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 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(y/x)=\arg(x+yi)}
, and
θ
=
arctan
(
x
2
+
y
2
/
z
)
=
arccos
(
z
/
r
)
{\displaystyle \theta =\arctan({\sqrt {x^{2}+y^{2}}}/z)=\arccos(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:
⟨
x
,
y
,
z
⟩
3
=
⟨
(
3
z
2
−
x
2
−
y
2
)
x
(
x
2
−
3
y
2
)
x
2
+
y
2
,
(
3
z
2
−
x
2
−
y
2
)
y
(
3
x
2
−
y
2
)
x
2
+
y
2
,
z
(
z
2
−
3
x
2
−
3
y
2
)
⟩
{\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:
v
n
:=
r
n
⟨
sin
(
p
θ
)
cos
(
q
ϕ
)
,
sin
(
p
θ
)
sin
(
q
ϕ
)
,
cos
(
p
θ
)
⟩
{\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
v
n
:=
r
n
⟨
sin
(
f
(
θ
,
ϕ
)
)
cos
(
g
(
θ
,
ϕ
)
)
,
sin
(
f
(
θ
,
ϕ
)
)
sin
(
g
(
θ
,
ϕ
)
)
,
cos
(
f
(
θ
,
ϕ
)
)
⟩
{\displaystyle {\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.