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
v
=
⟨
x
,
y
,
z
⟩
in ℝ3 is
v
n
:=
r
n
⟨
sin
(
n
θ
)
cos
(
n
ϕ
)
,
sin
(
n
θ
)
sin
(
n
ϕ
)
,
cos
(
n
θ
)
⟩
where
r
=
x
2
+
y
2
+
z
2
,
ϕ
=
arctan
(
y
/
x
)
=
arg
(
x
+
y
i
)
, and
θ
=
arctan
(
x
2
+
y
2
/
z
)
=
arccos
(
z
/
r
)
.
The Mandelbulb is then defined as the set of those
c
in ℝ3 for which the orbit of
⟨
0
,
0
,
0
⟩
under the iteration
v
↦
v
n
+
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
)
⟩
.
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
)
⟩
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
)
)
⟩
for functions f and g.
Other formulae come from identities which parametrise the sum of squares to give a power of the sum of squares such as:
(
x
2
−
y
2
−
z
2
)
2
+
(
2
x
z
)
2
+
(
2
x
y
)
2
=
(
x
2
+
y
2
+
z
2
)
2
which we can think of as a way to square a triplet of numbers so that the modulus is squared. So this gives, for example:
x
→
x
2
−
y
2
−
z
2
+
x
0
y
→
2
x
z
+
y
0
z
→
2
x
y
+
z
0
or various other permutations. This 'quadratic' formula can be applied several times to get many power-2 formulae.
Other formulae come from identities which parametrise the sum of squares to give a power of the sum of squares such as:
(
x
3
−
3
x
y
2
−
3
x
z
2
)
2
+
(
y
3
−
3
y
x
2
+
y
z
2
)
2
+
(
z
3
−
3
z
x
2
+
z
y
2
)
2
=
(
x
2
+
y
2
+
z
2
)
3
which we can think of as a way to cube a triplet of numbers so that the modulus is cubed. So this gives:
x
→
x
3
−
3
x
(
y
2
+
z
2
)
+
x
0
or other permutations.
y
→
−
y
3
+
3
y
x
2
−
y
z
2
+
y
0
z
→
z
3
−
3
z
x
2
+
z
y
2
+
z
0
for example. This reduces to the complex fractal
w
→
w
3
+
w
0
when z=0 and
w
→
w
¯
3
+
w
0
when y=0.
There are several ways to combine two such `cubic` transforms to get a power-9 transform which has slightly more structure.
Another way to create Mandelbulbs with cubic symmetry is by taking the complex iteration formula
z
→
z
4
m
+
1
+
z
0
for some integer m and adding terms to make it symmetrical in 3 dimensions but keeping the cross-sections to be the same 2 dimensional fractal. (The 4 comes from the fact that
i
4
=
1
.) For example, take the case of
z
→
z
5
+
z
0
. In two dimensions where
z
=
x
+
i
y
this is:
x
→
x
5
−
10
x
3
y
2
+
5
x
y
4
+
x
0
y
→
y
5
−
10
y
3
x
2
+
5
y
x
4
+
y
0
This can be then extended to three dimensions to give:
x
→
x
5
−
10
x
3
(
y
2
+
A
y
z
+
z
2
)
+
5
x
(
y
4
+
B
y
3
z
+
C
y
2
z
2
+
B
y
z
3
+
z
4
)
+
D
x
2
y
z
(
y
+
z
)
+
x
0
y
→
y
5
−
10
y
3
(
z
2
+
A
x
z
+
x
2
)
+
5
y
(
z
4
+
B
z
3
x
+
C
z
2
x
2
+
B
z
x
3
+
x
4
)
+
D
y
2
z
x
(
z
+
x
)
+
y
0
z
→
z
5
−
10
z
3
(
x
2
+
A
x
y
+
y
2
)
+
5
z
(
x
4
+
B
x
3
y
+
C
x
2
y
2
+
B
x
y
3
+
y
4
)
+
D
z
2
x
y
(
x
+
y
)
+
z
0
for arbitrary constants A,B,C and D which give different Mandelbulbs (usually set to 0). The case
z
→
z
9
gives a Mandelbulb most similar to the first example where n=9. A more pleasing result for the fifth power is got basing it on the formula:
z
→
−
z
5
+
z
0
.
This fractal has cross-sections of the power 9 Mandelbrot fractal. It has 32 small bulbs sprouting from the main sphere. It is defined by, for example:
x
→
x
9
−
36
x
7
(
y
2
+
z
2
)
+
126
x
5
(
y
2
+
z
2
)
2
−
84
x
3
(
y
2
+
z
2
)
3
+
9
x
(
y
2
+
z
2
)
4
+
x
0
y
→
y
9
−
36
y
7
(
z
2
+
x
2
)
+
126
y
5
(
z
2
+
x
2
)
2
−
84
y
3
(
z
2
+
x
2
)
3
+
9
y
(
z
2
+
x
2
)
4
+
y
0
z
→
z
9
−
36
z
7
(
x
2
+
y
2
)
+
126
z
5
(
x
2
+
y
2
)
2
−
84
z
3
(
x
2
+
y
2
)
3
+
9
z
(
x
2
+
y
2
)
4
+
z
0
These formula can be written in a shorter way:
x
→
1
2
(
x
+
i
y
2
+
z
2
)
9
+
1
2
(
x
−
i
y
2
+
z
2
)
9
+
x
0
and equivalently for the other coordinates.
A perfect spherical formula can be defined as a formula:
(
x
,
y
,
z
)
→
(
f
(
x
,
y
,
z
)
+
x
0
,
g
(
x
,
y
,
z
)
+
y
0
,
h
(
x
,
y
,
z
)
+
z
0
)
where
(
x
2
+
y
2
+
z
2
)
n
=
f
(
x
,
y
,
z
)
2
+
g
(
x
,
y
,
z
)
2
+
h
(
x
,
y
,
z
)
2
where f,g and h are nth power rational trinomials and n is an integer. The cubic fractal above is an example.
In the Disney movie Big Hero 6, the emotional climax takes place in the middle of a wormhole, which is represented by the stylized interior of a Mandelbulb.
The 2016 film Doctor Strange shows Mandelbulbs in some of the dimensions the Ancient One transports Strange to when displaying her power.
