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
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	ridiculously long boat	Partial list	179,011			8,879,327
22	672,172	cis-mirrored worm	Partial list	462,086			26,943,317
23	1,646,147	ludicrously long boat	Partial list	1,184,882
24	4,051,732	lake 2	Partial list	3,069,135
25	9,971,377	very^9 long boat	Partial list	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.

See also

• List of common still lifes
• List of pseudo still lifes
• List of still lifes
• List of strict 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

External links

• Still life at Wikipedia

• Still life at the Life Lexicon

• Pseudo still life at the Life Lexicon

• Quasi still life at the Life Lexicon

• Life Object Counts at Mark D. Niemiec's Life Page