White and Nylander's formula for the "nth power" of the vector in ℝ3 is
The Mandelbulb is then defined as the set of those in ℝ3 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.
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 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
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
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
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.