Notes

Chapter 3: The World of Simple Programs

Section 2: More Cellular Automata


Rule 150

This rule can be viewed as an analog of rule 90 in which the values of three cells, rather than two, are added modulo 2. Corresponding to the result on page 870 for rule 90, the number of black cells at row t in the pattern from rule 150 is given by

Apply[Times,Map[(2^#+2 - (-1)^#+2)/3 &, Cases[Split[IntegerDigits[t, 2]], k:{1 ..}:>Length[k]]]]

There are a total of 2^m Fibonacci[m+2] black cells in the pattern obtained up to step 2^m, implying fractal dimension Log[2, 1 + Sqrt[5]]. (See also page 956.)

The value at step t in the column immediately adjacent to the center is the nested sequence discussed on page 892 and given by Mod[IntegerExponent[t,2], 2]. The cell at position n on row t turns out to be given by Mod[GegenbauerC[n, -t, -1/2],2], as discussed on page 612.

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