ABSTRACT
This chapter describes the geometrical pattern of zeroes and ones obtained by reducing modulo two each element of Pascal's triangle formed from binomial coefficients. Pascal's triangle modulo two appears in the analysis of the structures generated by the evolution of a class of systems known as "cellular automata". The self-similarity of the patterns discussed leads to self-similarity in the natural structures generated. This "self-similarity" continues down to the smallest triangles. At each stage, one upright triangle from the pattern could be magnified by one or more factors of two to obtain essentially the complete pattern. The limiting pattern obtained from Pascal's triangle modulo two is thus "self-similar" or "scale invariant," and may be considered to exhibit the same structure at all length scales. Many examples of other "self-similar" figures are given in. Successive rows in the triangle are generalized to planes in the pyramid, with each plane carrying a square grid of integers.
