How are fractals made?

Physicist: There are a lot of ways.

A fractal is just a structure that stays interesting no matter how far you zoom in. More than that, generally you can’t tell how far you’ve zoomed in (it looks the same at many different size scales). Making up new fractals is surprisingly easy.

The most famous one (that you’ll find on posters in dorm rooms) is the “Mandelbrot set”, followed by the “Sierpinski Triangle”. I haven’t taken a survey, but these seem to be the most popular.

Sierpinski Triangle.

It’s pretty straightforward how the Sierpinski triangle is made: make a triangle, put three triangles in its corners, put three triangles in those triangle’s corners, etc. This sort of idea: take a shape and then populate it with smaller versions of itself, is arguably the most common way to generate fractals.

The Mandelbrot set on the other hand involves complex numbers and computing power that wasn’t available until the ’80s.

The Mandelbrot set is the black region. When you zoom in on the edge of the black area you'll find that the same patterns show up over and over no matter how far you zoom in.

To determine if a point is in the Mandelbrot set, start with the recursion:

z_n+1 = z_n² + c

z₀ =0

This means “square what you’ve got, add c, then take the result, square it, add c, then take the result, …”

For different values of c the string of numbers you get out does different things. For example;

for c = 1, you get: 0, 1, 2, 5, 26, 677, … (that’s 0, 1=0²+1, 2=1²+1, 5=2²+1, …)

for c = -0.5, you get: 0, -0.5, -0.25, -0.4375, -0.30859375, -0.404769897, …

The Mandelbrot set is defined as the set of values of c that lead to strings of numbers that stay bounded. So, c=-0.5 is in the set because the string of numbers it makes stays in more or less the same place (it stays between -0.5 and 0 forever). But c=1 is not in the set because its string of numbers blows up.

It’s a little more complicated because you actually consider complex numbers (which is why the picture you get is in a plane).