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: $Z_{n+1} = Z^2_{n}+C$

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

With the Buddhabrot set, we can add a few more axes. Instead of just using $C_i$ and $C_r$ , we also use the real and imaginary components of $Z_0$ as axes. Whereas in the Mandelbrot set, $Z_0$ is assumed to be $0+0i$ , 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: $C_i, C_r, Z_i,$ and $Z_r$ .

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: