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Is there a way to compute the limit of the ratio (number of black cells)/(number of white cells), in the rule 110 or rule 30 automaton? With initial state = 1 black cell.

Simulation of first 120000 rows shows a quite stable total density of 0.592..., and row-density 0.592...

Here is average density of some consecutive columns of height some thousands: How to explain the apparent periodicity? These are quite clearly converging, how to calculate the exact values? (0.62499..==5/8 ??)

0.6249983636387438, 0.5937438636452892, 0.5312544999934545, 0.5937569545353388, 0.624991818193719, 0.6249983636387438, 0.5937569545353388, 0.5312414091034049, 0.5937438636452892, 0.6250049090837686, 0.6249983636387438, 0.5937504090903141, 0.5312479545484298, 0.5937373182002644, 0.624991818193719, 0.6250049090837686, 0.5937569545353388, 0.5312479545484298, 0.5937438636452892, 0.6250049090837686, 0.6250114545287934, 0.5937504090903141, 0.5312479545484298, 0.5937504090903141, 0.6249983636387438, 0.6250114545287934, 0.5937504090903141, 0.5312414091034049, 0.5937634999803637, 0.6250114545287934, 0.6249983636387438, 0.5937438636452892

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    I assume you're only counting within a finite length of cells. Also, one might consider that the frequency might not even stabilize, and its asymptotic evolution will probably depend on choice of initial state.2012-01-20
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    New find, column density very stable, but differs alot between different columns!! Column density is periodic, with period 5, and takes on 3 different values.2012-01-21
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    These column densities appear to be 5/8, 19/32, and 17/32.2012-10-28

2 Answers 2