Rule 30

Rule 30 is an elementary cellular automaton introduced by Stephen Wolfram in 1983.wikipedia
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Conus textile

cloth of gold cone
For instance, a pattern resembling Rule 30 appears on the shell of the widespread cone snail species Conus textile.
The color pattern of its shell resembles a cellular automaton named Rule 30.

Cellular automaton

cellular automataCACell games (cellular automaton)
Using Wolfram's classification scheme, Rule 30 is a Class III rule, displaying aperiodic, chaotic behaviour.
He published his first paper in Reviews of Modern Physics investigating elementary cellular automata (Rule 30 in particular) in June 1983.

Wolfram code

binary-decimal notationRule 37R
Rule 30 is so named because 30 is the smallest Wolfram code which describes its rule set (as described below).
Notable rules in this class include rule 30, rule 110, and rule 184.

Cambridge North railway station

Cambridge NorthCambridge Science Park railway stationCambridge Science Park station
The Cambridge North railway station is decorated with architectural panels displaying the evolution of Rule 30 (or equivalently under black-white reversal, Rule 135).
The cladding of the building features a pierced design derived from Rule 30, a cellular automaton introduced by Stephen Wolfram in 1983.

Rule 90

*Other elementary cellular automata: Rule 30, Rule 110, and Rule 184

Rule 184

cellular automaton 184
*Rule 30, Rule 90, and Rule 110, other one-dimensional cellular automata with different behavior

Elementary cellular automaton

elementary cellular automata
Rule 30 is an elementary cellular automaton introduced by Stephen Wolfram in 1983.

Stephen Wolfram

WolframS. WolframStephen
Rule 30 is an elementary cellular automaton introduced by Stephen Wolfram in 1983.

Chaos theory

chaoticchaoschaotic behavior
Using Wolfram's classification scheme, Rule 30 is a Class III rule, displaying aperiodic, chaotic behaviour.

Random number generation

random number generatorrandom numberrandom numbers
Rule 30 has also been used as a random number generator in Mathematica, and has also been proposed as a possible stream cipher for use in cryptography.

Wolfram Mathematica

MathematicaWolframwebMathematica
Rule 30 has also been used as a random number generator in Mathematica, and has also been proposed as a possible stream cipher for use in cryptography.

Stream cipher

stream cypherstream ciphersstream
Rule 30 has also been used as a random number generator in Mathematica, and has also been proposed as a possible stream cipher for use in cryptography.

Cryptography

cryptographiccryptographercryptology
Rule 30 has also been used as a random number generator in Mathematica, and has also been proposed as a possible stream cipher for use in cryptography.

Robert L. Devaney

Bob DevaneyDevaneyDevaney, Robert L.
Wolfram based his classification of Rule 30 as chaotic based primarily on its visual appearance, and it was later shown to meet more rigorous definitions of chaos proposed by Devaney and Knudson.

Butterfly effect

sensitive dependence on initial conditionsa long drawn out chain of eventsbutterfly
In particular, according to Devaney's criteria, Rule 30 displays sensitive dependence on initial conditions (two initial configurations that differ only in a small number of cells rapidly diverge), its periodic configurations are dense in the space of all configurations, according to the Cantor topology on the space of configurations (there is a periodic configuration with any finite pattern of cells), and it is mixing (for any two finite patterns of cells, there is a configuration containing one pattern that eventually leads to a configuration containing the other pattern).

Cantor space

Cantor topology2^{\omega}
In particular, according to Devaney's criteria, Rule 30 displays sensitive dependence on initial conditions (two initial configurations that differ only in a small number of cells rapidly diverge), its periodic configurations are dense in the space of all configurations, according to the Cantor topology on the space of configurations (there is a periodic configuration with any finite pattern of cells), and it is mixing (for any two finite patterns of cells, there is a configuration containing one pattern that eventually leads to a configuration containing the other pattern).

Mixing (mathematics)

mixingstrong mixingtopological mixing
In particular, according to Devaney's criteria, Rule 30 displays sensitive dependence on initial conditions (two initial configurations that differ only in a small number of cells rapidly diverge), its periodic configurations are dense in the space of all configurations, according to the Cantor topology on the space of configurations (there is a periodic configuration with any finite pattern of cells), and it is mixing (for any two finite patterns of cells, there is a configuration containing one pattern that eventually leads to a configuration containing the other pattern).

Pseudorandom number generator

pseudo-random number generatorPRNGpseudorandom
Stephen Wolfram proposed using its center column as a pseudorandom number generator (PRNG); it passes many standard tests for randomness, and Wolfram previously used this rule in the Mathematica product for creating random integers.

Matthew Cook

Cook, Matthew
A less trivial example, found by Matthew Cook, is any input pattern consisting of infinite repetitions of the pattern '00001000111000', with repetitions optionally being separated by six ones.

Chi-squared test

chi-square testchi-squared statisticChi-squared
Sipper and Tomassini have shown that as a random number generator Rule 30 exhibits poor behavior on a chi squared test when applied to all the rule columns as compared to other cellular automaton-based generators.

Conway's Game of Life

Game of LifeConway's LifeConway’s Game of Life
The design was described by its architect as inspired by Conway's Game of Life, a different cellular automaton studied by Cambridge mathematician John Horton Conway, but is not actually based on Life.

John Horton Conway

John H. ConwayJohn ConwayConway
The design was described by its architect as inspired by Conway's Game of Life, a different cellular automaton studied by Cambridge mathematician John Horton Conway, but is not actually based on Life.

Randomness tests

Randomness testtest for randomnesspractical tests for random-number generators
Stephen Wolfram used randomness tests on the output of Rule 30 to examine its potential for generating random numbers, though it was shown to have an effective key size far smaller than its actual size and to perform poorly on a chi-squared test.