Chapter 9: Fundamental Physics

Section 3: Irreversibility and the Second Law of Thermodynamics

Periods in rule 37R

With a system of size n, the maximum possible repetition period is 22n. In actuality, however, the periods are considerably shorter. With all cells 0 on one step, and a block of nonzero cells on the next step, the periods are for example: {1}: 21; {1,1}: 3n-8; {1,0,1}: 666; {1,1,1}: 3n-8; {1,0,0,1}: irregular (<24n; peaks at 6j+1); {1,0,0,1,0,1}: irregular (≲2n; 857727 for n=26; 13705406 for n=100). With completely random initial conditions, there are great fluctuations, but a typical period is around 2n/3.

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