The Mandelbulb is a three-dimensional fractal, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009.
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 v = ⟨ x , y , z ⟩ in ℝ3 is
v n := r n ⟨ sin ( n θ ) cos ( n ϕ ) , sin ( n θ ) sin ( n ϕ ) , cos ( n θ ) ⟩ where
r = x 2 + y 2 + z 2 ,
ϕ = arctan ( y / x ) = arg ( x + y i ) , and
θ = arctan ( x 2 + y 2 / z ) = arccos ( z / r ) .
The Mandelbulb is then defined as the set of those c in ℝ3 for which the orbit of ⟨ 0 , 0 , 0 ⟩ under the iteration v ↦ v n + c 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:
⟨ x , y , z ⟩ 3 = ⟨ ( 3 z 2 − x 2 − y 2 ) x ( x 2 − 3 y 2 ) x 2 + y 2 , ( 3 z 2 − x 2 − y 2 ) y ( 3 x 2 − y 2 ) x 2 + y 2 , z ( z 2 − 3 x 2 − 3 y 2 ) ⟩ .
The Mandelbulb given by the formula above is actually one in a family of fractals given by parameters (p,q) given by:
v n := r n ⟨ sin ( p θ ) cos ( q ϕ ) , sin ( p θ ) sin ( q ϕ ) , cos ( p θ ) ⟩ 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
v n := r n ⟨ sin ( f ( θ , ϕ ) ) cos ( g ( θ , ϕ ) ) , sin ( f ( θ , ϕ ) ) sin ( g ( θ , ϕ ) ) , cos ( f ( θ , ϕ ) ) ⟩ for functions f and g.
Other formulae come from identities which parametrise the sum of squares to give a power of the sum of squares such as:
( x 2 − y 2 − z 2 ) 2 + ( 2 x z ) 2 + ( 2 x y ) 2 = ( x 2 + y 2 + z 2 ) 2 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:
x → x 2 − y 2 − z 2 + x 0 y → 2 x z + y 0 z → 2 x y + z 0 or various other permutations. This 'quadratic' formula can be applied several times to get many power-2 formulae.
Other formulae come from identities which parametrise the sum of squares to give a power of the sum of squares such as:
( x 3 − 3 x y 2 − 3 x z 2 ) 2 + ( y 3 − 3 y x 2 + y z 2 ) 2 + ( z 3 − 3 z x 2 + z y 2 ) 2 = ( x 2 + y 2 + z 2 ) 3 which we can think of as a way to cube a triplet of numbers so that the modulus is cubed. So this gives:
x → x 3 − 3 x ( y 2 + z 2 ) + x 0 or other permutations.
y → − y 3 + 3 y x 2 − y z 2 + y 0 z → z 3 − 3 z x 2 + z y 2 + z 0 for example. This reduces to the complex fractal w → w 3 + w 0 when z=0 and w → w ¯ 3 + w 0 when y=0.
There are several ways to combine two such `cubic` transforms to get a power-9 transform which has slightly more structure.
Another way to create Mandelbulbs with cubic symmetry is by taking the complex iteration formula z → z 4 m + 1 + z 0 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 i 4 = 1 .) For example, take the case of z → z 5 + z 0 . In two dimensions where z = x + i y this is:
x → x 5 − 10 x 3 y 2 + 5 x y 4 + x 0 y → y 5 − 10 y 3 x 2 + 5 y x 4 + y 0 This can be then extended to three dimensions to give:
x → x 5 − 10 x 3 ( y 2 + A y z + z 2 ) + 5 x ( y 4 + B y 3 z + C y 2 z 2 + B y z 3 + z 4 ) + D x 2 y z ( y + z ) + x 0 y → y 5 − 10 y 3 ( z 2 + A x z + x 2 ) + 5 y ( z 4 + B z 3 x + C z 2 x 2 + B z x 3 + x 4 ) + D y 2 z x ( z + x ) + y 0 z → z 5 − 10 z 3 ( x 2 + A x y + y 2 ) + 5 z ( x 4 + B x 3 y + C x 2 y 2 + B x y 3 + y 4 ) + D z 2 x y ( x + y ) + z 0 for arbitrary constants A,B,C and D which give different Mandelbulbs (usually set to 0). The case z → z 9 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: z → − z 5 + z 0 .
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:
x → x 9 − 36 x 7 ( y 2 + z 2 ) + 126 x 5 ( y 2 + z 2 ) 2 − 84 x 3 ( y 2 + z 2 ) 3 + 9 x ( y 2 + z 2 ) 4 + x 0 y → y 9 − 36 y 7 ( z 2 + x 2 ) + 126 y 5 ( z 2 + x 2 ) 2 − 84 y 3 ( z 2 + x 2 ) 3 + 9 y ( z 2 + x 2 ) 4 + y 0 z → z 9 − 36 z 7 ( x 2 + y 2 ) + 126 z 5 ( x 2 + y 2 ) 2 − 84 z 3 ( x 2 + y 2 ) 3 + 9 z ( x 2 + y 2 ) 4 + z 0 These formula can be written in a shorter way:
x → 1 2 ( x + i y 2 + z 2 ) 9 + 1 2 ( x − i y 2 + z 2 ) 9 + x 0 and equivalently for the other coordinates.
A perfect spherical formula can be defined as a formula:
( x , y , z ) → ( f ( x , y , z ) + x 0 , g ( x , y , z ) + y 0 , h ( x , y , z ) + z 0 ) where
( x 2 + y 2 + z 2 ) n = f ( x , y , z ) 2 + g ( x , y , z ) 2 + h ( x , y , z ) 2 where f,g and h are nth power rational trinomials and n is an integer. The cubic fractal above is an example.
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.The 2016 film Doctor Strange shows Mandelbulbs in some of the dimensions the Ancient One transports Strange to when displaying her power.
