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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.

Ruis's formula for the "nth power" of the vector in 3 is to see at

The exponentiation term can be defined by: {x,y,z} ^n = (r^n) { cos(n*φ) * cos(n*θ), sin(n*φ) * cos(n*θ), sin(n*θ)} where r = sqrt (x^2 + y^2 + z^2) and r1 = sqrt (x^2 + y^2)

As we define θ as the angle in z-r1-space and φ as the angle in x-y-space then θ = atan2 (z, r1) so θ = atan2 (z, sqrt (x^2 + y^2)) and φ = atan2 (y, x)

The addition term in z -> z^n + c is similar to standard complex addition, and is simply defined by: (x,y,z} + {a,b,c) = {x+a, y+b, z+c} The rest of the algorithm is similar to the 2D Mandelbrot!

Summary Formula 3D Mandelbulb, Juliusbulb and Juliabulb

r = sqrt (x^2 + y^2 + z^2)

θ = atan2 (z, sqrt(x^2 + y^2)

φ = atan2 (y, x)

newx = (r^n) * cos(n*φ) * cos(n*θ)

newy = (r^n) * sin(n*φ) * cos(n*θ)

newz = (r^n) * sin(n*θ)

where n is the order of the 3D Mandelbulb, Juliusbulb/Juliabulb.

Quadratic formula[edit]

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 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 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[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 obtained by basing it on the formula .

Fractal based on z → −z5

Power-nine formula[edit]

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[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. ^ 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.
  2. ^ 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. doi:10.1145/2775280.2792521. ISBN 9781450336369.
  3. ^ 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. the Fractal Navigator by Jules Ruis

External links[edit]