Mandelbulb

Other formulae come from identities parametrising the sum of squares to give a power of the sum of squares, such as

${\displaystyle (x^{3}-3xy^{2}-3xz^{2})^{2}+(y^{3}-3yx^{2}+yz^{2})^{2}+(z^{3}-3zx^{2}+zy^{2})^{2}=(x^{2}+y^{2}+z^{2})^{3},}$ 

which we can think of as a way to cube a triplet of numbers so that the modulus is cubed. So this gives, for example,

${\displaystyle x\to x^{3}-3x(y^{2}+z^{2})+x_{0}}$ 

${\displaystyle y\to -y^{3}+3yx^{2}-yz^{2}+y_{0}}$ 

${\displaystyle z\to z^{3}-3zx^{2}+zy^{2}+z_{0}}$ 

or other permutations.

This reduces to the complex fractal ${\displaystyle w\to w^{3}+w_{0}}$  when z = 0 and ${\displaystyle w\to {\overline {w}}^{3}+w_{0}}$  when y = 0.

There are several ways to combine two such "cubic" transforms to get a power-9 transform, which has slightly more structure.

Another way to create Mandelbulbs with cubic symmetry is by taking the complex iteration formula ${\displaystyle z\to z^{4m+1}+z_{0}}$  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 ${\displaystyle i^{4}=1}$ .) For example, take the case of ${\displaystyle z\to z^{5}+z_{0}}$ . In two dimensions, where ${\displaystyle z=x+iy}$ , this is

${\displaystyle x\to x^{5}-10x^{3}y^{2}+5xy^{4}+x_{0},}$ 

${\displaystyle y\to y^{5}-10y^{3}x^{2}+5yx^{4}+y_{0}.}$ 

This can be then extended to three dimensions to give

${\displaystyle x\to x^{5}-10x^{3}(y^{2}+Ayz+z^{2})+5x(y^{4}+By^{3}z+Cy^{2}z^{2}+Byz^{3}+z^{4})+Dx^{2}yz(y+z)+x_{0},}$ 

${\displaystyle y\to y^{5}-10y^{3}(z^{2}+Axz+x^{2})+5y(z^{4}+Bz^{3}x+Cz^{2}x^{2}+Bzx^{3}+x^{4})+Dy^{2}zx(z+x)+y_{0},}$ 

${\displaystyle z\to z^{5}-10z^{3}(x^{2}+Axy+y^{2})+5z(x^{4}+Bx^{3}y+Cx^{2}y^{2}+Bxy^{3}+y^{4})+Dz^{2}xy(x+y)+z_{0}}$ 

for arbitrary constants A, B, C and D, which give different Mandelbulbs (usually set to 0). The case ${\displaystyle z\to z^{9}}$  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 ${\displaystyle z\to -z^{5}+z_{0}}$ .

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,

${\displaystyle x\to x^{9}-36x^{7}(y^{2}+z^{2})+126x^{5}(y^{2}+z^{2})^{2}-84x^{3}(y^{2}+z^{2})^{3}+9x(y^{2}+z^{2})^{4}+x_{0},}$ 

${\displaystyle y\to y^{9}-36y^{7}(z^{2}+x^{2})+126y^{5}(z^{2}+x^{2})^{2}-84y^{3}(z^{2}+x^{2})^{3}+9y(z^{2}+x^{2})^{4}+y_{0},}$ 

${\displaystyle z\to z^{9}-36z^{7}(x^{2}+y^{2})+126z^{5}(x^{2}+y^{2})^{2}-84z^{3}(x^{2}+y^{2})^{3}+9z(x^{2}+y^{2})^{4}+z_{0}.}$ 

These formula can be written in a shorter way:

${\displaystyle x\to {\frac {1}{2}}\left(x+i{\sqrt {y^{2}+z^{2}}}\right)^{9}+{\frac {1}{2}}\left(x-i{\sqrt {y^{2}+z^{2}}}\right)^{9}+x_{0}}$ 

and equivalently for the other coordinates.

A perfect spherical formula can be defined as a formula

${\displaystyle (x,y,z)\to {\big (}f(x,y,z)+x_{0},g(x,y,z)+y_{0},h(x,y,z)+z_{0}{\big )},}$ 

where

${\displaystyle (x^{2}+y^{2}+z^{2})^{n}=f(x,y,z)^{2}+g(x,y,z)^{2}+h(x,y,z)^{2},}$ 

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

• In the 2014 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.^[2][3]
• In the 2018 science fiction horror film Annihilation, an extraterrestrial being appears in the form of a partial Mandelbulb.^[4]
• In the webcomic Unsounded the spirit realm of the khert is represented by a stylized golden mandelbulb.

• Mandelbox
• List of fractals by Hausdorff dimension

6. http://www.fractal.org the Fractal Navigator by Jules Ruis

• for the first use of the Mandelbulb formula on www.fractal.org website Jules Ruis
• Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal, on Daniel White's website
• Several variants of the Mandelbulb, on Paul Nylander's website
• An opensource fractal renderer that can be used to create images of the Mandelbulb
• Formula for Mandelbulb/Juliabulb/Juliusbulb by Jules Ruis
• Mandelbulb/Juliabulb/Juliusbulb with examples of real 3D objects
• Video : View of the Mandelbulb
• Video : Exploring Mandelbulb. 3D Fractal Animation
• The discussion thread in Fractalforums.com that led to the Mandelbulb
• Video fly through of an animated Mandelbulb world
• Open-source Mandelbulber v2 software - Explore trigonometric, hyper-complex, Mandelbox, IFS, and many other 3D fractals.