Mandelbulb

The Mandelbulb is a three-dimensional analogue of the Mandelbrot set, constructed by Daniel White and Paul Nylander using spherical coordinates.^[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. 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 3D vector $\langle x,y,z\rangle$ is

\langle x,y,z\rangle ^{n}=r^{n}\langle \sin(n\theta )\cos(n\phi ),\sin(n\theta )\sin(n\phi ),\cos(n\theta )\rangle

where

{\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\phi &=\arctan(y/x)=\arg(x+yi)\\{\rm {and\ }}\theta &=\arctan({\sqrt {x^{2}+y^{2}}}/z)=\arccos(z/r).\end{aligned}}

They use the iteration $z\mapsto z^{n}+c$ where $z^{n}$ is defined as above and $a+b$ is a vector addition.^[2] For n > 3, the result is a 3-dimensional bulb-like structure with fractal surface detail and a number of "lobes" controlled by the parameter n. Many of their graphic renderings use n = 8. The equations can be simplified into rational polynomials for every odd n, so it has been speculated by some^[who?] that odd-numbered powers are more elegant. For example n = 3 is simplified to:

\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

Quadratic formula

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}+(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:

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

y\rightarrow 2xz+y_{0}

z\rightarrow 2xy+z_{0}

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

Cubic formula

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

(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:

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

or other permutations.

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

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

for example. Which reduces to the complex fractal $w\rightarrow w^{3}+w_{0}$ when z=0 and $w\rightarrow {\overline {w}}^{3}+w_{0}$ 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 $z\rightarrow z^{4m+1}+z_{0}$ 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 $i^{4}=1$ .) For example, take the case of $z\rightarrow z^{5}+z_{0}$ . In two dimensions where $z=x+iy$ this is:

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

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

This can be then extended to three dimensions to give:

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}

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}

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 $z\rightarrow z^{9}$ gives a Mandelbulb most similar to the first example where n=9. An more pleasing result for the fifth power is got basing it on the formula: $z\rightarrow -z^{5}+z_{0}$ .

Power nine formula

Fractal with z^9 Mandelbrot cross sections

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: