Greg Klein's Blog

Buddhabrot 4-Dimensional Rotation

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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:

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

Written by gregklein

May 16, 2011 at 1:55 am

Posted in fractal

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