Rule 90


Rule 90 is one of the elementary cellular automaton rules introduced by Stephen Wolfram in 1983 (Wolfram 1983, 2002). It specifies the next color in a cell, depending on its color and its immediate neighbors. Its rule outcomes are encoded in the binary representation 90=01011010_2. This rule is illustrated above together with the evolution of a single black cell it produces after 15 steps (Wolfram 2002, p. 55).

Starting with a single black cell, successive generations are given by interpreting the numbers 1, 5, 17, 85, 257, 1285, 4369, 21845, ... (OEIS A038183) in binary, namely as 1, 101, 10001, 1010101, 100000001, ... (OEIS A070886).

Rule 90 is amphichiral, and its complement is rule 165.


The fractal produced by this rule was described by Sierpiński in 1915 and appearing in Italian art from the 13th century (Wolfram 2002, p. 43). It is therefore also known as the Sierpiński sieve, Sierpiński gasket, or Sierpiński triangle. The binomial coefficient (m; n) mod 2 can be computed using the XOR operation n XOR m, making Pascal's triangle mod 2 very easy to construct. Moreover, coloring all odd numbers black and even numbers white in Pascal's triangle produces a Sierpiński sieve (Guy 1990; Wolfram 2002, p. 870).

Rule 90 animation

Rule 90 is one of the eight additive elementary cellular automata (Wolfram 2002, p. 952).

See also

Additive Cellular Automaton, Elementary Cellular Automaton, Rule 30, Rule 50, Rule 54, Rule 60, Rule 62, Rule 94, Rule 102, Rule 110, Rule 126, Rule 150, Rule 158, Rule 182, Rule 188, Rule 190, Rule 220, Rule 222, Sierpiński Sieve

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Guy, R. K. "The Second Strong Law of Small Numbers." Math. Mag. 63, 3-20, 1990.Sloane, N. J. A. Sequences A038183 and A070886 in "The On-Line Encyclopedia of Integer Sequences."Wolfram, S. "Statistical Mechanics of Cellular Automata." Rev. Mod. Phys. 55, 601-644, 1983.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 90, 55, 870, and 952, 2002.

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Rule 90

Cite this as:

Weisstein, Eric W. "Rule 90." From MathWorld--A Wolfram Web Resource.

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