# Still life

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

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.

## 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.

Live cells Strict still lifes Pseudo still lifes Quasi still lifes
Count (A019473) Examples List Count (A056613) Examples List Count Examples List
1 0 0 0
2 0 0 0
3 0 0 0
4 2 block, tub Full list 0 0
5 1 boat Full list 0 0
6 5 beehive, ship Full list 0 0
7 4 eater 1, loaf Full list 0 0
8 9 canoe, pond Full list 1 bi-block Full list 6
9 10 hat, integral sign Full list 1 block on boat Full list 13
10 25 boat-tie, loop Full list 7 bi-boat Partial list 57
11 46 elevener Full list 16 141
12 121 honeycomb, table on table Full list 55 465
13 240 sesquihat Partial list 110 1,224
14 619 fourteener, paperclip Partial list 279 3,956
15 1,353 moose antlers Partial list 620 11,599
16 3,286 bi-cap, scorpion Partial list 1,645 pond on pond Partial list 36,538
17 7,773 twin hat Partial list 4,067 107,415
18 19,044 dead spark coil Partial list 10,843 327,250
19 45,759 eater 2 Partial list 27,250 972,040
20 112,243 spiral Partial list 70,637 2,957,488
21 273,188 tri-loaf 2 Partial list 179,011 8,879,327
22 672,172 cis-mirrored worm Partial list 462,086 26,943,317
23 1,646,147 1,184,882
24 4,051,732 lake 2 Partial list 3,069,135
25 9,971,377 7,906,676
26 24,619,307 Mickey Mouse Partial list 20,463,274
27 60,823,008 hat siamese vase Partial list 52,816,265
28 150,613,157 O quad-loaf Partial list 136,655,095
29 373,188,952 xs29_cc0s2ticz330fgkc Partial list 353,198,379
30 926,068,847 Clips Partial list 914,075,620
31 2,299,616,637 Aries betwixt two blocks Partial list 2,364,815,358
32 5,716,948,683 Inflected 30-great sym Partial list 6,123,084,116 triple pseudo still life Partial list
33 14,223,867,298 15,851,861,075
34 35,422,864,104 Partial list 41,058,173,683 quad pseudo still life Partial 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.

## 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