binomial
coefficient
$$\left( {\begin{array}{c} r \\ c \\ \end{array}} \right) = \frac{{r!}}{{c!\left( {r - c} \right)!}}$$
Binary Description of the Sierpinski Gasket
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:
To understand why Pascal's triangle mod 2 resembles the Sierpinski gasket, let $P_n$ denote the first $2^n$ rows of the triangle and consider what happens in going from $P_n$ to $P_{n+1}$.
$P_3$
Here are the first 8 rows of Pascal's triangle. Notice that it has the
identical pattern in each of the three corners that surround a black
triangle of even numbers.
$P_4$
Here we see that the three corners of $P_4$ contain identical copies
of $P_3$ each scaled by 1/2, surrounding a solid black triangle of
even numbers.
$P_5$
Again, the three corners of $P_5$ contain identical copies of
$P_4$ each scaled by 1/2, surrounding a solid black triangle of
even numbers.
$P_6$
The three corners of $P_6$ contain identical copies of
$P_5$ 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.
