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A ray-traced image of the 3D Mandelbulb
for the iteration vv8 + c.

The Mandelbulb is a three-dimensional fractal, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009.[1]

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

, and

The Mandelbulb is then defined as the set of those in 3 for which the orbit of under the iteration is bounded.[2] 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.

Quadratic formula[edit]

Other formulae come from identities which parametrise the sum of squares to give a power of the sum of squares such as:

which we can think of as a way to square a triplet of numbers so that the modulus is squared. So this gives, for example:

or various other permutations. This 'quadratic' formula can be applied several times to get many power-2 formulae.

Cubic formula[edit]

Cubic fractal

Other formulae come from identities which parametrise 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:

or other permutations.

for example. 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[edit]

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 got basing it on the formula: .

Fractal based on z->-z^5

Power nine formula[edit]

Fractal with z^9 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[edit]

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[edit]

See also[edit]


  1. ^ "Hypercomplex fractals". 
  2. ^ "Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal".  see "formula" section
  3. ^ Desowitz, Bill (January 30, 2015). "Immersed in Movies: Going Into the 'Big Hero 6' Portal". Animation Scoop. Indiewire. Retrieved May 3, 2015. 
  4. ^ 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: 52:1–52:1. doi:10.1145/2775280.2792521. ISBN 9781450336369. 
  5. ^ 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. 

External links[edit]