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

${\displaystyle {\mathbf {v} }^{n}:=r^{n}\langle \sin(n\theta )\cos(n\phi ),\sin(n\theta )\sin(n\phi ),\cos(n\theta )\rangle }$

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

The Mandelbulb is then defined as the set of those ${\displaystyle {\mathbf {c} }}$ in ℝ³ for which the orbit of ${\displaystyle \langle 0,0,0\rangle }$ under the iteration ${\displaystyle {\mathbf {v} }\mapsto {\mathbf {v} }^{n}+{\mathbf {c} }}$ is bounded.^[2] 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:

${\displaystyle \langle x,y,z\rangle ^{3}=\left\langle \ {\frac {(3z^{2}-x^{2}-y^{2})x(x^{2}-3y^{2})}{x^{2}+y^{2}}},{\frac {(3z^{2}-x^{2}-y^{2})y(3x^{2}-y^{2})}{x^{2}+y^{2}}},z(z^{2}-3x^{2}-3y^{2})\right\rangle }$.

The Mandelbulb given by the formula above is actually one in a family of fractals given by parameters (p,q) given by:

${\displaystyle {\mathbf {v} }^{n}:=r^{n}\langle \sin(p\theta )\cos(q\phi ),\sin(p\theta )\sin(q\phi ),\cos(p\theta )\rangle }$

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

${\displaystyle {\mathbf {v} }^{n}:=r^{n}\langle \sin(f(\theta ,\phi ))\cos(g(\theta ,\phi )),\sin(f(\theta ,\phi ))\sin(g(\theta ,\phi )),\cos(f(\theta ,\phi ))\rangle }$

for functions f and g.

Quadratic formula

Other formulae come from identities which parametrise the sum of squares to give a power of the sum of squares such as:

${\displaystyle (x^{2}-y^{2}-z^{2})^{2}+(2xz)^{2}+(2xy)^{2}=(x^{2}+y^{2}+z^{2})^{2}}$

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:

${\displaystyle x\rightarrow x^{2}-y^{2}-z^{2}+x_{0}}$

${\displaystyle y\rightarrow 2xz+y_{0}}$

${\displaystyle z\rightarrow 2xy+z_{0}}$

or various other permutations. This 'quadratic' formula can be applied several times to get many power-2 formulae.

Cubic formula

Other formulae come from identities which parametrise the sum of squares to give a power of the sum of squares such as:

${\displaystyle (x^{3}-3xy^{2}-3xz^{2})^{2}+(y^{3}-3yx^{2}+yz^{2})^{2}+(z^{3}-3zx^{2}+zy^{2})^{2}=(x^{2}+y^{2}+z^{2})^{3}}$

which we can think of as a way to cube a triplet of numbers so that the modulus is cubed. So this gives:

${\displaystyle x\rightarrow x^{3}-3x(y^{2}+z^{2})+x_{0}}$

or other permutations.

${\displaystyle y\rightarrow -y^{3}+3yx^{2}-yz^{2}+y_{0}}$

${\displaystyle z\rightarrow z^{3}-3zx^{2}+zy^{2}+z_{0}}$

for example. This reduces to the complex fractal ${\displaystyle w\rightarrow w^{3}+w_{0}}$ when z=0 and ${\displaystyle w\rightarrow {\overline {w}}^{3}+w_{0}}$ when y=0.

There are several ways to combine two such `cubic` transforms to get a power-9 transform which has slightly more structure.

Quintic formula

Another way to create Mandelbulbs with cubic symmetry is by taking the complex iteration formula ${\displaystyle z\rightarrow z^{4m+1}+z_{0}}$ 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 ${\displaystyle i^{4}=1}$.) For example, take the case of ${\displaystyle z\rightarrow z^{5}+z_{0}}$. In two dimensions where ${\displaystyle z=x+iy}$ this is:

${\displaystyle x\rightarrow x^{5}-10x^{3}y^{2}+5xy^{4}+x_{0}}$

${\displaystyle y\rightarrow y^{5}-10y^{3}x^{2}+5yx^{4}+y_{0}}$

This can be then extended to three dimensions to give:

${\displaystyle x\rightarrow x^{5}-10x^{3}(y^{2}+Ayz+z^{2})+5x(y^{4}+By^{3}z+Cy^{2}z^{2}+Byz^{3}+z^{4})+Dx^{2}yz(y+z)+x_{0}}$

${\displaystyle y\rightarrow y^{5}-10y^{3}(z^{2}+Axz+x^{2})+5y(z^{4}+Bz^{3}x+Cz^{2}x^{2}+Bzx^{3}+x^{4})+Dy^{2}zx(z+x)+y_{0}}$

${\displaystyle z\rightarrow z^{5}-10z^{3}(x^{2}+Axy+y^{2})+5z(x^{4}+Bx^{3}y+Cx^{2}y^{2}+Bxy^{3}+y^{4})+Dz^{2}xy(x+y)+z_{0}}$

for arbitrary constants A,B,C and D which give different Mandelbulbs (usually set to 0). The case ${\displaystyle z\rightarrow z^{9}}$ 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: ${\displaystyle z\rightarrow -z^{5}+z_{0}}$.

Power nine formula

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:

${\displaystyle x\rightarrow x^{9}-36x^{7}(y^{2}+z^{2})+126x^{5}(y^{2}+z^{2})^{2}-84x^{3}(y^{2}+z^{2})^{3}+9x(y^{2}+z^{2})^{4}+x_{0}}$

${\displaystyle y\rightarrow y^{9}-36y^{7}(z^{2}+x^{2})+126y^{5}(z^{2}+x^{2})^{2}-84y^{3}(z^{2}+x^{2})^{3}+9y(z^{2}+x^{2})^{4}+y_{0}}$

${\displaystyle z\rightarrow z^{9}-36z^{7}(x^{2}+y^{2})+126z^{5}(x^{2}+y^{2})^{2}-84z^{3}(x^{2}+y^{2})^{3}+9z(x^{2}+y^{2})^{4}+z_{0}}$

These formula can be written in a shorter way:

${\displaystyle x\rightarrow {\frac {1}{2}}(x+i{\sqrt {y^{2}+z^{2}}})^{9}+{\frac {1}{2}}(x-i{\sqrt {y^{2}+z^{2}}})^{9}+x_{0}}$

and equivalently for the other coordinates.

Spherical formula

A perfect spherical formula can be defined as a formula:

${\displaystyle (x,y,z)\rightarrow (f(x,y,z)+x_{0},g(x,y,z)+y_{0},h(x,y,z)+z_{0})}$

where

${\displaystyle (x^{2}+y^{2}+z^{2})^{n}=f(x,y,z)^{2}+g(x,y,z)^{2}+h(x,y,z)^{2}}$

where f,g and h are nth power rational trinomials and n is an integer. The cubic fractal above is an example.

Uses in media

• 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.^[3]

See also

• Mandelbox
• Multicomplex fractals
• List of fractals by Hausdorff dimension