Still life

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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 strict still lifes.

Strict still lifes

A strict still life is a still life that is either connected (i.e., has no islands), 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] 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.
Pseudo still life that can be partitioned into three still lifes, but not two. RLE: here
Pseudo still life that 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 constellation is not, strictly speaking, a still life; instead, the term is used for patterns composed of two or more non-interacting objects. This contrasts with pseudo 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.

Enumerating still lifes

The number of strict and pseudo still lifes that exist for a given number of cells has been enumerated up to 24. 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) and Simon Ekström (25-28 cells). The values in the pseudo still life table were enumerated by Mark Niemiec (1-24 cells) and Simon Ekström (25-28 cells). The values in the tables below are given by Sloane's OEISicon light 11px.pngA019473 and OEISicon light 11px.pngA056613, respectively.

Live cells # of strict still lifes Examples List
1 0
2 0
3 0
4 2 block, tub Full list
5 1 boat Full list
6 5 beehive, ship Full list
7 4 eater 1, loaf Full list
8 9 canoe, pond Full list
9 10 hat, integral sign Full list
10 25 boat-tie, loop Full list
11 46 elevener Full list
12 121 honeycomb, table on table Partial list
13 240 sesquihat Partial list
14 619 fourteener, paperclip Partial list
15 1,353 moose antlers Partial list
16 3,286 bi-cap, scorpion Partial list
17 7,773 twin hat Partial list
18 19,044 dead spark coil Partial list
19 45,759 eater 2 Partial list
20 112,243 spiral Partial list
21 273,188 very^7 long boat Partial list
22 672,172 cis-mirrored worm Partial list
23 1,646,147 very^8 long boat Partial list
24 4,051,732 lake 2 Partial list
25 9,971,377 very^9 long boat Partial list
26 24,619,307 Mickey Mouse Partial list
27 60,823,008
28 150,613,157 Tetraloaf I Partial list
29 373,188,952
30 926,068,847
31 2,299,616,637
Live cells # of pseudo still lifes Examples List
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 1 bi-block Complete list
9 1 block on boat Complete list
10 7 bi-boat Partial list
11 16
12 55
13 110
14 279
15 620
16 1,645 pond on pond Partial list
17 4,067
18 10,843
19 27,250
20 70,637
21 179,011
22 462,086
23 1,184,882
24 3,069,135
25 7,906,676
26 20,463,274
27 52,816,265
28 136,655,095
29 353,198,379
30 914,075,620
31 2,364,815,358

See also

References

  1. 1.0 1.1 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.

External links