Chapter 2: The Crucial Experiment

Section 1: How Do Simple Programs Behave?

Self-similarity [of rule 90]

The pattern generated by rule 90 after a given number of steps has the property that it is identical to what one would get by going twice as many steps, and then keeping only every other row and column. After 2m steps the triangular region outlined by the pattern contains altogether 4m cells, but only 3m of these are black. In the limit of an infinite number of steps one gets a fractal known as a Sierpiński pattern (see page 934), with fractal dimension Log[2, 3] 1.59 (see page 933). Nesting occurs in all cellular automata with additive rules (see page 955).

Image Source Notebooks:

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