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When the initial density of black cells has any value less than 50%, only white stripes ever survive. … When the initial density of black cells is less than 50%, all regions of black eventually disappear, and the system becomes completely white. … If the initial density of black cells is less than 50%, then only white stripes ultimately survive.

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All patterns that repeat in 10 or less steps under evolution according to rule 30.

And unless some digits effectively never matter, this process cannot normally take less steps than there are digits in its input.
… The pictures on the right show how colors of particular cells in these patterns can be found with much less computational effort.

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The behavior of the code 20 cellular automaton from the top of the facing page for all initial conditions with black cells in a region of size less than nine.

If 2r is less than q , the digit generated at that step is 0, and r is replaced by 2r . … With this procedure, the value of r is always less than q .

And in a 10×10 array the chance of finding a pattern where the fraction of squares that violate the constraints is even less than 50% is only one in a thousand, while the chance of finding a pattern where the fraction is less than 25% is one in four trillion.

But as the arrangement of particles in the system becomes more random, this attempt becomes less and less successful.

This simple rule produces randomness through the mechanism of intrinsic randomness generation, and this randomness in turn leads to a pattern of growth that takes on an increasingly smooth more-or-less circular form.

Young children, animals and the insane are typically held less responsible for their actions. And in a moral theory of law this can be understood in my approach as a consequence of the computations they do being less sophisticated—so that their outcome is less free of the environment and of their underlying rules. (In a pragmatic theory the explanation would presumably be that less sophisticated computations would not be up to the task of handling the elaborate system of incentives that laws had defined.)

The underlying density of particles is approximately 1 per cell, or 1/6 the maximum possible—a density which more or less minimizes the viscosity of the fluid.