Agnes Scott College
Larry Riddle, Agnes Scott College

 

binomial
coefficient
$$\left( {\begin{array}{*{20}{c}} r \\ c \\ \end{array}} \right) = \frac{{r!}}{{c!\left( {r - c} \right)!}}$$

Binary Description of the Sierpinski Gasket

Pascal's triangle mod 2 Does the figure to the right appear to be the Sierpinski gasket? Actually, it is a representation of Pascal's triangle (mod 2). The first 9 rows of Pascal's triangle consist of the following binomial coefficients:

Pascal's triangle

The figure represents the first 128 rows of Pascal's triangle. The coefficients in the triangle that are odd are displayed as red boxes. The coefficients in the triangle that are even are displayed as black boxes.

To understand why Pascal's triangle mod 2 resembles the Sierpinski gasket, let Pn denote the first 2n rows of the triangle and consider what happens in going from Pn to Pn+1.

8 rows of Pascal triangle
P3

Here are the first 8 rows of Pascal's triangle. Notice that it has the identical pattern in each of the three corners which surround a black triangle of even numbers.

16 rows of Pascal triangle
P4

Here we see that the three corners of P4 contain identical copies of P3 each scaled by 1/2, surrounding a solid black triangle of even numbers.

32 rows
P5

Again, the three corners of P5 contain identical copies of P4 each scaled by 1/2, surrounding a solid black triangle of even numbers.

64 rows
P6

The three corners of P6 contain identical copies of P5 each scaled by 1/2, surrounding a solid black triangle of even numbers.

The odd binomial coefficients form a discrete variant of the Sierpinski gasket in which the iterative process of going from one stage to the next is the same as with the gasket: make 3 copies, each scaled by 1/2, and translate to the appropriate corner. In the "limit", the Pascal triangle with an infinite number of rows would have the same self-similarity property as the Sierpinski gasket.

Here is a proof of the mathematical justification for this behavior.

References

  1. Fine, N.J. "Binomial coefficients modulo a prime," Mathematical Association of America Monthly, December 1947, 589-592.
  2. Peitgen, Heinz-Otto, Hartmut Jurgens and Dietmar Saupe. Fractals for the Classroom, Part One: Introduction to Fractals and Chaos, Springer-Verlag New York, Inc. 1990.
  3. Stewart, Ian. "Four encounters with Sierpinski's Gasket," Mathematical Intelligencer 17 (1995), No. 1, 52-64.
  4. Wolfram, Stephen. "Geometry of Binomial Coefficients," Mathematical Association of America Monthly, November 1984, 566-570.