Mandelbulb

The Mandelbulb is a three-dimensional fractal, constructed for the first time in 1997 by Jules Ruis and in 2009 further developed by Daniel White and Paul Nylander using spherical coordinates.

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 and bicomplex numbers.

White and Nylander's formula for the "''n''th power" of the vector $\mathbf v = \langle x, y, z\rangle$ in is

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

where
: $r = \sqrt,$
: $\phi = \arctan\frac = \arg(x + yi),$
: $\theta = \arctan\frac = \arccos\frac.$

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

: $\langle x, y, z\rangle^3 = \left\langle\frac, \frac, z (z^2 - 3x^2 - 3y^2)\right\rangle.$

The Mandelbulb given by the formula above is actually one in a family of fractals given by parameters (''p'', ''q'') given by

: $\mathbf v^n := r^n \langle\sin(p\theta) \cos(q\phi), \sin(p\theta) \sin(q\phi), \cos(p\theta)\rangle.$

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

: $\mathbf v^n := r^n \big\langle\sin\big(f(\theta, \phi)\big) \cos\big(g(\theta, \phi)\big), \sin\big(f(\theta, \phi)\big) \sin\big(g(\theta, \phi)\big), \cos\big(f(\theta, \phi)\big)\big\rangle$

for functions ''f'' and ''g''.

Cubic formula

Other formulae come from identities parametrising the sum of squares to give a power of the sum of squares, such as
:$(x^3 - 3xy^2 - 3xz^2)^2 + (y^3 - 3yx^2 + yz^2)^2 + (z^3 - 3zx^2 + zy^2)^2 = (x^2 + y^2 + z^2)^3,$
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,
: $x \to x^3 - 3x (y^2 + z^2) + x_0$
: $y \to -y^3 + 3 y x^2 - y z^2 + y_0$
: $z \to z^3 - 3 z x^2 + z y^2 + z_0$
or other permutations.

This reduces to the complex fractal $w \to w^3 + w_0$ when ''z'' = 0 and $w \to \overline^3 + w_0$ 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

Another way to create Mandelbulbs with cubic symmetry is by taking the complex iteration formula $z \to z^ + z_0$ 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 $z \to z^5 + z_0$. In two dimensions, where $z = x + iy$, this is
: $x \to x^5 - 10 x^3 y^2 + 5 x y^4 + x_0,$
: $y \to y^5 - 10 y^3 x^2 + 5 y x^4 + y_0.$

This can be then extended to three dimensions to give
: $x \to x^5 - 10 x^3 (y^2 + A y z + z^2) + 5 x (y^4 + B y^3 z + C y^2 z^2 + B y z^3 + z^4) + D x^2 y z (y+z) + x_0,$
: $y \to y^5 - 10 y^3 (z^2 + A x z + x^2) + 5 y (z^4 + B z^3 x + C z^2 x^2 + B z x^3 + x^4) + D y^2 z x (z+x)+ y_0,$
: $z \to z^5 - 10 z^3 (x^2 + A x y + y^2) + 5 z (x^4 + B x^3 y + C x^2 y^2 + B x y^3 + y^4) + D z^2 x y (x+y) +z_0$

for arbitrary constants ''A'', ''B'', ''C'' and ''D'', which give different Mandelbulbs (usually set to 0). The case $z \to z^9$ 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 $z \to -z^5 + z_0$.

Power-nine formula

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,

: $x \to x^9 - 36 x^7 (y^2 + z^2) + 126 x^5 (y^2 + z^2)^2 - 84 x^3 (y^2 + z^2)^3 + 9 x (y^2 + z^2)^4 + x_0,$
: $y \to y^9 - 36 y^7 (z^2 + x^2) + 126 y^5 (z^2 + x^2)^2 - 84 y^3 (z^2 + x^2)^3 + 9 y (z^2 + x^2)^4 + y_0,$
: $z \to z^9 - 36 z^7 (x^2 + y^2) + 126 z^5 (x^2 + y^2)^2 - 84 z^3 (x^2 + y^2)^3 + 9 z (x^2 + y^2)^4 + z_0.$

These formula can be written in a shorter way:
: $x \to \frac \left(x + i\sqrt\right)^9 + \frac \left(x - i\sqrt\right)^9 + x_0$
and equivalently for the other coordinates.

Spherical formula

A perfect spherical formula can be defined as a formula
: $(x,y,z) \to \big(f(x, y, z) + x_0, g(x, y, z) + y_0, h(x, y, z) + z_0\big),$
where
: $(x^2 + y^2 + z^2)^n = f(x, y, z)^2 + g(x, y, z)^2 + h(x, y, z)^2,$
where ''f'', ''g'' and ''h'' are ''n''th-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.
* In the 2018 science fiction horror film ''Annihilation'', an extraterrestrial being appears in the form of a partial Mandelbulb.
* In the webcomic ''Unsounded'' the spirit realm of the kerht is represented by a stylized golden mandelbulb.

See also

* Mandelbox
* List of fractals by Hausdorff dimension

References

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

External links

for the first use of the Mandelbulb formula on www.fractal.org website Jules Ruis


* ttp://www.bugman123.com/Hypercomplex/index.html 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

{{Mathematical art

Fractals