Todd Rowland
Wolfram Research
Maryland
Registered: Oct 2003
Posts: 115 
phase transitions
That certainly is an interesting looking combination of what is called complex behavior and nested behavior.
Like other observations of prime number behavior, there are some things here which are not as extraordinary as they seem, while there are some interesting things happening.
It turns out that the complex region and the nested region have more to do with the rule, the absolute value of the differences, than anything else. A sequence of random odd numbers produces the same thing as the sequence of prime numbers.
It is an interesting rule, which I haven't seen before. It seems that a random initial condition leads to a rule 60 pattern with two values. Zero and the greatest common denominator. After one step, the sequence of odd numbers becomes all even, except for the first entry. The common denominator in the resulting region is 2, and it quickly becomes 0's and 2's, operating under exactly the Rule 60 elementary cellular automaton.
The greatest common denominator dominates, and that odd first entry, in this case a 1, creates a region which is all 0's and 1's. Note that the absolute difference between 1 and either 0 or 2 is always 1. That is what is on the diagonal, a sequence of 1's.
In this case, the region in the odd phase is nested because it is literally following Rule 60 from a single black cell. See p.58
Since Rule 60 is good at preserving the randomness of its initial conditions, it is probably the case that there are different phases to the absolute value of the differences, corresponding to common denominators. It also obeys a maximal principle, and is like a diffusion, though the number of particles is not conserved. See attached image, which starts with random multiples of 2, followed by random numbers, then by multipls of 3, all between 1 and 120.
One feature of the evolution that is related to the prime numbers is the size of the gaps which seem to correspond to the size of the multicolored features.
Incidentally, the convention is to display an evolution to a rule by showing the initial condition as the top row. In Mathematica code, your picture looks like Rest[NestList[Abs[Differences[Prepend[#,0]]]&, Array[Prime, 200],200]].
Todd Rowland has attached this image:
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