elementary CA

Elementary cellular automaton

The simplest class of one-dimensional CA: two states (0/1), three-cell neighborhood (left, center, right).

Parameters: $S=2$ states, $N=3$ neighborhood size, Number of possible rules: $2^{2^3}=256$

Each cell examines its left neighbor, itself, and its right neighbor - giving $2^3=8$ possible neighborhood configurations:

111  110  101  100  011  010  001  000

Each configuration maps to a new state (0 or 1) for the center cell:

Rules are numbered using wolfram code, which encodes the rule’s behavior as a binary number read from the next-state outputs.
The $i$-th bit (from right) of the Wolfram code specifies the next state when the neighborhood configuration equals $i$ in binary:

rule 110

Current pattern	111	110	101	100	011	010	001	000
New state	0	1	1	0	1	1	1	0

Reading the next states as binary: 01101110 = $110_{10}$

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https://mathworld.wolfram.com/ElementaryCellularAutomaton.html
https://en.wikipedia.org/wiki/Rule_110

All 256 rules starting from a single black cell:

![[elementary CA-1762632781862.webp]]
![[elementary CA-1762632794076.webp]]
![[elementary CA-1762632805012.webp]]
![[elementary CA-1762632823043.webp]]
![[elementary CA-1762632658970.webp]]