Geometry of Binomial Coefficients

This note describes the geometrical pattern of zeroes and ones obtained by reducing modulo two each element of Pascal's triangle formed from binomial coefficients. When an infinite number of rows of Pascal's triangle are included, the limiting pattern is found to be "self-similar," and is characterized by a "fractal dimension" log2 3. Analysis of the pattern provides a simple derivation of the result that the number of even binomial coefficients in the nth row of Pascal's triangle is 2#1 (n), where #1 (n) is a function which gives the number of occurrences of the digit 1 in the binary representation of the integer n. 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." (See [1], [2], [3] for further details and references.) These systems have been investigated as simple mathematical models for natural processes (such as snowflake growth) which exhibit the phenomenon of "self-organization." The self-similarity of the patterns discussed below leads to self-similarity in the natural structures generated. Figure 1 shows the first few rows of Pascal's triangle, together with the figure obtained by reducing each element modulo two, and indicating ones by black squares and zeroes by white (blank) squares. Figure 2 gives sixty-four rows of Pascal's