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

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 $\mathbf {v} =\langle x,y,z\rangle$ in $ℝ 3$ is

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

where

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

\phi =\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 $ℝ 3$ for which the orbit of $\langle 0,0,0\rangle$ under the iteration $\mathbf {v} \mapsto \mathbf {v} ^{n}+\mathbf {c}$ 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:

\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

\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

\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

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}-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,

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

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

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

or other permutations.

This reduces to the complex fractal $w\to w^{3}+w_{0}$ when z = 0 and $w\to {\overline {w}}^{3}+w_{0}$ 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

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}+5xy^{4}+x_{0},

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

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},

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 $z\to z^{9}$ 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}}\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.

Power-nine fractal detail

Spherical formula

A perfect spherical formula can be defined as a formula

(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

(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.[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 kerht is represented by a stylized golden mandelbulb.

References

"Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal". see "formula" section.
Desowitz, Bill (January 30, 2015). "Immersed in Movies: Going Into the 'Big Hero 6' Portal". Animation Scoop. Indiewire. Archived from the original on May 3, 2015. Retrieved May 3, 2015.
Hutchins, David; Riley, Olun; Erickson, Jesse; Stomakhin, Alexey; Habel, Ralf; Kaschalk, Michael (2015). "Big Hero 6: Into the portal". ACM SIGGRAPH 2015 Talks. SIGGRAPH '15. New York, NY, USA: ACM. pp. 52:1. doi:10.1145/2775280.2792521. ISBN 9781450336369. S2CID 7488766.
Gaudette, Emily (February 26, 2018). "What Is Area X and the Shimmer in 'Annihilation'? VFX Supervisor Explains the Horror Film's Mathematical Solution". Newsweek. Retrieved March 9, 2018.

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

External links

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] "Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal". see "formula" section.

[2] Desowitz, Bill (January 30, 2015). "Immersed in Movies: Going Into the 'Big Hero 6' Portal". Animation Scoop. Indiewire. Archived from the original on May 3, 2015. Retrieved May 3, 2015.

[3] Hutchins, David; Riley, Olun; Erickson, Jesse; Stomakhin, Alexey; Habel, Ralf; Kaschalk, Michael (2015). "Big Hero 6: Into the portal". ACM SIGGRAPH 2015 Talks. SIGGRAPH '15. New York, NY, USA: ACM. pp. 52:1. doi:10.1145/2775280.2792521. ISBN 9781450336369. S2CID 7488766.

[4] Gaudette, Emily (February 26, 2018). "What Is Area X and the Shimmer in 'Annihilation'? VFX Supervisor Explains the Horror Film's Mathematical Solution". Newsweek. Retrieved March 9, 2018.

Fractals
Characteristics	Fractal dimensions Assouad Box-counting Higuchi Correlation Hausdorff Packing Topological Recursion Self-similarity
Iterated function system	Barnsley fern Cantor set Koch snowflake Menger sponge Sierpinski carpet Sierpinski triangle Apollonian gasket Fibonacci word Space-filling curve Blancmange curve De Rham curve Minkowski Dragon curve Hilbert curve Koch curve Lévy C curve Moore curve Peano curve Sierpiński curve Z-order curve String T-square n-flake Vicsek fractal Hexaflake Gosper curve Pythagoras tree Weierstrass function
Strange attractor	Multifractal system
L-system	Fractal canopy Space-filling curve H tree
Escape-time fractals	Burning Ship fractal Julia set Filled Newton fractal Douady rabbit Lyapunov fractal Mandelbrot set Misiurewicz point Multibrot set Newton fractal Tricorn Mandelbox Mandelbulb
Rendering techniques	Buddhabrot Orbit trap Pickover stalk
Random fractals	Brownian motion Brownian tree Brownian motor Fractal landscape Lévy flight Percolation theory Self-avoiding walk
People	Michael Barnsley Georg Cantor Bill Gosper Felix Hausdorff Desmond Paul Henry Gaston Julia Helge von Koch Paul Lévy Aleksandr Lyapunov Benoit Mandelbrot Hamid Naderi Yeganeh Lewis Fry Richardson Wacław Sierpiński
Other	"How Long Is the Coast of Britain?" Coastline paradox Fractal art List of fractals by Hausdorff dimension The Fractal Geometry of Nature (1982 book) The Beauty of Fractals (1986 book) Chaos: Making a New Science (1987 book) Kaleidoscope Chaos theory