# Rule 30

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

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### 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 110

**Rule 110 cellular automaton**

### 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**

### 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.