Buddhabrot 4-Dimensional Rotation

I’ve been playing with fractals recently. More specifically, the Buddhabrot fractal.

The Buddhabrot set is closely related to the Mandelbrot set:

The mandelbrot set is displayed on the complex plane, where one axis is the real component of (we’ll call it ) and the other is the imaginary component of (a.k.a ).

With the Buddhabrot set, we can add a few more axes. Instead of just using and , we also use the real and imaginary components of as axes. Whereas in the Mandelbrot set, is assumed to be , in our 4D Buddhabrot set, we make these values variable as well.

So instead of the two dimensional image you’re used to seeing of the Mandelbrot set, we’ve instead got 4 axes: and .

Which leads us to our next problem — how do you visualise a 4 dimensional object on a computer? Well, it’s difficult. You obviously can’t just draw an image of it, or even just show a 3 dimensional picture of it. Instead, we show it rotation in 4 dimensional space.

So, here’s the result:

I’ve made all of this code open source — here’s the project page.