The Results

Can you animate the Mandelbulb?

Yes, of course! It only takes a huge amount of time. This one rendered in about an hour, it animates the power from 1 to 20:

The Code

What is a Mandelbulb?

A mandelbulb is a 3D Fractal, which was formulated in 2009 by Paul Nylander. It takes the known approach of \(z_{n+1} \to z_n^2+c\). But he defines the nth power of a vector \(z\) (\(z^n\)) a little more complex:

Let \(v = (x,y,z)\) be a vector in \(\mathbb{R}\), then \(v^n := r^2 \cdot (\sin(n\theta)\cos(n\phi), sin(n\theta)sin(n\phi),cos(n\theta))\) where \[ r = |x| \\ \phi = \arctan(y/x) \\ \theta = \arctan\left(\frac{\sqrt{x^2+y^2}}{z}\right) \]

The 3D Mandelbulb is now defined as the set of points, where the orbit of \(z_n\) is bounded (the vector does not grow indefinitely).

This is where the power comes in (mentioned in the image descriptions). We can look at Mandelbulbs with powers that are not 2.

What happens when the power is < 1?

Short answer: Weird stuff. Long answer: I don't know!

here are some renders with weird powers:

Mandelbulb with power 0.2

Mandelbulb with power -2. Looks like a giant, interdimensional jellyfish.