Mandelbulb

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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. However, this set does not exhibit detail at all scales like the 2D Mandelbrot set does.

White and Nylander's formula for the "nth power" of the vector in 3 is

where
,
, 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

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

In Popular Culture[edit]

  • In the Disney movie Big Hero 6, the emotional climax takes place in the middle of a wormhole, which is represented by the stylized interior of a Mandelbulb.[3]

See also[edit]

References[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. 

External links[edit]