In mathematics and computability theory, an elementary cellular automaton is a one-dimensional cellular automaton where there are two possible states (labeled 0 and 1) and the rule to determine the state of a cell in the next generation depends only on the current state of the cell and its two immediate neighbors. There is an elementary cellular automaton (rule 110, defined below) which is capable of universal computation, and as such it is one of the simplest possible models of computation.

There are 8 = 2³ possible configurations for a cell and its two immediate neighbors. The rule defining the cellular automaton must specify the resulting state for each of these possibilities so there are 256 = 2²³ possible elementary cellular automata. Stephen Wolfram proposed a scheme, known as the Wolfram code, to assign each rule a number from 0 to 255 which has become standard. Each possible current configuration is written in order, 111, 110, ..., 001, 000, and the resulting state for each of these configurations is written in the same order and interpreted as the binary representation of an integer. This number is taken to be the rule number of the automaton. For example, 110_d=01101110₂. So rule 110 is defined by the transition rule:

Although there are 256 possible rules, many of these are trivially equivalent to each other up to a simple transformation of the underlying geometry. The first such transformation is reflection through a vertical axis and the result of applying this transformation to a given rule is called the mirrored rule. These rules will exhibit the same behavior up to reflection through a vertical axis, and so are equivalent in a computational sense.

For example, if the definition of rule 110 is reflected through a vertical line, the following rule (rule 124) is obtained:

Rules which are the same as their mirrored rule are called amphichiral. Of the 256 elementary cellular automata, 64 are amphichiral.

The second such transformation is to exchange the roles of 0 and 1 in the definition. The result of applying this transformation to a given rule is called the complementary rule. For example, if this transformation is applied to rule 110, we get the following rule

and, after reordering, we discover that this is rule 137:

There are 16 rules which are the same as their complementary rules.

Finally, the previous two transformations can be applied successively to a rule to obtain the mirrored complementary rule. For example, the mirrored complementary rule of rule 110 is rule 193. There are 16 rules which are the same as their mirrored complementary rules.

Of the 256 elementary cellular automata, there are 88 which are inequivalent under these transformations.

It turns out that reflection and complementation are automorphisms of the monoid of one-dimensional cellular automata, as they both preserve composition.^[1]

One method used to study these automata is to follow its history with an initial state of all 0s except for a single cell with a 1. When the rule number is even (so that an input of 000 does not compute to a 1) it makes sense to interpret state at each time, t, as an integer expressed in binary, producing a sequence a(t) of integers. In many cases these sequences have simple, closed form expressions or have a generating function with a simple form. The following rules are notable:

These images depict space-time diagrams, in which each row of pixels shows the cells of the automaton at a single point in time, with time increasing downwards. They start with an initial automaton state in which a single cell, the pixel in the center of the top row of pixels, is in state 1 and all other cells are 0.

• Rule 0
• Rule 1
• Rule 2
• Rule 3
• Rule 4
• Rule 5
• Rule 6
• Rule 7
• Rule 8
• Rule 9
• Rule 10
• Rule 11
• Rule 12
• Rule 13
• Rule 14
• Rule 15
• Rule 16
• Rule 17
• Rule 18
• Rule 19
• Rule 20
• Rule 21
• Rule 22
• Rule 23
• Rule 24
• Rule 25
• Rule 26
• Rule 27
• Rule 28
• Rule 29
• Rule 30
• Rule 31
• Rule 32
• Rule 33
• Rule 34
• Rule 35
• Rule 36
• Rule 37
• Rule 38
• Rule 39
• Rule 40
• Rule 41
• Rule 42
• Rule 43
• Rule 44
• Rule 45
• Rule 46
• Rule 47
• Rule 48
• Rule 49
• Rule 50
• Rule 51
• Rule 52
• Rule 53
• Rule 54
• Rule 55
• Rule 56
• Rule 57
• Rule 58
• Rule 59
• Rule 60
• Rule 61
• Rule 62
• Rule 63
• Rule 64
• Rule 65
• Rule 66
• Rule 67
• Rule 68
• Rule 69
• Rule 70
• Rule 71
• Rule 72
• Rule 73
• Rule 74
• Rule 75
• Rule 76
• Rule 77
• Rule 78
• Rule 79
• Rule 80
• Rule 81
• Rule 82
• Rule 83
• Rule 84
• Rule 85
• Rule 86
• Rule 87
• Rule 88
• Rule 89
• Rule 90
• Rule 91
• Rule 92
• Rule 93
• Rule 94
• Rule 95
• Rule 96
• Rule 97
• Rule 98
• Rule 99

A second way to investigate the behavior of these automata is to examine its history starting with a random state. This behavior can be better understood in terms of Wolfram classes. Wolfram gives the following examples as typical rules of each class.^[3]

• Class 1: Cellular automata which rapidly converge to a uniform state. Examples are rules 0, 32, 160 and 232.
• Class 2: Cellular automata which rapidly converge to a repetitive or stable state. Examples are rules 4, 108, 218 and 250.
• Class 3: Cellular automata which appear to remain in a random state. Examples are rules 22, 30, 126, 150, 182.
• Class 4: Cellular automata which form areas of repetitive or stable states, but also form structures that interact with each other in complicated ways. An example is rule 110. Rule 110 has been shown to be capable of universal computation.^[4]

Each computed result is placed under that result's source creating a two-dimensional representation of the system's evolution.

In the following gallery, this evolution from random initial conditions is shown for each of the 88 inequivalent rules. Below each image is the rule number used to produce the image, and in brackets the rule numbers of equivalent rules produced by reflection or complementing are included, if they exist.^[5] As mentioned above, the reflected rule would produce a reflected image, while the complementary rule would produce an image with black and white swapped.

• Rule 0 (255)
• Rule 1 (127)
• Rule 2 (16, 191, 247)
• Rule 3 (17, 63, 119)
• Rule 4 (223)
• Rule 5 (95)
• Rule 6 (20, 159, 215)
• Rule 7 (21, 31, 87)
• Rule 8 (64, 239, 253)
• Rule 9 (65, 111, 125)
• Rule 10 (80, 175, 245)
• Rule 11 (47, 81, 117)
• Rule 12 (68, 207, 221)
• Rule 13 (69, 79, 93)
• Rule 14 (84, 143, 213)
• Rule 15 (85)
• Rule 18 (183)
• Rule 19 (55)
• Rule 22 (151)
• Rule 23
• Rule 24 (66, 189, 231)
• Rule 25 (61, 67, 103)
• Rule 26 (82, 167, 181)
• Rule 27 (39, 53, 83)
• Rule 28 (70, 157, 199)
• Rule 29 (71)
• Rule 30 (86, 135, 149)
• Rule 32 (251)
• Rule 33 (123)
• Rule 34 (48, 187, 243)
• Rule 35 (49, 59, 115)
• Rule 36 (219)
• Rule 37 (91)
• Rule 38 (52, 155, 211)
• Rule 40 (96, 235, 249)
• Rule 41 (97, 107, 121)
• Rule 42 (112, 171, 241)
• Rule 43 (113)
• Rule 44 (100, 203, 217)
• Rule 45 (75, 89, 101)
• Rule 46 (116, 139, 209)
• Rule 50 (179)
• Rule 51
• Rule 54 (147)
• Rule 56 (98, 185, 227)
• Rule 57 (99)
• Rule 58 (114, 163, 177)
• Rule 60 (102, 153, 195)
• Rule 62 (118, 131, 145)
• Rule 72 (237)
• Rule 73 (109)
• Rule 74 (88, 173, 229)
• Rule 76 (205)
• Rule 77
• Rule 78 (92, 141, 197)
• Rule 90 (165)
• Rule 94 (133)
• Rule 104 (233)
• Rule 105
• Rule 106 (120, 169, 225)
• Rule 108 (201)
• Rule 110 (124, 137, 193)
• Rule 122 (161)
• Rule 126 (129)
• Rule 128 (254)
• Rule 130 (144, 190, 246)
• Rule 132 (222)
• Rule 134 (148, 158, 214)
• Rule 136 (192, 238, 252)
• Rule 138 (174, 208, 244)
• Rule 140 (196, 206, 220)
• Rule 142 (212)
• Rule 146 (182)
• Rule 150
• Rule 152 (188, 194, 230)
• Rule 154 (166, 180, 210)
• Rule 156 (198)
• Rule 160 (250)
• Rule 162 (176, 186, 242)
• Rule 164 (218)
• Rule 168 (224, 234, 248)
• Rule 170 (240)
• Rule 172 (202, 216, 228)
• Rule 178
• Rule 184 (226)
• Rule 200 (236)
• Rule 204
• Rule 232