What is "mandelbulb"

The Mandelbulb is a three-dimensional fractal, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009.

A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers. It is possible to construct Mandelbrot sets in 4 dimensions using quaternions. However, this set does not exhibit detail at all scales like the 2D Mandelbrot set does.

White and Nylander's formula for the "nth power" of the vector ${\mathbf v} = \langle x, y, z\rangle$ in is

$${\mathbf v}^n := r^n\langle\sin(n\theta)\cos(n\phi),\sin(n\theta)\sin(n\phi),\cos(n\theta)\rangle$$

where
$r=\sqrt{x^2+y^2+z^2}$,
ϕ = arctan(y/x) = arg(x + yi), and
$\theta=\arctan(\sqrt{x^2+y^2}/z)=\arccos(z/r)$.
The Mandelbulb is then defined as the set of those ${\mathbf c}$ in for which the orbit of ⟨0, 0, 0⟩ under the iteration ${\mathbf v} \mapsto {\mathbf v}^n+{\mathbf c}$ is bounded. For n > 3, the result is a 3-dimensional bulb-like structure with fractal surface detail and a number of "lobes" depending on n. Many of their graphic renderings use n = 8. However, the equations can be simplified into rational polynomials when n is odd. For example, in the case n = 3, the third power can be simplified into the more elegant form:

$$\langle x, y, z\rangle^3 = \left\langle\ \frac{(3z^2-x^2-y^2)x(x^2-3y^2)}{x^2+y^2} ,\frac{(3z^2-x^2-y^2)y(3x^2-y^2)}{x^2+y^2},z(z^2-3x^2-3y^2)\right\rangle$$
.

The Mandelbulb given by the formula above is actually one in a family of fractals given by parameters (p,q) given by:

${\mathbf v}^n := r^n\langle\sin(p\theta)\cos(q\phi),\sin(p\theta)\sin(q\phi),\cos(p\theta)\rangle$

Since p and q do not necessarily have to equal n for the identity |v|=|v| to hold. More general fractals can be found by setting

$${\mathbf v}^n := r^n\langle\sin(f(\theta,\phi))\cos(g(\theta,\phi)),\sin(f(\theta,\phi))\sin(g(\theta,\phi)),\cos(f(\theta,\phi))\rangle$$

for functions f and g.