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 2^{m} steps the triangular region outlined by the pattern contains altogether 4^{m} cells, but only 3^{m} 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).