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Mandelbulb (745 views - Animation & Rendering )

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 and bicomplex numbers. White and Nylander's formula for the "nth power" of the vector v = ⟨ x , y , z ⟩ {\displaystyle {\mathbf {v} }=\langle x,y,z\rangle } in ℝ3 is v n := r n ⟨ sin ⁡ ( n θ ) cos ⁡ ( n ϕ ) , sin ⁡ ( n θ ) sin ⁡ ( n ϕ ) , cos ⁡ ( n θ ) ⟩ {\displaystyle {\mathbf {v} }^{n}:=r^{n}\langle \sin(n\theta )\cos(n\phi ),\sin(n\theta )\sin(n\phi ),\cos(n\theta )\rangle } where r = x 2 + y 2 + z 2 {\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}}} , ϕ = arctan ⁡ ( y / x ) = arg ⁡ ( x + y i ) {\displaystyle \phi =\arctan(y/x)=\arg(x+yi)} , and θ = arctan ⁡ ( x 2 + y 2 / z ) = arccos ⁡ ( z / r ) {\displaystyle \theta =\arctan({\sqrt {x^{2}+y^{2}}}/z)=\arccos(z/r)} . The Mandelbulb is then defined as the set of those c {\displaystyle {\mathbf {c} }} in ℝ3 for which the orbit of ⟨ 0 , 0 , 0 ⟩ {\displaystyle \langle 0,0,0\rangle } under the iteration v ↦ v n + c {\displaystyle {\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: ⟨ 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 ) ⟩ {\displaystyle \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 } . 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 ) ⟩ {\displaystyle {\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 v n := r n ⟨ sin ⁡ ( f ( θ , ϕ ) ) cos ⁡ ( g ( θ , ϕ ) ) , sin ⁡ ( f ( θ , ϕ ) ) sin ⁡ ( g ( θ , ϕ ) ) , cos ⁡ ( f ( θ , ϕ ) ) ⟩ {\displaystyle {\mathbf {v} }^{n}:=r^{n}\langle \sin(f(\theta ,\phi ))\cos(g(\theta ,\phi )),\sin(f(\theta ,\phi ))\sin(g(\theta ,\phi )),\cos(f(\theta ,\phi ))\rangle } for functions f and g.
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Mandelbulb

Mandelbulb

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 in 3 is

where
,
, and
.

The Mandelbulb is then defined as the set of those in 3 for which the orbit of under the iteration 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 |vn|=|v|n to hold. More general fractals can be found by setting

for functions f and g.

Quadratic formula

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.

Cubic formula

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 when z=0 and 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 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 .) For example, take the case of . In two dimensions where 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 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: .

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:

These formula can be written in a shorter way:

and equivalently for the other coordinates.

Spherical formula

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.

Uses in media

  • 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



This article uses material from the Wikipedia article "Mandelbulb", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia

Animation & Rendering

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