Conserved Quantities and Continuum Phenomena

Reversibility is one general feature that appears to exist in the basic laws of physics. Another is conservation of various quantities—so that for example in the evolution of any closed physical system, total values of quantities like energy and electric charge appear always to stay the same.

With most rules, systems like cellular automata do not usually exhibit such conservation laws. But just as with reversibility, it turns out to be possible to find rules that for example conserve the total number of black cells appearing on each step.

Among elementary cellular automata with just two colors and nearest-neighbor rules, the only types of examples are the fairly trivial ones shown in the pictures below.

Elementary cellular automata whose evolution conserves the total number of black cells. The behavior of the rules shown here is simple enough that in each case it is fairly obvious how the number of black cells manages to stay the same on every step.

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From Stephen Wolfram: A New Kind of Science [citation]