Mandelbulb

The Mandelbulb is a three-dimensional fractal, constructed for the first time in 1997 by Jules Ruis and further developed in 2009 by Daniel White and Paul Nylander using spherical coordinates.

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 and bicomplex numbers.

White and Nylander's formula for the "nth power" of the vector ${\displaystyle \mathbf {v} =\langle x,y,z\rangle }$ in ℝ³ is

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

where

${\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}},}$

${\displaystyle \phi =\arctan {\frac {y}{x}}=\arg(x+yi),}$

${\displaystyle \theta =\arctan {\frac {\sqrt {x^{2}+y^{2}}}{z}}=\arccos {\frac {z}{r}}.}$

The Mandelbulb is then defined as the set of those ${\displaystyle \mathbf {c} }$ in ℝ³ for which the orbit of ${\displaystyle \langle 0,0,0\rangle }$ under the iteration ${\displaystyle \mathbf {v} \mapsto \mathbf {v} ^{n}+\mathbf {c} }$ is bounded.^[1] 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:

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

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

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

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

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

for functions f and g.