Balanced polygamma function

File:Power 8 mandelbulb fractal overview.jpg

A ray-traced image of the 3D Mandelbulb
for the iteration z Template:Mapsto z⁸ + c.

The Mandelbulb is a three-dimensional analogue of the Mandelbrot set, constructed 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. 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 $⟨x,y,z⟩$ is

$⟨x,y,z{⟩}^n=r^n⟨\sin (n\theta )\cos (n\varphi ),\sin (n\theta )\sin (n\varphi ),\cos (n\theta )⟩$

where

$\begin{array}{cc}r& =\sqrt{x^2+y^2+z^2}\\ \varphi & =\arctan (y/x)=\arg (x+yi)\\ \mathrm{a}\mathrm{n}\mathrm{d}\theta & =\arctan (\sqrt{x^2+y^2}/z)=\arccos (z/r).\end{array}$

They use the iteration $z\mapsto z^n+c$ where $z^n$ is defined as above and $a+b$ is a vector addition. 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 someTemplate:Who that odd-numbered powers are more elegant. For example n = 3 is simplified to:

$⟨x,y,z{⟩}^3=⟨\frac{(3z^2-x^2-y^2)x(x^2-3y^2)}{x^2+y^2},\frac{(3z^2-x^2-y^2)y(3x^2-y^2)}{x^2+y^2},z(z^2-3x^2-3y^2)⟩$

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:

$x\to x^2-y^2-z^2+x_0$

$y\to 2xz+y_0$

$z\to 2xy+z_0$

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

Cubic formula

File:Cubic fractal.jpg

Cubic fractal

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

$(x^3-3x{y}^2-3x{z}^2{)}^2+(y^3-3y{x}^2+y{z}^2{)}^2+(z^3-3z{x}^2+z{y}^2{)}^2=(x^2+y^2+z^2{)}^3$

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

$x\to x^3-3x(y^2+z^2)+x_0$

or other permutations.

$y\to -y^3+3y{x}^2-y{z}^2+y_0$

$z\to z^3-3z{x}^2+z{y}^2+z_0$

for example. Which reduces to the complex fractal $w\to w^3+w_0$ when z=0 and $w\to {\bar{w}}^3+w_0$ 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

File:Quintic fractal.jpg

Quintic Mandelbulb

File:Quintic fractal2.jpg

Quintic Mandelbulb with C=2

Another way to create Mandelbulbs with cubic symmetry is by taking the complex iteration formula $z\to z^{4m+1}+z_0$ 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 $z\to z^5+z_0$. In two dimensions where $z=x+iy$ this is:

$x\to x^5-10x^3{y}^2+5x{y}^4+x_0$

$y\to y^5-10y^3{x}^2+5y{x}^4+y_0$

This can be then extended to three dimensions to give:

$x\to x^5-10x^3(y^2+Ayz+z^2)+5x(y^4+B{y}^{3}z+C{y}^2{z}^2+By{z}^3+z^4)+D{x}^{2}yz(y+z)+x_0$

$y\to y^5-10y^3(z^2+Axz+x^2)+5y(z^4+B{z}^{3}x+C{z}^2{x}^2+Bz{x}^3+x^4)+D{y}^{2}zx(z+x)+y_0$

$z\to z^5-10z^3(x^2+Axy+y^2)+5z(x^4+B{x}^{3}y+C{x}^2{y}^2+Bx{y}^3+y^4)+D{z}^{2}xy(x+y)+z_0$

for arbitrary constants A,B,C and D which give different Mandelbulbs (usually set to 0). The case $z\to z^9$ 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: $z\to -z^5+z_0$.

File:Cube3dmandel.jpg

Fractal based on z->-z^5

Power nine formula

File:Z9fractal.jpg

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:

$x\to x^9-36x^7(y^2+z^2)+126x^5(y^2+z^2{)}^2-84x^3(y^2+z^2{)}^3+9x(y^2+z^2{)}^4+x_0$

$y\to y^9-36y^7(z^2+x^2)+126y^5(z^2+x^2{)}^2-84y^3(z^2+x^2{)}^3+9y(z^2+x^2{)}^4+y_0$

$z\to z^9-36z^7(x^2+y^2)+126z^5(x^2+y^2{)}^2-84z^3(x^2+y^2{)}^3+9z(x^2+y^2{)}^4+z_0$

These formula can be written in a shorter way:

$x\to \frac{1}{2}(x+i\sqrt{y^2+z^2}{)}^9+\frac{1}{2}(x-i\sqrt{y^2+z^2}{)}^9+x_0$

and equivalently for the other coordinates.

File:Mandelblob.jpg

Power nine fractal detail

Spherical formula

A perfect spherical formula can be defined as a formula:

$(x,y,z)\to (f(x,y,z)+x_0,g(x,y,z)+y_0,h(x,y,z)+z_0)$

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.

See also

• Mandelbox
• List of fractals by Hausdorff dimension

References

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• Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal, on Daniel White's website
• Several variants of the Mandelbulb, on Paul Nylander's website
• An opensource fractal renderer that can be used to create images of the Mandelbulb
• Formula for Mandelbulb/Juliabulb/Juliusbulb by Jules Ruis
• Mandelbulb/Juliabulb/Juliusbulb with examples of real 3D objects
• Video : View of the Mandelbulb
• The discussion thread in Fractalforums.com that lead to the Mandelbulb
• Video fly through of an animated Mandelbulb world