The Burning Ship Fractal
This is the burning ship fractal. You know what’s insane? The picture above is created just by iterating one function.
How it works (super briefly)
(For this section, I’m going to assume you know a bit about complex numbers, the complex plane, and recursive functions)
Complex numbers have a real part and an imaginary part. Re(x) is the real part of complex number x, and Im(x) is the imaginary part of x. A complex number can be expressed as x = Re(x) + i*Im(x). If x=3+2i for example, Re(x)=3 and Im(x)=2.
Pick some complex number, which we’ll call c. Then, iterate this function:
Keep iterating the function over and over. When you do this, there are two things that could happen. The sequence (the value of each zₙ₊₁) could keep growing and growing, shooting off to infinity (in math terms the sequence diverges). Or the number gets stuck in a loop1. If the sequence gets stuck in a loop, we’ll color the point c black. If the sequence diverges, we don’t color it black.
Repeat this for every complex number (aka every possible value of c) and you get a picture of the burning ship fractal.
What’s a fractal?
A fractal is a shape with complicated structure no matter how much you zoom in.
Here’s what I mean by that. If you look at, say, a sin wave, the more you zoom into it, the simpler the shape gets.
But if you zoom in on a Sierpiński triangle, the shape never gets simpler. The Sierpiński triangle repeats itself infinitely, so the complexity stays the same as you zoom in. The Sierpiński triangle is a fractal.
Usually when people think of fractals, they think of “self-similar” shapes. Self-similar means the shape has an infinitely repeating pattern. The Sierpiński triangle is self-similar, along with the Mandelbrot set, the Koch snowflake, and (a personal favorite) the Gosper curve, aka the “flowsnake.”
But even though fractals are usually self-similar, they don’t have to be. Examples of fractals that aren’t self-similar include the Lyapunov fractal, the coastline of Great Britain (yes really), and the burning ship fractal above.
Thanks to Mikey for showing this to me.
Other Related Stuff
One example: if c= -i, the sequence goes {0, -i, -1-i, i, -1-i, i…} and goes back and forth between i and (-1-i) forever.