Learning Processing: A Beginner's Guide to Programming Images ...

Mathematics 217
In 1975, Benoit Mandelbrot coined the term fractal to describe self-similar shapes found in nature.
Much of the stuff we encounter in our physical world can be described by idealized geometrical forms—a
postcard has a rectangular shape, a ping-pong ball is spherical, and so on. However, many naturally
occurring structures cannot be described by such simple means. Some examples are snowfl akes, trees,
coastlines, and mountains. Fractals provide a geometry for describing and simulating these types of
self-similar shapes (by “ self-similar ” we mean no matter how “ zoomed out ” or “ zoomed in, ” the shape
ultimately appears the same). One process for generating these shapes is known as recursion.
We know that a function can call another function. We do this whenever we call any function inside
of the draw( ) function. But can a function call itself? Can draw( ) call draw( ) ? In fact, it can (although
calling draw( ) from within draw( ) is a terrible example, since it would result in an infi nite loop).
Functions that call themselves are recursive and are appropriate for solving diff erent types of problems.
Th is occurs in mathematical calculations; the most common example of this is “ factorial. ”
Th e factorial of any number n , usually written as n !, is defi ned as:
n ! � n * n – 1 * . . . . * 3 * 2 * 1
0! � 1
We could write a function to calculate factorial using a for loop in Processing :
int factorial(int n) {
int f = 1;
for (int i = 0; i < n; i + + ) {
f = f * (i + 1);
}
return f;
}
If you look closely at how factorial works, however, you will notice something interesting. Let’s examine
4! and 3!
4! � 4 * 3 * 2 * 1
3! � 3 * 2 * 1
therefore. . . 4! � 4 * 3!
We can describe this in more general terms. For any positive integer n :
n ! � n * ( n – 1)!
1! � 1
Written in English:
Th e factorial of N is defi ned as N times the factorial of N – 1.