Cellular automaton

A two-dimensional cyclic cellular automaton with 16 states, on a 300×300 torus, after 400 steps starting from a random initial configuration. At this stage in the automaton's evolution, three different types of pattern can be seen: random fields similar to the starting configuration, large blocks of color, and spirals. Eventually the spirals will take over the entire field.

A cellular automaton is a 4-tuple $\left(Z,S,N,f\right)$, where:

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} is a lattice.
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a finite set of cell states or values.
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is the finite neighborhood.
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is the transition function.

The lattice is a grid filled with cells, each cell is uniquely identified by its coordinates and value. The cellular automaton evolves via the transition function with respect to the discrete time variable.

The Rule 184 cellular automaton, run for 128 steps each from three random initial confgurations with density 25%, 50%, and 75%.

Lattice-Gas Cellular Automatons can be used to model physical systems such as fluid and biological systems. Mathematica uses a cellular automatons as a random number generator.

Further reading

Cellular Automaton Fluids: Basic Theory, Stephen Wolfram, 1986.

Random Sequence Generation by Cellular Automata, Stephen Wolfram, 1986.

Thermodynamics and Hydrodynamics of Cellular Automata, Stephen Wolfram, 1985.

Cryptography with Cellular Automata, Stephen Wolfram, 1986.