Jason Cawley
Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
Posts: 712 
Certainly, there are fine questions here. Since white triangles form in rule 30 below lines of white cells, you can simply look for blocks of white n cells long. Some facts about this behavior discussed in the book can be found in notes on page 871 and page 1127.
The first makes the claim that the probability of a block of white n cells long seems to go as 2^n, as you would expect if whites and blacks were independent and random. It also notes the first run of 10 whites occurs on step 67 from the simple initial, and length 20 first occurs at step 515.
The second note considers the step at which every block up to a given length has occurred at least once, instead, and gives the figures for lengths up to 10.
One might readily extend either question to all initials (parameterized by width, say) rather than the simplest initial condition, and push for much greater block lengths. Does the 2^n conjecture basically hold? How long do gaps between step numbers to get every type of block get, from n to n+1? We know special initials e.g. periodic for rule 30, can and do give atypical behaviors for the rule from most or random initials. Are there identifiable classes of initials that have atypical behavior in the size of white triangles that occur in them, or are nearly all initials similar in their "white run length spectrum"?
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