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.
Note that period 1 is often abbreviated p1 connoting stable. In the context of logic circuitry, this tends to mean that a mechanism is constructed from Herschel conduits that contain only still lifes as catalysts.
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 mirrored 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
Mirrored 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, mirrored 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. 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.
Bi-block is a pseudo still life
because each block is stable by itself
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.
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
(highlighted in red) 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 yellow; 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), Simon Ekström and "Apple Bottom" (29-30 cells), and Nathaniel Johnston (31-34 cells).
- The values in the pseudo still life table were enumerated by Mark Niemiec (1-24 cells), Simon Ekström (25-28 cells), Simon Ekström and "Apple Bottom" (29-30 cells), and Nathaniel Johnston (31-34 cells).
- The values in the quasi still life table below were originally computed by Mark Niemiec (8-20 cells) and Simon Ekström (21-22 cells).
|Live cells||Strict still lifes||Pseudo still lifes||Quasi still lifes|
|Count ( A019473)||Examples||List||Count ( A056613)||Examples||List||Count ( A330283)||Examples||List|
|4||2||block, tub||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||cis-bi-boat||Full list||57|
|12||121||honeycomb, mirrored table||Full list||55||tri-block||Partial list||465|
|14||619||fourteener, paperclip||Partial list||279||3,956|
|15||1,353||moose antlers||Partial list||620||11,599|
|16||3,286||mirrored 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|
|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|
|24||4,051,732||lake 2||Partial list||3,069,135|
|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|
|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|
|34||35,422,864,104||Cis-mirrored elf-shoe||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.
- List of common still lifes
- List of pseudo still lifes
- List of still lifes
- List of strict still lifes
- Nivasch, Gabriel (July, 2001). "Still lifes". Retrieved on March 23, 2016.
- 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.
- Cook, Matthew. "Still Life". Mathematical Sciences Research Institute.
- Simon Ekström (January 2, 2017). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
- Simon Ekström (January 3, 2017). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
- Simon Ekström (January 7, 2017). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
- Simon Ekström (January 14, 2017). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
- Nathaniel Johnston (March 27, 2017). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
- Nathaniel Johnston (May 25, 2017). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
- Nathaniel Johnston (April 5, 2019). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
- Nathaniel Johnston (January 9, 2020). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
- Mark Niemiec (January 15, 2017). Re: Enumerating Still Lifes (in C) (discussion thread) at the ConwayLife.com forums
- Still life at Wikipedia