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 ${\displaystyle {\mathbf {v} }=\langle x,y,z\rangle }$ in ℝ³ is

where
${\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}}}$,
${\displaystyle \phi =\arctan(y/x)=\arg(x+yi)}$, and
${\displaystyle \theta =\arctan({\sqrt {x^{2}+y^{2}}}/z)=\arccos(z/r)}$.

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