The Mandelbulb is a three-dimensional fractal 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 v = ⟨ x , y , z ⟩ {\displaystyle \mathbf {v} =\langle x,y,z\rangle } in ℝ3 is
v n := r n ⟨ sin ( n θ ) cos ( n ϕ ) , sin ( n θ ) sin ( n ϕ ) , cos ( n θ ) ⟩ , {\displaystyle \mathbf {v} ^{n}:=r^{n}\langle \sin(n\theta )\cos(n\phi ),\sin(n\theta )\sin(n\phi ),\cos(n\theta )\rangle ,}
where
r = x 2 + y 2 + z 2 , {\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}},}
ϕ = arctan y x = arg ( x + y i ) , {\displaystyle \phi =\arctan {\frac {y}{x}}=\arg(x+yi),}
θ = arctan x 2 + y 2 z = arccos z r . {\displaystyle \theta =\arctan {\frac {\sqrt {x^{2}+y^{2}}}{z}}=\arccos {\frac {z}{r}}.}
The Mandelbulb is then defined as the set of those c {\displaystyle \mathbf {c} } in ℝ3 for which the orbit of ⟨ 0 , 0 , 0 ⟩ {\displaystyle \langle 0,0,0\rangle } under the iteration v ↦ v n + c {\displaystyle \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:
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