Mandelbulb

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

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

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

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

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.^[1] 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

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

for functions f and g.

Cubic formula

Other formulae come from identities parametrising 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, for example,

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

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

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

or other permutations.

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

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

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

This can be then extended to three dimensions to give

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

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

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

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

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,

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

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

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

These formula can be written in a shorter way:

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

and equivalently for the other coordinates.

Spherical formula

A perfect spherical formula can be defined as a formula

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

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 2014 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.^[2][3]
• In the 2018 science fiction horror film Annihilation, an extraterrestrial being appears in the form of a partial Mandelbulb.^[4]
• 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

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