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Rule 28


ElementaryCARule28

Rule 28 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 28=00011100_2. This rule is illustrated above together with the evolution of a single black cell it produces after 15 steps (Wolfram 2002, p. 55).

The mirror image, complement, and mirror complement are rules 70, 199, and 157, respectively. For initial conditions consisting of a single black cell, it is equivalent to rule 156, since both have binary representations of the form x0011100_2. The mirror image, complement, and mirrored complement of rule 156 are rules 198, 198, and 156, respectively (in other words, rule 156 is invariant under the combined complementing and mirror imaging operations).

Starting with a single black cell, successive generations n=0, 1, ... are given by interpreting the Jacobsthal numbers J_n, namely 1, 3, 5, 11, 21, 43, 85, 171, 341, ... (OEIS A001045), in binary and ignoring leading zeros, namely 1, 11, 101, 1011, 10101, 101011, ... (OEIS A070909). Computation of the nth generation is therefore computationally reducible for an initial configuration consisting of a single black cell by computing

 a(n)=J_(n+2)=1/3[2^(n+2)-(-1)^n].

See also

Elementary Cellular Automaton, Rule 30, Rule 50, Rule 54, Rule 60, Rule 62, Rule 90, Rule 94, Rule 102, Rule 110, Rule 126, Rule 150, Rule 158, Rule 182, Rule 188, Rule 190, Rule 220, Rule 222

Related Wolfram sites

http://atlas.wolfram.com/01/01/28/

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References

Sloane, N. J. A. Sequences A001045/M2482 and A070909 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 952, 2002.

Referenced on Wolfram|Alpha

Rule 28

Cite this as:

Weisstein, Eric W. "Rule 28." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Rule28.html

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