Have you wondered what my profile picture is this whole time? That picture is called the Mandelbrot fractal. It’s formed from the Mandelbrot set.

To see if a number, n,  is in the Mandelbrot set, you have to perform an infinitely recursive function: square n, add n, and repeat (although you will always add the same value n that you started with). If the number does not trend towards infinity (or negative infinity, I guess), then it’s in the Mandelbrot set.

For example, if I start with -1, I square it and add -1 to get 0. I repeat the process to get-1 again.

All of the numbers that are in the Mandelbrot set have an absolute value less than or equal to 2. If you pick a point outside of that number, the cycle is guaranteed to trend towards infinity. This is because for every number, m, that’s greater than 2, m squared is greater than 2m. If I square m and subtract m, I’m guaranteed to get a number with a larger absolute value, so this trends towards infinity. This doctor describes numbers outside of radius 2 as “blowing up”.

 

Inside the black blobs, things are fairly simple. At the edges, things get fuzzy. All those little smudges in the graph are smaller graphs that are similar to the original Mandelbrot set.

This is black and white, though. Real Mandelbrot fractals have color. This is a color GIF zooming in on the Mandelbrot set.

 

What do the colors mean? Dr. Krieger says that they describe how quickly numbers grow. If a number grows more quickly than other numbers, it gets a different color. This helps illustrate the boundary of the set (kind of).

You can make amazing pictures with this fractal, much better  than any of this garbage. This is why I chose the Mandelbrot set as my picture.