View source for Mandelbulb

{{Short description|Three-dimensional fractal}}
{{More citations needed|date=December 2024}}
[[File:Visit of the Mandelbulb (4K UHD; 50FPS).webm|right|thumb|A 4K UHD 3D Mandelbulb video]]
[[File:Power 8 mandelbulb fractal overview.jpg|right|thumb|A [[Ray marching|ray-marched]] image of the 3D Mandelbulb
for the iteration ''v'' {{mapsto}} ''v''<sup>8</sup> + ''c'']]

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

A [[canonical form|canonical]] 3-dimensional [[Mandelbrot set]] does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of [[complex number]]s. It is possible to construct Mandelbrot sets in 4 dimensions using [[quaternion]]s and [[Bicomplex number|bicomplex numbers]]. 

White and Nylander's formula for the "''n''th power" of the vector <math>\mathbf v = \langle x, y, z\rangle</math> in {{math|ℝ<sup>3</sup>}} is

: <math>\mathbf v^n := r^n \langle\sin(n\theta) \cos(n\phi), \sin(n\theta) \sin(n\phi), \cos(n\theta)\rangle,</math>

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

The Mandelbulb is then defined as the set of those <math>\mathbf c</math> in {{math|ℝ<sup>3</sup>}} for which the orbit of <math>\langle 0, 0, 0\rangle</math> under the iteration <math>\mathbf v \mapsto \mathbf v^n + \mathbf c</math> is bounded.<ref>{{cite web
|url=http://www.skytopia.com/project/fractal/2mandelbulb.html
|title=Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal}} see "formula" section.</ref> For ''n''&nbsp;>&nbsp;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''&nbsp;=&nbsp;8. However, the equations can be simplified into rational polynomials when ''n'' is [[parity (mathematics)|odd]]. For example, in the case ''n''&nbsp;=&nbsp;3, the third power can be simplified into the [[mathematical beauty|more elegant]] form:

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

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

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

Since ''p'' and ''q'' do not necessarily have to equal ''n'' for the identity |''v<sup>n</sup>''|&nbsp;=&nbsp;|''v''|<sup>''n''</sup> to hold, more general fractals can be found by setting

: <math>\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</math>

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

==Cubic formula==
[[File:cubic fractal.jpg|thumb|Cubic fractal]]
Other formula come from identities parametrising the sum of squares to give a power of the sum of squares, such as
:<math>
\left(x^3 - 3xy^2 - 3xz^2 \right)^2 + \left(y^3 - 3yx^2 + yz^2 \right)^2 + \left(z^3 - 3zx^2 + zy^2 \right)^2 = \left(x^2 + y^2 + z^2 \right)^3,</math>
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,
: <math>x \to x^3 - 3x \left(y^2 + z^2 \right) + x_0</math>
: <math>y \to -y^3 + 3 y x^2 - y z^2  + y_0</math>
: <math>z \to z^3 - 3 z x^2 + z y^2 + z_0</math>
or other permutations.

This reduces to the complex fractal <math>w \to w^3 + w_0</math> when ''z''&nbsp;=&nbsp;0 and <math>w \to \overline{w}^3 + w_0</math> when ''y''&nbsp;=&nbsp;0.

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

==Quintic formula==

[[File:Quintic fractal.jpg|thumb|Quintic Mandelbulb]]
[[File:Quintic fractal2.jpg|thumb|Quintic Mandelbulb with ''C''&nbsp;=&nbsp;2]]

Another way to create Mandelbulbs with cubic symmetry is by taking the complex iteration formula <math>z \to z^{4m+1} + z_0</math> 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 <math>i^4 = 1</math>.) For example, take the case of <math>z \to z^5 + z_0</math>. In two dimensions, where <math>z = x + iy</math>, this is
: <math>x \to x^5 - 10 x^3 y^2 + 5 x y^4 + x_0,</math>
: <math>y \to y^5 - 10 y^3 x^2 + 5 y x^4 + y_0.</math>

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

for arbitrary constants ''A'', ''B'', ''C'' and ''D'', which give different Mandelbulbs (usually set to 0). The case <math>z \to z^9</math> gives a Mandelbulb most similar to the first example, where ''n''&nbsp;=&nbsp;9. A more pleasing result for the fifth power is obtained by basing it on the formula <math>z \to -z^5 + z_0</math>.
[[File:Cube3dmandel.jpg|thumb|Fractal based on ''z''&nbsp;→&nbsp;−''z''<sup>5</sup>]]

==Power-nine formula==

[[File:Z9fractal.jpg|thumb|Fractal with ''z''<sup>9</sup> 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,

: <math>x \to x^9 - 36 x^7 \left(y^2 + z^2 \right) + 126 x^5 \left(y^2 + z^2 \right)^2 - 84 x^3 \left(y^2 + z^2 \right)^3 + 9 x \left(y^2 + z^2 \right)^4 + x_0,</math>
: <math>y \to y^9 - 36 y^7 \left(z^2 + x^2 \right) + 126 y^5 \left(z^2 + x^2 \right)^2 - 84 y^3 \left(z^2 + x^2 \right)^3 + 9 y \left(z^2 + x^2 \right)^4 + y_0,</math>
: <math>z \to z^9 - 36 z^7 \left(x^2 + y^2 \right) + 126 z^5 \left(x^2 + y^2 \right)^2 - 84 z^3 \left(x^2 + y^2 \right)^3 + 9 z \left(x^2 + y^2 \right)^4 + z_0.</math>

These formula can be written in a shorter way:
: <math>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</math>
and equivalently for the other coordinates.

[[File:Mandelblob.jpg|thumb|Power-nine fractal detail]]

==Spherical formula==
A perfect spherical formula can be defined as a formula
: <math>
(x,y,z) \to \big(f(x, y, z) + x_0, g(x, y, z) + y_0, h(x, y, z) + z_0\big),
</math>
where
: <math>
\left(x^2 + y^2 + z^2 \right)^n = f(x, y, z)^2 + g(x, y, z)^2 + h(x, y, z)^2,
</math>
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 ==
<!-- Examples should be cited to a _secondary_ source (not the example itself) -->
* In the 2014 animated film ''[[Big Hero 6 (film)|Big Hero 6]]'', the climax takes place in the middle of a [[wormhole]], which is represented by the stylized interior of a Mandelbulb.<ref>{{Cite web |url = http://blogs.indiewire.com/animationscoop/immersed-in-movies-going-into-the-big-hero-6-portal-20150130 |title = Immersed in Movies: Going Into the 'Big Hero 6' Portal |date = January 30, 2015 |access-date = May 3, 2015 |website = Animation Scoop |publisher = Indiewire |last = Desowitz |first = Bill |archive-url = https://web.archive.org/web/20150503145129/http://blogs.indiewire.com/animationscoop/immersed-in-movies-going-into-the-big-hero-6-portal-20150130 |archive-date = May 3, 2015 |url-status = dead}}</ref><ref>{{Cite book |last1=Hutchins |first1=David |last2=Riley |first2=Olun |last3=Erickson |first3=Jesse |last4=Stomakhin |first4=Alexey |last5=Habel |first5=Ralf |last6=Kaschalk |first6=Michael |title=ACM SIGGRAPH 2015 Talks |chapter=Big Hero 6: Into the portal |date=2015 |series=SIGGRAPH '15 |location=New York, NY, USA |publisher=ACM |pages=52:1 |doi=10.1145/2775280.2792521 |isbn=9781450336369|s2cid=7488766 }}</ref>
* In the 2018 [[science fiction film|science fiction]] [[horror film]] ''[[Annihilation (film)|Annihilation]]'', an [[Extraterrestrial_life|extraterrestrial being]] appears in the form of a partial Mandelbulb.<ref>{{cite web |url=http://www.newsweek.com/annihilation-meaning-shimmer-area-x-lighthouse-818834 |title=What Is Area X and the Shimmer in 'Annihilation'? VFX Supervisor Explains the Horror Film's Mathematical Solution |last=Gaudette  |first=Emily |date=February 26, 2018 |publisher=[[Newsweek]] |access-date=March 9, 2018 }}</ref>
* In the [[webcomic]] ''Unsounded'' the spirit realm of the khert is represented by a stylized golden mandelbulb.
* In the 2013-2020 drama show Marvel's Agents of S.H.E.I.L.D, season 7, episode 13, "What We're Fighting For", the beginning shows the S.H.E.I.L.D. team travel through the Quantum Realm whose opening appearance from the outside resembles the end of an internally rotating mandelbulb. 
* In the 2016 film Marvel's Doctor Strange, one of the dimensions Strange is shown by the Ancient One depicts the top end of an expanding mandelbulb. 
* In the 2020 film 2067, as the main character is arriving in the future, the distorted land he descends upon resembles a distorted and morphing mandelbulb.

==See also==

* [[Mandelbox]]
* [[List of fractals by Hausdorff dimension]]

== References ==
{{Reflist}}
6.  http://www.fractal.org the Fractal Navigator by Jules Ruis

== External links ==
{{commons}}
* [http://www.fractal.org/BBM15.pdf for the first use of the Mandelbulb formula on www.fractal.org website Jules Ruis]
* [http://www.skytopia.com/project/fractal/mandelbulb.html Mandelbulb: The Unravelling of the Real 3D Mandelbrot Fractal, on Daniel White's website]
* [http://www.bugman123.com/Hypercomplex/index.html Several variants of the Mandelbulb, on Paul Nylander's website]
* [http://sourceforge.net/projects/mandelbulber/ An opensource fractal renderer that can be used to create images of the Mandelbulb]
* [http://www.fractal.org/Formula-Mandelbulb.pdf Formula for Mandelbulb/Juliabulb/Juliusbulb by Jules Ruis]
* [http://www.fractal.org/Mandelbulb.pdf Mandelbulb/Juliabulb/Juliusbulb with examples of real 3D objects]
* [https://www.youtube.com/watch?v=82xDvdgrzx0 Video : View of the Mandelbulb] 
* [http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/ The discussion thread in Fractalforums.com that led to the Mandelbulb]
* [https://www.youtube.com/watch?v=VB-XUoDqYfs Video fly through of an animated Mandelbulb world]
* [https://github.com/buddhi1980/mandelbulber2/ Open-source Mandelbulber v2 software - Explore trigonometric, hyper-complex, Mandelbox, IFS, and many other 3D fractals.]
* [https://www.youtube.com/watch?v=oY6yNjRT7Mo Silent video: Explore the Mandelbulb (2024 rendering)] 



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[[Category:Fractals]]