Mandelbulb

Short description: Three-dimensional fractal

File:Visit of the Mandelbulb (4K UHD; 50FPS).webm

A ray-traced image of the 3D Mandelbulb for the iteration v ↦ v⁸ + c

The Mandelbulb is a three-dimensional fractal, constructed for the first time in 1997 by Jules Ruis and in 2009 further developed 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 and bicomplex numbers.

White and Nylander's formula for the "nth power" of the vector $\mathbf{𝐯}=⟨x,y,z⟩$ in ℝ³ is

${\mathbf{𝐯}}^n:=r^n⟨\sin (n\theta )\cos (n\varphi ),\sin (n\theta )\sin (n\varphi ),\cos (n\theta )⟩,$

where

$r=\sqrt{x^2+y^2+z^2},$

$\varphi =\arctan \frac{y}{x}=\arg (x+yi),$

$\theta =\arctan \frac{\sqrt{x^2+y^2}}{z}=\arccos \frac{z}{r}.$

The Mandelbulb is then defined as the set of those $\mathbf{𝐜}$ in ℝ³ for which the orbit of $⟨0,0,0⟩$ under the iteration $\mathbf{𝐯}\mapsto {\mathbf{𝐯}}^n+\mathbf{𝐜}$ 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:

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

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

${\mathbf{𝐯}}^n:=r^n⟨\sin (p\theta )\cos (q\varphi ),\sin (p\theta )\sin (q\varphi ),\cos (p\theta )⟩.$

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

${\mathbf{𝐯}}^n:=r^n⟨\sin (f(\theta ,\varphi ))\cos (g(\theta ,\varphi )),\sin (f(\theta ,\varphi ))\sin (g(\theta ,\varphi )),\cos (f(\theta ,\varphi ))⟩$

for functions f and g.

Cubic formula

Cubic fractal

Other formulae come from identities parametrising 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, for example,

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

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

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

or other permutations.

This reduces to the complex fractal $w\to w^3+w_0$ when z = 0 and $w\to \stackrel{3}{\stackrel{‾}{w}}+w_0$ 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

Quintic Mandelbulb

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. A more pleasing result for the fifth power is obtained by basing it on the formula $z\to -z^5+z_0$.

Fractal based on z → −z⁵

Power-nine formula

Fractal with z⁹ 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.

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.

Uses in media

• In the 2014 computer-animated film Big Hero 6, the climax takes place in the middle of a wormhole, which is represented by the stylized interior of a Mandelbulb.
• In the 2018 science fiction horror film Annihilation, an extraterrestrial being appears in the form of a partial Mandelbulb.
• In the webcomic Unsounded the spirit realm of the khert is represented by a stylized golden mandelbulb.

See also

• Mandelbox
• List of fractals by Hausdorff dimension

References

6. http://www.fractal.org the Fractal Navigator by Jules Ruis

External links

• for the first use of the Mandelbulb formula on www.fractal.org website Jules Ruis
• 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
• Video : Exploring Mandelbulb. 3D Fractal Animation
• The discussion thread in Fractalforums.com that led to the Mandelbulb
• Video fly through of an animated Mandelbulb world
• Open-source Mandelbulber v2 software - Explore trigonometric, hyper-complex, Mandelbox, IFS, and many other 3D fractals.