Almond Bulb

Mandelbulb image by: v ⁸  ↦ v + c

The almond bulb fractal is a three-dimensional fractal . It was constructed in 2009 by Daniel White and Paul Nylander. For this purpose, a conventional model bread was subjected to a spherical coordinate transformation.

mathematics

Play media file

Flight around the Almond Bulb

A three-dimensional Mandelbrot set in normal form does not exist, because there is no 3-dimensional analogue of the complex plane (only higher-dimensional number systems such as quaternions or dimensions of other hyper-complex numbers).

White and Nylander's formula for the nth exponent of the vector in a Cartesian coordinate system ( ℝ ³ ) is ${\displaystyle {\mathbf {v} }=\langle x,y,z\rangle }$

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

under the use of

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

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

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

The Mandelbulb is then defined as the set of c-values ​​in ℝ ³ for which the orbit is limited by under the iteration . For n  > 3 there is a 3-dimensional pear-like structure with fractal surface details and a number of "lobes" depending on n . Many graphic renderings use the value 8 for n . The equations can be simplified into rational polynomials if n is odd. For the case n = 3, this can be transformed into the following, simplified form: ${\displaystyle \langle 0,0,0\rangle }$ ${\displaystyle {\mathbf {v} }\mapsto {\mathbf {v} }^{n}+{\mathbf {c} }}$ ${\displaystyle {\mathbf {v} }\mapsto {\mathbf {v} }^{3}}$

${\displaystyle \langle x,y,z\rangle ^{3}=\left\langle \!{\frac {(3z^{2}-x^{2}-y^{2})x(x^{2}-3y^{2})}{x^{2}+y^{2}}},\;{\frac {(3z^{2}-x^{2}-y^{2})y(3x^{2}-y^{2})}{x^{2}+y^{2}}},\;z(z^{2}-3x^{2}-3y^{2})\right\rangle }$.

More generally, one can use corresponding fractals (in addition to n also depending on p and q ) for the mapping

${\displaystyle {\mathbf {v} }^{n}:=r^{n}\langle \sin(p\theta )\cos(q\phi ),\;\sin(p\theta )\sin(q\phi ),\;\cos(p\theta )\rangle }$

construct, where p and q do not have to be equal to n in order to | v ​ⁿ | = | v | ⁿ to meet. Even more general fractals can be used with iteration

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

being found.

Similarity to the Mandelbrot crowd

Play media file

Similarity to the crowd visualized

Through certain transformations of the Mandelbulb fractal one can guess a similarity to the Mandelbrot set . If you cut the fractal in the middle in the case of n = 2 , you can see the classic Mandelbrot set.

Trivia

• In the computer animation film Baymax , released in 2014, a scene takes place in the center of a wormhole that resembles the stylized interior of an almond bulb.
• An alien in the science fiction horror film Extinction as part of an almond bulb.
• The spirit realm of the Kerht in the webcomic Unsounded is represented as a golden almond bulb .

See also

• Almond box
• Almond Bulber (fractal generating program; named after the Almond Bulb)

Related Links

Commons : Mandelbulb  - Album with pictures, videos and audio files

• Mandelbulb: The Unraveling of the Real 3D Mandelbrot Fractal, on Daniel White's website
• Several variants of the Almond Bulb on Nylander's website
• A fractal generating program which, among other things, renders the almond bulb.
• Formulas for Almond Bulb / Juliabulb / Juliusbulb by Jules Ruis
• Mandelbulb / Juliabulb / Juliusbulb with examples of real 3D objects
• Video: View of the Almond Bulb
• Discussion on Fractalforums.com on the topic of Almond Bulb
• Animated flight around the almond bulb

Individual evidence

1. ^ Richard Rosenman: Hypercomplex Fractals. BUGMAN, March 7, 2009, accessed July 9, 2020 . 
2. ↑ Mandelbulb: The Unraveling of the Real 3D Mandelbrot Fractal . see "Formula" area
3. ↑ David Hutchins, Olun Riley, Jesse Erickson, Alexey Stomakhin, Ralf Habel, Michael Kaschalk: Big Hero 6 | ACM SIGGRAPH 2015 Talks. Retrieved July 9, 2020 . 
4. ↑ Emily Gaudette: What Is Area X and the Shimmer in 'Annihilation'? VFX Supervisor Explains the Horror Film's Mathematical Solution. NewsWeek, February 26, 2018, accessed February 26, 2018 . 
5. ↑ Unsounded. Retrieved July 9, 2020 .