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 {\mathbf v} = \langle x, y, z\rangle in 3 is

{\mathbf v}^n := r^n\langle\sin(n\theta)\cos(n\phi),\sin(n\theta)\sin(n\phi),\cos(n\theta)\rangle

where
r=\sqrt{x^2+y^2+z^2},
\phi=\arctan(y/x)=\arg (x+yi), and
\theta=\arctan(\sqrt{x^2+y^2}/z)=\arccos(z/r).

The Mandelbulb then defined as the set of those {\mathbf c} in 3 for which the orbit of \langle 0, 0, 0\rangle under the iteration {\mathbf v} \mapsto {\mathbf v}^n+{\mathbf c} 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:

\langle x, y, z\rangle^3 = \left\langle\ \frac{(3z^2-x^2-y^2)x(x^2-3y^2)}{x^2+y^2} ,\frac{(3z^2-x^2-y^2)y(3x^2-y^2)}{x^2+y^2},z(z^2-3x^2-3y^2)\right\rangle.

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:


(x^2-y^2-z^2)^2+(2 x z)^2+(2xy)^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\rightarrow  x^2-y^2-z^2+x_0

y\rightarrow  2 x z+y_0

z\rightarrow  2 x y +z_0

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

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:


(x^3-3xy^2-3xz^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\rightarrow  x^3 - 3 x (y^2 + z^2) + x_0

or other permutations.


y\rightarrow 
-y^3 + 3 y x^2 - y z^2  + y_0

z\rightarrow  
z^3 - 3 z x^2 + z y^2 + z_0

for example. Which reduces to the complex fractal w\rightarrow w^3+w_0 when z=0 and w\rightarrow \overline{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.

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 z\rightarrow z^{4m+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\rightarrow z^5 + z_0. In two dimensions where z=x+iy this is:


x\rightarrow x^5-10 x^3 y^2 + 5 x y^4 + x_0

y\rightarrow y^5-10 y^3 x^2 + 5 y x^4 + y_0

This can be then extended to three dimensions to give:


x\rightarrow 
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\rightarrow 
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\rightarrow  
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 \rightarrow z^9 gives a Mandelbulb most similar to the first example where n=9. An more pleasing result for the fifth power is got basing it on the formula: z\rightarrow -z^5 + z_0.

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:


x\rightarrow 
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\rightarrow 
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\rightarrow 
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\rightarrow 
\frac{1}{2}(x+i\sqrt{y^2+z^2})^9+\frac{1}{2}(x-i\sqrt{y^2+z^2})^9+x_0

and equivalently for the other coordinates.

Power nine fractal detail

Spherical formula[edit]

A perfect spherical formula can be defined as a formula:


(x,y,z)\rightarrow( 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 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]