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 (see table on table above). Some pseudo still lifes have also been found that can be partitioned into three or more stable subpatterns, but can not be partitioned into two stable subpatterns (see the second and third images below).

Bi-block is a pseudo still life because each block is stable by itself.
Triple pseudo still life can be partitioned into three still lifes, but not two.
Quad pseudo still life can be partitioned into four still lifes, but not two or three.

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.[1][2]

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) and Mark Niemiec (15-24 cells). The values in the pseudo still life table were enumerated by Mark Niemiec. The values in the tables below are given by Sloane's A019473 and A056613, 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 Partial list
12 121 honeycomb, table on table Partial list
13 240 sesquihat Partial list
14 619 fourteener, paperclip Partial list
15 1353 moose antlers Partial list
16 3286 bi-cap, scorpion Partial list
17 7773 twin hat Partial list
18 19044 dead spark coil Partial list
19 45759 eater 2 Partial list
20 112243 spiral Partial list
21 273188
22 672172
23 1646147
24 4051711 lake 2 Partial list
Live cells # of pseudo still lifes Examples
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 1 bi-block
9 1 block on boat
10 7
11 16
12 55
13 110
14 279
15 620
16 1645 pond on pond
17 4067
18 10843
19 27250
20 70637
21 179011
22 462086
23 1184882
24 3068984

See also


  1. 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. 
  2. Cook, Matthew. "Still Life". Mathematical Sciences Research Institute.

External links