 TOPICS  # Rule 102 Rule 102 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 . This rule is illustrated above together with the evolution of a single black cell it produces after 15 steps (OEIS A075439; Wolfram 2002, p. 55).

Starting with a single black cell, successive generations are given by interpreting the numbers 1, 6, 20, 120, 272, 1632, 5440, 32640, ... (OEIS A117998) in binary. Omitting trailing zeros (the right cells in step of the triangle are always 0) gives 1, 3, 5, 15, 17, 51, 85, 255, 257, 771, ... (OEIS A001317), which are simply the previous numbers divided by and which have binary representation 1, 11; 101, 1111, 10001, ... (OEIS A047999). Surprisingly, this is precisely the Sierpiński sieve.

The mirror image is rule 60, the complement is rule 153, and the mirrored complement is rule 195. Rule 102 is one of the eight additive elementary cellular automata (Wolfram 2002, p. 953).

Additive Cellular Automaton, Elementary Cellular Automaton, Rule 30, Rule 50, Rule 54, Rule 60, Rule 62, Rule 90, Rule 94, Rule 110, Rule 126, Rule 150, Rule 158, Rule 182, Rule 188, Rule 190, Rule 220, Rule 222, Sierpiński Sieve

## Related Wolfram sites

http://atlas.wolfram.com/01/01/102/

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## References

Sloane, N. J. A. Sequences A001317/M2495, A047999, A075439, and A117998 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. 55, 90, and 953, 2002.

Rule 102

## Cite this as:

Weisstein, Eric W. "Rule 102." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Rule102.html