366:
all finite patterns in rules beginning with B1, grow in all directions rather than remaining of bounded size, with a front that moves at the speed of light. Thus, the remaining "interesting" rules are the ones beginning with B3 (Game of Life, Highlife, Morley, 2x2, Day&Night) or beginning with B0 (and not including S8, as otherwise the dual can be studied instead).
149:
115:
open-source cellular automaton package and in the RLE format for storing cellular automaton patterns, a rule is written in the form By/Sx where x and y are the same as in the MCell notation. Thus, in this notation, Conway's Game of Life is denoted B3/S23. The "B" in this format stands for "birth" and
365:
Any automaton of the above form that does not include any of B0, B1, B2 or B3 cannot support movement or expansion of patterns because any cell outside a rectangular building box containing the pattern has at most three on neighbours. Most finite patterns in rules whose notation begins with B2, and
84:
number the binary representation of which has bits that correspond to each possible number of neighbors and state of a cell; the bits of this number are zero or one accordingly as a cell with that neighborhood is dead or alive in the next generation. The other two notations unpack the same sequence
374:
There are other cellular automata which are inspired by the Game of Life, but which do not fit the definition of "life-like" given in this article, because their neighborhoods are larger than the Moore neighborhood, or they are defined on three-dimensional lattices, or they use a different lattice
409:
rules include one or more "dying" states cells switch to instead of instantly dying. The most famous examples in this category are the rules "Brian's Brain" (B2/S/3) and "Star Wars" (B2/S345/4). Random patterns in these two rules feature a large variety of spaceships and rakes with a speed of c,
427:
of Evan's Larger Than Life CA, in the limit as the neighborhood radius goes to infinity, while the lattice spacing goes to zero. Technically, they are not cellular automata at all, because the underlying "space" is the continuous
Euclidean plane
272:
If a pattern is composed of 2x2 blocks, it will continue to evolve in the same form; grouping these blocks into larger powers of two leads to the same behavior, but slower. Has complex oscillators of high periods as well as a small glider.
416:
is a family of cellular automata studied by Kellie
Michele Evans. They have very large radius neighborhoods, but perform "birth/death" thresholding similar to Conway's life. These automata have eerily organic "glider" and "blinker"
444:
is a family of continuous cellular automata created by Bert Wang-Chak Chan. The space, time and states of the Game of Life are generalized to continuous domains, using large neighborhoods, fractional updates, and real number states,
64:(B3/S23), the most famous cellular automaton, which meets all of these criteria. Many different terms are used to describe this class. It is common to refer to it as the "Life family" or to simply use phrases like "similar to Life".
362:-1) must have exactly one neighbor, and will become on in the next step. Similarly, the pattern must grow at each step in each of the four diagonal directions. Thus, any nonempty starting pattern leads to explosive growth.
260:
Forms large diamonds with chaotically fluctuating boundaries. First studied by Dean
Hickerson, who in 1993 offered a $ 50 prize to find a pattern that fills space with live cells; the prize was won in 1999 by David Bell.
48:
In each time step of the automaton, the new state of a cell can be expressed as a function of the number of adjacent cells that are in the alive state and of the cell's own state; that is, the rule is
221:
Also known as
Inkspot or Flakes. Cells that become alive never die. It combines chaotic growth with more structured ladder-like patterns that can be used to simulate arbitrary Boolean circuits.
198:
All patterns are phoenixes, meaning that every live cell immediately dies, and many patterns lead to explosive chaotic growth. However, some engineered patterns with complex behavior are known.
156:
There are 2 = 262,144 possible Life-like rules, only a small fraction of which have been studied in any detail. In the descriptions below, all rules are specified in Golly/RLE format.
92:
In the notation used by Mirek's
Cellebration, a rule is written as a string x/y where each of x and y is a sequence of distinct digits from 0 to 8, in numerical order. The presence of a digit
338:
Any automaton of the above form that contains the element B1 (e.g. B17/S78, or B145/S34) will always be explosive for any finite pattern: at any step, consider the cell (
208:
This rule supports a small self-replicating pattern which, when combined with a small glider pattern, causes the glider to bounce back and forth in a pseudorandom walk.
805:
1015:
Vichniac, Gérard Y. (1986), "Cellular automata models of disorder and organization", in
Bienenstock, E.; Fogelman Soulié, F.; Weisbuch, G. (eds.),
943:
830:
655:
584:
515:
294:
1303:
1085:
125:
141:
1072:, Collection Aléa-Saclay: Monographs and Texts in Statistical Physics, Cambridge University Press, Cambridge, pp. 37–38,
133:
38:
Each cell of the automaton has two states (conventionally referred to as "alive" and "dead", or alternatively "on" and "off")
335:, including some rules with B0 in which the background of the field of cells alternates between live and dead at each step.
108:
live neighbors becomes alive in the next generation. For instance, in this notation, Conway's Game of Life is denoted 23/3.
994:
401:
behave identically under rotation and reflection. There are 2≈5.07*10 rules of this kind, including outer-totalistic rules.
281:
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that behave differently in different directions. There are 2≈1.34*10 rules of this kind, including isotropic rules.
249:, until computer simulation found that larger patterns tend to explode. Has many small oscillators and spaceships.
193:
86:
72:
There are three standard notations for describing these rules, that are similar to each other but incompatible.
772:(October 1970), "Mathematical Games - The fantastic combinations of John Conway's new solitaire game "life"",
246:
229:
61:
28:
448:
Carter Bays has proposed a variety of generalizations of the Game of Life to three-dimensional CA defined on
1279:
1207:
Bays, Carter (2006), "A note about the discovery of many new rules for the game of three-dimensional life",
1135:
Evans, Kellie
Michele (2003), "Larger than Life: threshold-range scaling of Life's coherent structures",
322:
45:; it consists of the eight adjacent cells to the one under consideration and (possibly) the cell itself.
1227:
Bays, Carter (2007), "The discovery of glider guns in a game of life for the triangular tessellation",
1109:
Sapin, Emmanuel (2010), "Larger than Life: threshold-range scaling of Life's coherent structures", in
1144:
1038:
923:
485:
456:). Bays has also studied two-dimensional life-like CA with triangular or hexagonal neighborhoods.
1179:
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796:
633:
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42:
20:
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Pivato, Marcus (2007), "RealLife: the continuum limit of Larger than Life cellular automata",
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651:
580:
574:
511:
1247:
Bays, Carter (2005), "A note on the game of life in hexagonal and pentagonal tessellations",
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Named after
Stephen Morley; also called Move. Supports very high-period and slow spaceships.
185:'s replicating automaton: every pattern is eventually replaced by multiple copies of itself.
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Also called the twisted majority rule. Symmetric under on-off reversal. Approximates the
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Symmetric under on-off reversal. Has engineered patterns with highly complex behavior.
182:
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746:
573:
Wuensche, Andrew (2011), "16.10 The game-of-Life and other Life-like rules – rcode",
100:
live neighbors survives into the next generation of the pattern, and the presence of
672:
77:
904:
Johnston, Nathaniel (2010), "The B36/S125 "2x2" Life-Like
Cellular Automaton", in
1122:
1024:
935:
847:
823:
The
Recursive Universe: Cosmic Complexity and the Limits of Scientific Knowledge
688:
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350:-coordinate among cells that are on, and among such cells the one with minimum
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1000:
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Several more rules are listed and described in the MCell rule list and by
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1054:
863:
Gravner, Janko; Griffeath, David (1998), "Cellular automaton growth on
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453:
81:
148:
16:
Type of cellular automaton with similarities to Conway's Game of Life
1288:
124:
1068:
Chopard, Bastien; Droz, Michel (1998), "2.2.4 The annealing rule",
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638:
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140:
139:
131:
123:
1019:, NATO ASI Series, vol. 20, Springer-Verlag, pp. 3–20,
132:
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depend on the configuration of live cells in the neighborhood.
624:(2010), "Growth and decay in life-like cellular automata", in
476:; Packard, N. H. (1985), "Two-dimensional cellular automata",
286:
Similar to Life but with a small self-replicating pattern.
727:
Cellular Automata Machines: A New Environment for Modeling
1289:
Game of Life - Conway and Variants - Online Software Tool
35:
The array of cells of the automaton has two dimensions.
410:
often crashing and combining into even more objects.
245:
Was initially thought to be a stable alternative to
128:
Chaotic diamonds in the Diamoeba (B35678/S5678) rule
89:of characters that is more easily read by a human.
1272:"Totalistic Growth Rules with Moore Neighborhood"
60:This class of cellular automata is named for the
979:Day & Night - An Interesting Variant of Life
555:Cellular Automaton Rules Lexicon — Family: Life
325:on the boundaries between live and dead cells.
1070:Cellular automata modeling of physical systems
1017:Disordered Systems and Biological Organization
73:
576:Exploring Discrete Dynamics: The DDLAB manual
8:
104:in the y string means that a dead cell with
96:in the x string means that a live cell with
436:. They have been studied by Marcus Pivato.
1041:(1993), "Lava lamps in the 21st century",
568:
566:
1183:
962:HighLife - An Interesting Variant of Life
917:
880:
637:
547:
806:Winning Ways for your Mathematical Plays
671:Silverman, Brian, "Changing the Rules",
545:
543:
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527:
332:
158:
147:
136:Exploding chaos in the Seeds (B2/S) rule
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31:) if it meets the following criteria:
867:: theorems, examples, and problems",
677:, Mathematical Association of America
614:
612:
610:
608:
606:
604:
602:
600:
598:
596:
41:The neighborhood of each cell is the
7:
510:, Westview Press, pp. 211–249,
825:, Contemporary Books, p. 134,
579:, Luniver Press, pp. 145–146,
747:"Life without Death is P-complete"
725:(1987), "1.2 Animate-by-numbers",
27:(in the sense of being similar to
14:
508:Cellular Automata and Complexity
869:Advances in Applied Mathematics
116:the "S" stands for "survival".
1115:Game of Life Cellular Automata
910:Game of Life Cellular Automata
809:(2nd ed.), A K Peters Ltd
630:Game of Life Cellular Automata
478:Journal of Statistical Physics
399:Isotropic non-totalistic rules
144:Conway's Game of Life (B3/S23)
120:A selection of Life-like rules
1:
1278:, Department of Mathematics,
1157:10.1016/S0167-2789(03)00155-6
912:, Springer, pp. 99–114,
1229:Journal of Cellular Automata
1172:Theoretical Computer Science
821:Poundstone, William (1985),
632:, Springer, pp. 71–98,
354:-coordinate. Then the cell (
217:Life without Death
111:In the notation used by the
74:Wolfram & Packard (1985)
1276:The Primordial Soup Kitchen
1123:10.1007/978-1-84996-217-9_9
1025:10.1007/978-3-642-82657-3_1
936:10.1007/978-1-84996-217-9_7
691:collected by Jason Summers.
648:10.1007/978-1-84996-217-9_6
432:, not the discrete lattice
1320:
1194:10.1016/j.tcs.2006.11.019
729:, MIT Press, pp. 6–7
701:Nivasch, Gabriel (2007),
234:Highly complex behavior.
1304:Cellular automaton rules
1078:10.1017/CBO9780511549755
993:Morley, Stephen (2005),
375:topology. For example:
171:Description and sources
160:Notable Life-like rules
1280:University of Wisconsin
882:10.1006/aama.1998.0599
558:, Mirek's Cellebration
153:
145:
137:
129:
1039:Pickover, Clifford A.
704:The photon/XOR system
323:curve-shortening flow
152:Anneal (B4678/S35678)
151:
143:
135:
127:
29:Conway's Game of Life
1117:, pp. 135–165,
674:The Virtual Computer
1149:2003PhyD..183...45E
1043:The Visual Computer
928:2010golc.book...99J
797:Conway, John Horton
774:Scientific American
490:1985JSP....38..901P
346:) that has minimum
161:
1270:Griffeath, David,
1055:10.1007/bf01900906
741:Griffeath, David;
689:Patterns for Seeds
552:Wójtowicz, Mirek,
498:10.1007/BF01010423
159:
154:
146:
138:
130:
68:Notation for rules
52:(sometimes called
43:Moore neighborhood
21:cellular automaton
1111:Adamatzky, Andrew
945:978-1-84996-216-2
906:Adamatzky, Andrew
846:Eisenmann, Jack,
832:978-0-8092-5202-2
743:Moore, Cristopher
657:978-1-84996-216-2
626:Adamatzky, Andrew
586:978-1-905986-31-6
517:978-0-201-62664-3
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484:(5–6): 901–946,
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370:Generalizations
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295:Day & Night
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78:Wolfram code
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62:Game of Life
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417:structures.
407:Generations
179:Replicator
461:References
304:B368/S245
1137:Physica D
919:1203.1644
801:Guy, R.K.
780:: 120–123
757:: 437–447
639:0911.2890
391:isotropic
266:B36/S125
257:Diamoeba
25:Life-like
1298:Category
803:(2004),
745:(1996),
506:(1994),
421:RealLife
282:HighLife
278:B36/S23
242:34 Life
239:B34/S34
76:use the
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