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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 and bicomplex numbers.
White and Nylander's formula for the "nth power" of the vector ${\mathbf {v} }=\langle x,y,z\rangle$ in ℝ^{3} is
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 is then defined as the set of those ${\mathbf {c} }$ in ℝ^{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:
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 |v^{n}|=|v|^{n} to hold. More general fractals can be found by setting
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:
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:
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:
which we can think of as a way to cube a triplet of numbers so that the modulus is cubed. So this gives:
or other permutations.
for example. This 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.
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:
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 $z\rightarrow z^{9}$ 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: $z\rightarrow -z^{5}+z_{0}$.
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:
These formula can be written in a shorter way:
and equivalently for the other coordinates.
A perfect spherical formula can be defined as a formula:
where
where f,g and h are nth power rational trinomials and n is an integer. The cubic fractal above is an example.
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