Mandelbulb
Introduction A 4K UHD 3D Mandelbulb videoray-traced image of the 3D Mandelbulbfor the iteration ''v''  ''v''8 + ''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 "''n''th power" of the vector  in  is: where: : : The Mandelbulb is then defined as the set of those  in  for which the orbit of  under the iteration  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:: The Mandelbulb given by the formula above is actually one in a family of fractals given by parameters (''p'', ''q'') given by: 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: for functions ''f'' and ''g''.
Cubic formula Cubic fractalOther formulae come from identities parametrising the sum of squares to give a power of the sum of squares, such as: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,: : : or other permutations.This reduces to the complex fractal  when ''z'' = 0 and  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 MandelbulbQuintic Mandelbulb with ''C'' = 2Another way to create Mandelbulbs with cubic symmetry is by taking the complex iteration formula  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 .) For example, take the case of . In two dimensions, where , this is: : This can be then extended to three dimensions to give: : : for arbitrary constants ''A'', ''B'', ''C'' and ''D'', which give different Mandelbulbs (usually set to 0). The case  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 .Fractal based on ''z'' → −''z''5
Power-nine formula Fractal with ''z''9 Mandelbrot cross-sectionsThis 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,: : : These formula can be written in a shorter way:: and equivalently for the other coordinates.Power-nine fractal detail
Spherical formula A perfect spherical formula can be defined as a formula: where: where ''f'', ''g'' and ''h'' are ''n''th-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.