A 4K UHD 3D Mandelbulb video
A ray-traced image of the 3D Mandelbulb for the iteration vv8 + 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 in 3 is


The Mandelbulb is then defined as the set of those in 3 for which the orbit of under the iteration 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:

The Mandelbulb given by the formula above is actually one in a family of fractals given by parameters (pq) 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 fractal

Other 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 Mandelbulb
Quintic Mandelbulb with C = 2

Another 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 → −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,

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 f, g and h are nth-power rational trinomials and n is an integer. The cubic fractal above is an example.

Uses in media

See also


  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.

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