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The Mandelbulb Graphical Pattern

The Mandelbulb is a three-dimensional fractal, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009.[1]

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 v=x,y,z in 3 is

vn:=rnsin(nθ)cos(nϕ),sin(nθ)sin(nϕ),cos(nθ)

where
r=x2+y2+z2,
ϕ=arctan(y/x)=arg(x+yi), and
θ=arctan(x2+y2/z)=arccos(z/r).