudo still life because the two blocks interact: the two dead cells between them are influenced by both.
Blockade is a constellation because the four blocks do not interact in any way.

Certain unstable (e.g. oscillating) patterns are sometimes also referred to as constellations. The term "stable constellation" is used to refer specifically to still life constellations.

Quasi still lifes

A stable constellation in which the constituent objects share dead cells, but where all cells that used to remain dead from under-population in the overall pattern still do so in the constituent objects, is called a quasi still life. In Conway's Life, this occurs when objects are diagonally adjacent (e.g. two blocks sharing a single diagonal neighbor), or when single protruding cells in two objects such as tubs share multiple neighbors.

Two blocks sharing a single diagonal neighbor, marked in green; this cell is dead from underpopulation, and remains so after separation.

The term "quasi still life" is due to Mark Niemiec.

Enumerating still lifes

The number of strict and pseudo still lifes that exist for a given number of cells has been enumerated up to 34, and the number of quasi still lifes for a given number of cells up to 22.

  • The values in the strict still life table below were originally computed by John Conway (4-7 cells), Robert Wainwright (8-10 cells), David Buckingham (11-13 cells), Peter Raynham (14 cells), Mark Niemiec (15-24 cells), Simon Ekström (25-28 cells),[4][5][6] Simon Ekström and "Apple Bottom" (29-30 cells),[7] and Nathaniel Johnston (31-34 cells).[8][9][10][11]
  • The values in the pseudo still life table were enumerated by Mark Niemiec (1-24 cells), Simon Ekström (25-28 cells),[4][5][6] Simon Ekström and "Apple Bottom" (29-30 cells),[7] and Nathaniel Johnston (31-34 cells).[8][9][10][11]
  • The values in the quasi still life table below were originally computed by Mark Niemiec (8-20 cells) and Simon Ekström (21-22 cells).[12]
Live cells Strict still lifes Pseudo still lifes Quasi still lifes
Count (A019473 ) Examples List Count (A056613 ) Examples List Count Examples List
1000
2000
3000
42block, tubFull list00
51boatFull list00
65beehive, shipFull list00
74eater 1, loafFull list00
89canoe, pondFull list1bi-blockFull list6
910hat, integral signFull list1block on boatFull list13
1025boat-tie, loopFull list7bi-boatPartial list57
1146elevenerFull list16141
12121honeycomb, table on tableFull list55465
13240sesquihatPartial list1101,224
14619fourteener, paperclipPartial list2793,956
151,353moose antlersPartial list62011,599
163,286bi-cap, scorpionPartial list1,645pond on pondPartial list36,538
177,773twin hatPartial list4,067107,415
1819,044dead spark coilPartial list10,843327,250
1945,759eater 2Partial list27,250972,040
20112,243spiralPartial list70,6372,957,488
21273,188ridiculously long boatPartial list179,0118,879,327
22672,172cis-mirrored wormPartial list462,08626,943,317
231,646,147ludicrously long boatPartial list1,184,882
244,051,732lake 2Partial list3,069,135
259,971,377very^9 long boatPartial list7,906,676
2624,619,307Mickey MousePartial list20,463,274
2760,823,008hat siamese vasePartial list52,816,265
28150,613,157O quad-loafPartial list136,655,095
29373,188,952xs29_cc0s2ticz330fgkcPartial list353,198,379
30926,068,847ClipsPartial list914,075,620
312,299,616,637Aries betwixt two blocksPartial list2,364,815,358
325,716,948,683Inflected 30-great symPartial list6,123,084,116triple pseudo still lifePartial list
3314,223,867,29815,851,861,075
3435,422,864,104Partial list41,058,173,683quad pseudo still lifePartial list

As the number of bits increases, these counts increase exponentially; the rate for strict still lifes is about O(2.46n), while for pseudo still lifes it is around O(2.56n), and approximately O(3.04n) for quasi still lifes.

gollark: I would join but I would have to update and it would be irritating.
gollark: I've never seen anyone actually die to nanobots before.
gollark: Guess it's the apocalypse?
gollark: Is there something wrong with OpenOS?
gollark: Have you considered installing potatOS on your datacentre thingy?

See also

References

  1. Nivasch, Gabriel (July, 2001). "Still lifes". Retrieved on March 23, 2016.
  2. Cook, Matthew (2003). "Still life theory". New Constructions in Cellular Automata: 93–118, Santa Fe Institute Studies in the Sciences of Complexity, Oxford University Press.
  3. Cook, Matthew. "Still Life". Mathematical Sciences Research Institute.
  4. Simon Ekström (January 2, 2017). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
  5. Simon Ekström (January 3, 2017). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
  6. Simon Ekström (January 7, 2017). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
  7. Simon Ekström (January 14, 2017). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
  8. Nathaniel Johnston (March 27, 2017). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
  9. Nathaniel Johnston (May 25, 2017). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
  10. Nathaniel Johnston (April 5, 2019). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
  11. Nathaniel Johnston (January 9, 2020). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
  12. Mark Niemiec (January 15, 2017). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
This article is issued from Conwaylife. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.