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In [[mathematics]] and [[Computability theory (computer science)|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. As such it is one of the simplest possible models of computation. Nevertheless, there is an elementary cellular automaton ([[rule 110]], defined below) which is capable of [[Turing completeness|universal computation]].


==The numbering system==
There are 8 = 2<sup>3</sup> 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 = 2<sup>2<sup>3</sup></sup> 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<sub>d</sub>=96<sub>d</sub>+14<sub>d</sub> written in binary is 01101110<sub>2</sub>. So rule 110 is defined by the transition rule:


{| class="wikitable" style="text-align: center"
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|-
| 111
| 110
| 101
| 100
| 011
| 010
| 001
| 000
! current pattern
! P=(L,C,R)
|-
| 0
| 1
| 1
| 0
| 1
| 1
| 1
| 0
! new state for center cell
! N<sub>110<sub>d</sub></sub>=(C+R+C*R+L*C*R)%2
|}
 
==Reflections and complements==
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:
{| class="wikitable" style="text-align: center"
|-
| 111
| 110
| 101
| 100
| 011
| 010
| 001
| 000
! current pattern
| P=(L,C,R)
|-
| 0
| 1
| 1
| 1
| 1
| 1
| 0
| 0
! new state for center cell
| N<sub>112<sub>d</sub>+12<sub>d</sub>=124<sub>d</sub></sub>=(L+C+L*C+L*C*R)%2
|}
 
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, the following rule (rule 137) obtained:
{| class="wikitable" style="text-align: center"
|-
! current pattern
| 111
| 110
| 101
| 100
| 011
| 010
| 001
| 000
|-
! new state for center cell
| 1
| 0
| 0
| 0
| 1
| 0
| 0
| 1
|}
 
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 8 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.
 
==Single 1 histories==
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:
 
===Rule 28===
The sequence generated is 1, 3, 5, 11, 21, 43, 85, 171, ... {{OEIS|id=A001045 }}. This is the sequence of [[Jacobsthal numbers]] and has generating function
 
:<math>\frac{1+2x}{(1+x)(1-2x)}</math>.
 
It has the closed form expression
:<math>a(t) = (4\cdot 2^t-(-1)^t)/3</math>
 
Note that rule 156 generates the same sequence.
 
===Rule 50===
The sequence generated is 1, 5, 21, 85, 341, 1365, 5461, 21845, ... {{OEIS|id=A002450}}. This has generating function
:<math>\frac{1}{(1-x)(1-4x)}</math>.
 
It has the closed form expression
:<math>a(t) = (4\cdot 4^t-1)/3</math>.
 
Note that rules 58, 114, 122, 178, 186, 242 and 250 generate the same sequence.
 
===Rule 54===
The sequence generated is 1, 7, 17, 119, 273, 1911, 4369, 30583, ... {{OEIS|id=A118108}}. This has generating function
:<math>\frac{1+7x}{(1-x^2)(1-16x^2)}</math>.
 
It has the closed form expression
:<math>a(t) = (22\cdot 4^t-6(-4)^t-4+3(-1)^t)/15</math>.
 
===Rule 60===
The sequence generated is 1, 3, 5, 15, 17, 51, 85, 255, ... {{OEIS|id=A001317}}. This can be obtained by taking successive rows of [[Pascal's triangle]] modulo 2 and interpreting them as integers in binary, which can be graphically represented by a [[Sierpinski triangle]].
 
===Rule 90===
{{main|Rule 90}}
The sequence generated is 1, 5, 17, 85, 257, 1285, 4369, 21845, ... {{OEIS|id=A038183}}. This can be obtained by taking successive rows of [[Pascal's triangle]] modulo 2 and interpreting them as integers in base 4. Note that rules 18, 26, 82, 146, 154, 210 and 218 generate the same sequence.
 
===Rule 94===
The sequence generated is 1, 7, 27, 119, 427, 1879, 6827, 30039, ... {{OEIS|id=A118101}}. This can be expressed as
 
:<math>a(t)  =
\begin{cases}
  1, & \mbox{if }t = 0 \\
  7, & \mbox{if }t = 1 \\
  (1+5\cdot 4^n)/3  , & \mbox{if }t\mbox{ is even }>0 \\
  (10+11\cdot 4^n)/6 , & \mbox{if }t\mbox{ is odd }>1
\end{cases}
</math>.
 
This has generating function
 
:<math>\frac{(1+2x)(1+5x-16x^4)}{(1-x^2)(1-16x^2)}</math>.
 
===Rule 102===
The sequence generated is 1, 6, 20, 120, 272, 1632, 5440, 32640, ... {{OEIS|id=A117998}}. This is simply the sequence generated by rule 60 (which is its mirror rule) multiplied by successive powers of 2.
 
===Rule 110===
{{main|Rule 110}}
 
===Rule 150===
The sequence generated is 1, 7, 21, 107, 273, 1911, 5189, 28123, ... {{OEIS|id=A038184}}. This can be obtained by taking the coefficients of the successive powers of (1+''x''+''x''<sup>2</sup>) modulo 2 and interpreting them as integers in binary.
 
===Rule 158===
The sequence generated is 1, 7, 29, 115, 477, 1843, 7645, 29491, ... {{OEIS|id=A118171}}. This has generating function
 
:<math>\frac{1+7x+12x^2-4x^3}{(1-x^2)(1-16x^2)}</math>.
 
===Rule 188===
The sequence generated is 1, 3, 5, 15, 29, 55, 93, 247, ... {{OEIS|id=A118173}}. This has generating function
 
:<math>\frac{1+3x+4x^2+12x^3+8x^4-8x^5}{(1-x^2)(1-16x^4)}</math>.
 
===Rule 190===
The sequence generated is 1, 7, 29, 119, 477, 1911, 7645, 30583, ... {{OEIS|id=A037576}}. This has generating function
 
:<math>\frac{1+3x}{(1-x^2)(1-4x)}</math>.
 
===Rule 220===
The sequence generated is 1, 3, 7, 15, 31, 63, 127, 255, ... {{OEIS|id=A000225}}. This is the sequence of [[Mersenne numbers]] and has generating function
 
:<math>\frac{1}{(1-x)(1-2x)}</math>.
 
It has the closed form expression
:<math>a(t) = 2\cdot 2^t-1</math>.
Note that rule 252 generates the same sequence.
 
===Rule 222===
The sequence generated is 1, 7, 31, 127, 511, 2047, 8191, 32767, ... {{OEIS|id=A083420}}. This is every other entry in the sequence of [[Mersenne numbers]] and has generating function
 
:<math>\frac{1+2x}{(1-x)(1-4x)}</math>.
 
It has the closed form expression
:<math>a(t) = 2\cdot 4^t-1</math>.
 
Note that rule 254 generates the same sequence.
 
<!--
===Images for rules 0-99===
 
These start with a single pixel.
 
<gallery widths="100px" heights="100px">
Image:WolframRule0.png|Rule 0
Image:WolframRule1.png|Rule 1
Image:WolframRule2.png|Rule 2
Image:WolframRule3.png|Rule 3
Image:WolframRule4.png|Rule 4
Image:WolframRule5.png|Rule 5
Image:WolframRule6.png|Rule 6
Image:WolframRule7.png|Rule 7
Image:WolframRule8.png|Rule 8
Image:WolframRule9.png|Rule 9
Image:WolframRule10.png|Rule 10
Image:WolframRule11.png|Rule 11
Image:WolframRule12.png|Rule 12
Image:WolframRule13.png|Rule 13
Image:WolframRule14.png|Rule 14
Image:WolframRule15.png|Rule 15
Image:WolframRule16.png|Rule 16
Image:WolframRule17.png|Rule 17
Image:WolframRule18.png|Rule 18
Image:WolframRule19.png|Rule 19
Image:WolframRule20.png|Rule 20
Image:WolframRule21.png|Rule 21
Image:WolframRule22.png|Rule 22
Image:WolframRule23.png|Rule 23
Image:WolframRule24.png|Rule 24
Image:WolframRule25.png|Rule 25
Image:WolframRule26.png|Rule 26
Image:WolframRule27.png|Rule 27
Image:WolframRule28.png|Rule 28
Image:WolframRule29.png|Rule 29
Image:WolframRule30.png|Rule 30
Image:WolframRule31.png|Rule 31
Image:WolframRule32.png|Rule 32
Image:WolframRule33.png|Rule 33
Image:WolframRule34.png|Rule 34
Image:WolframRule35.png|Rule 35
Image:WolframRule36.png|Rule 36
Image:WolframRule37.png|Rule 37
Image:WolframRule38.png|Rule 38
Image:WolframRule39.png|Rule 39
Image:WolframRule40.png|Rule 40
Image:WolframRule41.png|Rule 41
Image:WolframRule42.png|Rule 42
Image:WolframRule43.png|Rule 43
Image:WolframRule44.png|Rule 44
Image:WolframRule45.png|Rule 45
Image:WolframRule46.png|Rule 46
Image:WolframRule47.png|Rule 47
Image:WolframRule48.png|Rule 48
Image:WolframRule49.png|Rule 49
Image:WolframRule50.png|Rule 50
Image:WolframRule51.png|Rule 51
Image:WolframRule52.png|Rule 52
Image:WolframRule53.png|Rule 53
Image:WolframRule54.png|Rule 54
Image:WolframRule55.png|Rule 55
Image:WolframRule56.png|Rule 56
Image:WolframRule57.png|Rule 57
Image:WolframRule58.png|Rule 58
Image:WolframRule59.png|Rule 59
Image:WolframRule60.png|Rule 60
Image:WolframRule61.png|Rule 61
Image:WolframRule62.png|Rule 62
Image:WolframRule63.png|Rule 63
Image:WolframRule64.png|Rule 64
Image:WolframRule65.png|Rule 65
Image:WolframRule66.png|Rule 66
Image:WolframRule67.png|Rule 67
Image:WolframRule68.png|Rule 68
Image:WolframRule69.png|Rule 69
Image:WolframRule70.png|Rule 70
Image:WolframRule71.png|Rule 71
Image:WolframRule72.png|Rule 72
Image:WolframRule73.png|Rule 73
Image:WolframRule74.png|Rule 74
Image:WolframRule75.png|Rule 75
Image:WolframRule76.png|Rule 76
Image:WolframRule77.png|Rule 77
Image:WolframRule78.png|Rule 78
Image:WolframRule79.png|Rule 79
Image:WolframRule80.png|Rule 80
Image:WolframRule81.png|Rule 81
Image:WolframRule82.png|Rule 82
Image:WolframRule83.png|Rule 83
Image:WolframRule84.png|Rule 84
Image:WolframRule85.png|Rule 85
Image:WolframRule86.png|Rule 86
Image:WolframRule87.png|Rule 87
Image:WolframRule88.png|Rule 88
Image:WolframRule89.png|Rule 89
Image:WolframRule90.png|Rule 90
Image:WolframRule91.png|Rule 91
Image:WolframRule92.png|Rule 92
Image:WolframRule93.png|Rule 93
Image:WolframRule94.png|Rule 94
Image:WolframRule95.png|Rule 95
Image:WolframRule96.png|Rule 96
Image:WolframRule97.png|Rule 97
Image:WolframRule98.png|Rule 98
Image:WolframRule99.png|Rule 99
</gallery>
-->
<!--
<gallery widths="100px" heights="100px">
Image:WolframRule100.png|Rule 100
Image:WolframRule101.png|Rule 101
Image:WolframRule102.png|Rule 102
Image:WolframRule103.png|Rule 103
Image:WolframRule104.png|Rule 104
Image:WolframRule105.png|Rule 105
Image:WolframRule106.png|Rule 106
Image:WolframRule107.png|Rule 107
Image:WolframRule108.png|Rule 108
Image:WolframRule109.png|Rule 109
Image:WolframRule110.png|Rule 110
Image:WolframRule111.png|Rule 111
Image:WolframRule112.png|Rule 112
Image:WolframRule113.png|Rule 113
Image:WolframRule114.png|Rule 114
Image:WolframRule115.png|Rule 115
Image:WolframRule116.png|Rule 116
Image:WolframRule117.png|Rule 117
Image:WolframRule118.png|Rule 118
Image:WolframRule119.png|Rule 119
Image:WolframRule120.png|Rule 120
Image:WolframRule121.png|Rule 121
Image:WolframRule122.png|Rule 122
Image:WolframRule123.png|Rule 123
Image:WolframRule124.png|Rule 124
Image:WolframRule125.png|Rule 125
Image:WolframRule126.png|Rule 126
Image:WolframRule127.png|Rule 127
Image:WolframRule128.png|Rule 128
Image:WolframRule129.png|Rule 129
Image:WolframRule130.png|Rule 130
Image:WolframRule131.png|Rule 131
Image:WolframRule132.png|Rule 132
Image:WolframRule133.png|Rule 133
Image:WolframRule134.png|Rule 134
Image:WolframRule135.png|Rule 135
Image:WolframRule136.png|Rule 136
Image:WolframRule137.png|Rule 137
Image:WolframRule138.png|Rule 138
Image:WolframRule139.png|Rule 139
Image:WolframRule140.png|Rule 140
Image:WolframRule141.png|Rule 141
Image:WolframRule142.png|Rule 142
Image:WolframRule143.png|Rule 143
Image:WolframRule144.png|Rule 144
Image:WolframRule145.png|Rule 145
Image:WolframRule146.png|Rule 146
Image:WolframRule147.png|Rule 147
Image:WolframRule148.png|Rule 148
Image:WolframRule149.png|Rule 149
Image:WolframRule150.png|Rule 150
Image:WolframRule151.png|Rule 151
Image:WolframRule152.png|Rule 152
Image:WolframRule153.png|Rule 153
Image:WolframRule154.png|Rule 154
Image:WolframRule155.png|Rule 155
Image:WolframRule156.png|Rule 156
Image:WolframRule157.png|Rule 157
Image:WolframRule158.png|Rule 158
Image:WolframRule159.png|Rule 159
Image:WolframRule160.png|Rule 160
Image:WolframRule161.png|Rule 161
Image:WolframRule162.png|Rule 162
Image:WolframRule163.png|Rule 163
Image:WolframRule164.png|Rule 164
Image:WolframRule165.png|Rule 165
Image:WolframRule166.png|Rule 166
Image:WolframRule167.png|Rule 167
Image:WolframRule168.png|Rule 168
Image:WolframRule169.png|Rule 169
Image:WolframRule170.png|Rule 170
Image:WolframRule171.png|Rule 171
Image:WolframRule172.png|Rule 172
Image:WolframRule173.png|Rule 173
Image:WolframRule174.png|Rule 174
Image:WolframRule175.png|Rule 175
Image:WolframRule176.png|Rule 176
Image:WolframRule177.png|Rule 177
Image:WolframRule178.png|Rule 178
Image:WolframRule179.png|Rule 179
Image:WolframRule180.png|Rule 180
Image:WolframRule181.png|Rule 181
Image:WolframRule182.png|Rule 182
Image:WolframRule183.png|Rule 183
Image:WolframRule184.png|Rule 184
Image:WolframRule185.png|Rule 185
Image:WolframRule186.png|Rule 186
Image:WolframRule187.png|Rule 187
Image:WolframRule188.png|Rule 188
Image:WolframRule189.png|Rule 189
Image:WolframRule190.png|Rule 190
Image:WolframRule191.png|Rule 191
Image:WolframRule192.png|Rule 192
Image:WolframRule193.png|Rule 193
Image:WolframRule194.png|Rule 194
Image:WolframRule195.png|Rule 195
Image:WolframRule196.png|Rule 196
Image:WolframRule197.png|Rule 197
Image:WolframRule198.png|Rule 198
Image:WolframRule199.png|Rule 199
</gallery>
 
<gallery widths="100px" heights="100px">
Image:WolframRule200.png|Rule 200
Image:WolframRule201.png|Rule 201
Image:WolframRule202.png|Rule 202
Image:WolframRule203.png|Rule 203
Image:WolframRule204.png|Rule 204
Image:WolframRule205.png|Rule 205
Image:WolframRule206.png|Rule 206
Image:WolframRule207.png|Rule 207
Image:WolframRule208.png|Rule 208
Image:WolframRule209.png|Rule 209
Image:WolframRule210.png|Rule 210
Image:WolframRule211.png|Rule 211
Image:WolframRule212.png|Rule 212
Image:WolframRule213.png|Rule 213
Image:WolframRule214.png|Rule 214
Image:WolframRule215.png|Rule 215
Image:WolframRule216.png|Rule 216
Image:WolframRule217.png|Rule 217
Image:WolframRule218.png|Rule 218
Image:WolframRule219.png|Rule 219
Image:WolframRule220.png|Rule 220
Image:WolframRule221.png|Rule 221
Image:WolframRule222.png|Rule 222
Image:WolframRule223.png|Rule 223
Image:WolframRule224.png|Rule 224
Image:WolframRule225.png|Rule 225
Image:WolframRule226.png|Rule 226
Image:WolframRule227.png|Rule 227
Image:WolframRule228.png|Rule 228
Image:WolframRule229.png|Rule 229
Image:WolframRule230.png|Rule 230
Image:WolframRule231.png|Rule 231
Image:WolframRule232.png|Rule 232
Image:WolframRule233.png|Rule 233
Image:WolframRule234.png|Rule 234
Image:WolframRule235.png|Rule 235
Image:WolframRule236.png|Rule 236
Image:WolframRule237.png|Rule 237
Image:WolframRule238.png|Rule 238
Image:WolframRule239.png|Rule 239
Image:WolframRule240.png|Rule 240
Image:WolframRule241.png|Rule 241
Image:WolframRule242.png|Rule 242
Image:WolframRule243.png|Rule 243
Image:WolframRule244.png|Rule 244
Image:WolframRule245.png|Rule 245
Image:WolframRule246.png|Rule 246
Image:WolframRule247.png|Rule 247
Image:WolframRule248.png|Rule 248
Image:WolframRule249.png|Rule 249
Image:WolframRule250.png|Rule 250
Image:WolframRule251.png|Rule 251
Image:WolframRule252.png|Rule 252
Image:WolframRule253.png|Rule 253
Image:WolframRule254.png|Rule 254
Image:WolframRule255.png|Rule 255
</gallery>
-->
<!-- NB. Mathworld has formulas for rules 220 and 222 interchanged  -->
 
==Random initial state==
<!-- [[Rule 34]] links to this section -->
 
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.<ref>Stephan Wolfram, ''A New Kind of Science'' p223 ff.</ref>
* 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.<ref>[http://www30.wolframalpha.com/input/?i=rule+110 Rule 110 - Wolfram|Alpha]</ref>
 
Each computed result is placed under that results' source creating a two-dimensional representation of the system's evolution.  The 88 inequivalent rules are as follows, evolved from random initial conditions:
<gallery widths="100px" heights="101px">
Image:Rule0rand.png|Rule 0
Image:Rule1rand.png|Rule 1
Image:Rule2rand.png|Rule 2
Image:Rule3rand.png|Rule 3
Image:Rule4rand.png|Rule 4
Image:Rule5rand.png|Rule 5
Image:Rule6rand.png|Rule 6
Image:Rule7rand.png|Rule 7
Image:Rule8rand.png|Rule 8
Image:Rule9rand.png|Rule 9
Image:Rule10rand.png|Rule 10
Image:Rule11rand.png|Rule 11
Image:Rule12rand.png|Rule 12
Image:Rule13rand.png|Rule 13
Image:Rule14rand.png|Rule 14
Image:Rule15rand.png|Rule 15
Image:Rule18rand.png|Rule 18
Image:Rule19rand.png|Rule 19
Image:Rule22rand.png|Rule 22
Image:Rule23rand.png|Rule 23
Image:Rule24rand.png|Rule 24
Image:Rule25rand.png|Rule 25
Image:Rule26rand.png|Rule 26
Image:Rule27rand.png|Rule 27
Image:Rule28rand.png|Rule 28
Image:Rule29rand.png|Rule 29
Image:Rule30rand.png|[[Rule 30]]
Image:Rule32rand.png|Rule 32
Image:Rule33rand.png|Rule 33
Image:Rule34rand.png|Rule 34
Image:Rule35rand.png|Rule 35
Image:Rule36rand.png|Rule 36
Image:Rule37rand.png|Rule 37
Image:Rule38rand.png|Rule 38
Image:Rule40rand.png|Rule 40
Image:Rule41rand.png|Rule 41
Image:Rule42rand.png|Rule 42
Image:Rule43rand.png|Rule 43
Image:Rule44rand.png|Rule 44
Image:Rule45rand.png|Rule 45
Image:Rule46rand.png|Rule 46
Image:Rule50rand.png|Rule 50
Image:Rule51rand.png|Rule 51
Image:Rule54rand.png|Rule 54
Image:Rule56rand.png|Rule 56
Image:Rule57rand.png|Rule 57
Image:Rule58rand.png|Rule 58
Image:Rule60rand.png|Rule 60
Image:Rule62rand.png|Rule 62
Image:Rule72rand.png|Rule 72
Image:Rule73rand.png|Rule 73
Image:Rule74rand.png|Rule 74
Image:Rule76rand.png|Rule 76
Image:Rule77rand.png|Rule 77
Image:Rule78rand.png|Rule 78
Image:Rule90rand.png|[[Rule 90]]
Image:Rule94rand.png|Rule 94
Image:Rule104rand.png|Rule 104
Image:Rule105rand.png|Rule 105
Image:Rule106rand.png|Rule 106
Image:Rule108rand.png|Rule 108
Image:Rule110rand.png|[[Rule 110]]
Image:Rule122rand.png|Rule 122
Image:Rule126rand.png|Rule 126
Image:Rule128rand.png|Rule 128
Image:Rule130rand.png|Rule 130
Image:Rule132rand.png|Rule 132
Image:Rule134rand.png|Rule 134
Image:Rule136rand.png|Rule 136
Image:Rule138rand.png|Rule 138
Image:Rule140rand.png|Rule 140
Image:Rule142rand.png|Rule 142
Image:Rule146rand.png|Rule 146
Image:Rule150rand.png|Rule 150
Image:Rule152rand.png|Rule 152
Image:Rule154rand.png|Rule 154
Image:Rule156rand.png|Rule 156
Image:Rule160rand.png|Rule 160
Image:Rule162rand.png|Rule 162
Image:Rule164rand.png|Rule 164
Image:Rule168rand.png|Rule 168
Image:Rule170rand.png|Rule 170
Image:Rule172rand.png|Rule 172
Image:Rule178rand.png|Rule 178
Image:Rule184rand.png|[[Rule 184]]
Image:Rule200rand.png|Rule 200
Image:Rule204rand.png|Rule 204
Image:Rule232rand.png|Rule 232
</gallery>
 
===Unusual cases===
In some cases the behavior of a cellular automaton is not immediately obvious. For example, for Rule 62, interacting structures develop as in a Class 4. But in these interactions at least one of the structures is annihilated so the automaton eventually enters a repetitive state and the cellular automaton is Class 2.<ref>[http://www30.wolframalpha.com/input/?i=rule+62 Rule 62 - Wolfram|Alpha]</ref>
 
Rule 73 is Class 2<ref>[http://www30.wolframalpha.com/input/?i=rule+73 Rule 73 - Wolfram|Alpha]</ref> because any time there are two consecutive 1s surrounded by 0s, this feature is preserved in succeeding generations. This effectively creates walls which block the flow of information between different parts of the array. There are a finite number of possible configurations in the section between two walls so the automaton must eventually start repeating inside each section, though the period may be very long if the section is wide enough. These walls will form with probability 1 for completely random initial conditions. However, if the condition is added that the lengths of runs of consecutive 0s or 1s must always be odd, then the automaton displays Class 3 behavior since the walls can never form.
 
Rule 54 is Class 4,<ref>[http://www30.wolframalpha.com/input/?i=rule+54 Rule 54 - Wolfram|Alpha]</ref> but it remains unknown whether it is capable of universal computation. Interacting structures form, but structures that are useful for computation have yet to be found.<ref>''A New Kind of Science'' p697</ref>
 
==Summed or averaged histories==
A third way to investigate the behavior of these automata is to examine the summed or averaged histories starting from a given state over all 256 rules.<ref>[[Alex Wissner-Gross|A. D. Wissner-Gross]], "[http://www.alexwg.org/link?url=http%3A%2F%2Fwww.alexwg.org%2Fpublications%2FJCellAuto_4-27.pdf Pattern formation without favored local interactions]", ''Journal of Cellular Automata'' 4, 27-36 (2008).</ref> Surprisingly, despite the fact that no local interaction is favored by such a sum or average, rich patterns still emerge.
 
==References==
*{{MathWorld|title=Elementary Cellular Automaton|urlname=ElementaryCellularAutomaton}}
*{{MathWorld|title=Rule 30|urlname=Rule30}}
*{{MathWorld|title=Rule 50|urlname=Rule50}}
*{{MathWorld|title=Rule 54|urlname=Rule54}}
*{{MathWorld|title=Rule 60|urlname=Rule60}}
*{{MathWorld|title=Rule 62|urlname=Rule62}}
*{{MathWorld|title=Rule 90|urlname=Rule90}}
*{{MathWorld|title=Rule 94|urlname=Rule94}}
*{{MathWorld|title=Rule 102|urlname=Rule102}}
*{{MathWorld|title=Rule 110|urlname=Rule110}}
*{{MathWorld|title=Rule 126|urlname=Rule126}}
*{{MathWorld|title=Rule 150|urlname=Rule150}}
*{{MathWorld|title=Rule 158|urlname=Rule158}}
*{{MathWorld|title=Rule 182|urlname=Rule182}}
*{{MathWorld|title=Rule 188|urlname=Rule188}}
*{{MathWorld|title=Rule 190|urlname=Rule190}}
*{{MathWorld|title=Rule 220|urlname=Rule220}}
*{{MathWorld|title=Rule 222|urlname=Rule222}}
 
{{reflist}}
 
==External links==
{{Commons category|Elementary cellular automata}}
* [http://atlas.wolfram.com/01/01 "Elementary Cellular Automata" at the ''Wolfram Atlas of Simple Programs'']
* [http://ilmoeuro.ilmainenwebhotelli.com/eca/ Elementary cellular automaton demonstration (requires JavaScript and a modern browser)]
* [http://www.pouet.net/prod.php?which=60478 32 bytes long MS-DOS executable drawing by cellular automaton] ([[Rule 110]] by default)
 
[[Category:Cellular automata]]

Latest revision as of 17:12, 13 May 2014


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