Editing Mandelbrot set (section)

==Generalizations==
{{multiple image
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 | image2 = Mandelbrot set from powers 0.05 to 2.webm
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 | footer = Animations of the Multibrot set for ''d'' from 0 to 5 (left) and from 0.05 to 2 (right).
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[[File:Quaternion Julia x=-0,75 y=-0,14.jpg|thumb|A 4D Julia set may be projected or cross-sectioned into 3D, and because of this a 4D Mandelbrot is also possible.]]

===Multibrot sets===
[[Multibrot set]]s are bounded sets found in the [[complex plane]] for members of the general monic univariate [[polynomial]] family of recursions

:<math>z \mapsto z^d + c</math>.<ref>{{Cite conference|contribution=On Fibers and Local Connectivity of Mandelbrot and Multibrot Sets|last=Schleicher|first=Dierk|date=2004|title=Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 1|editor-last1=Lapidus|editor-first1=Michel L.|editor-last2=van Frankenhuijsen|editor-first2=Machiel|publisher=American Mathematical Society|url=https://books.google.com/books?id=uSpT729coosC|pages=477–517}}</ref>

For an [[integer]] ''d'', these sets are connectedness loci for the Julia sets built from the same formula. The full cubic connectedness locus has also been studied; here one considers the two-parameter recursion <math>z \mapsto z^3 + 3kz + c</math>, whose two [[critical point (mathematics)|critical points]] are the [[complex square root]]s of the parameter ''k''. A parameter is in the cubic connectedness locus if both critical points are stable.<ref>[[Rudy Rucker]]'s discussion of the CCM: [http://www.cs.sjsu.edu/faculty/rucker/cubic_mandel.htm CS.sjsu.edu]</ref> For general families of [[holomorphic function]]s, the ''boundary'' of the Mandelbrot set generalizes to the [[bifurcation locus]].{{Citation needed|date=July 2023}}

The [[Multibrot set]] is obtained by varying the value of the exponent ''d''. The article has a video that shows the development from ''d'' = 0 to 7, at which point there are 6 i.e. <math>(d-1)</math> lobes around the [[perimeter]]. In general, when ''d'' is a positive integer, the central region in each of these sets is always an [[epicycloid]] of <math>(d-1)</math> cusps. A similar development with negative integral exponents results in <math>(1-d)</math> clefts on the inside of a ring, where the main central region of the set is a [[hypocycloid]] of <math>(1-d)</math> cusps.{{Citation needed|date=July 2023}}

===Higher dimensions===
There is no perfect extension of the Mandelbrot set into 3D, because there is no 3D analogue of the complex numbers for it to iterate on. There is an extension of the complex numbers into 4 dimensions, the [[quaternion]]s, that creates a perfect extension of the Mandelbrot set and the Julia sets into 4 dimensions.<ref name="javier-barrallo"/> These can then be either [[cross section (geometry)|cross-sectioned]] or [[Projection mapping|projected]] into a 3D structure. The quaternion (4-dimensional) Mandelbrot set is simply a [[solid of revolution]] of the 2-dimensional Mandelbrot set (in the j-k plane), and is therefore uninteresting to look at.<ref name="javier-barrallo">{{cite web|last=Barrallo|first=Javier|date=2010|title=Expanding the Mandelbrot Set into Higher Dimensions|url=https://archive.bridgesmathart.org/2010/bridges2010-247.pdf|access-date=15 September 2021|website=BridgesMathArt}}</ref> Taking a 3-dimensional cross section at <math>d = 0\ (q = a + bi +cj + dk)</math> results in a solid of revolution of the 2-dimensional Mandelbrot set around the real axis.{{Citation needed|date=July 2023}}

===Other non-analytic mappings===
[[File:Mandelbar fractal from XaoS.PNG|left|thumb|Image of the [[Tricorn (mathematics)|Tricorn / Mandelbar fractal]]]]

The '''[[tricorn (mathematics)|tricorn]] fractal''', also called the '''Mandelbar set''', is the connectedness locus of the [[Antiholomorphic function|anti-holomorphic]] family <math>z \mapsto \bar{z}^2 + c</math>.<ref name=":6">{{Citation |last1=Inou |first1=Hiroyuki |title=Accessible hyperbolic components in anti-holomorphic dynamics |date=2022-03-23 |arxiv=2203.12156 |last2=Kawahira |first2=Tomoki}}</ref><ref>{{Citation |last1=Gauthier |first1=Thomas |title=Distribution of postcritically finite polynomials iii: Combinatorial continuity |date=2016-02-02 |arxiv=1602.00925 |last2=Vigny |first2=Gabriel}}</ref> It was encountered by [[John Milnor|Milnor]] in his study of parameter slices of real [[Cubic function|cubic polynomials]].{{Citation needed|date=March 2025}} It is not locally connected.<ref name=":6" /> This property is inherited by the connectedness locus of real cubic polynomials.{{Citation needed|date=March 2025}}

Another non-analytic generalization is the [[Burning Ship fractal]], which is obtained by iterating the following:

:<math>z \mapsto (|\Re \left(z\right)|+i|\Im \left(z\right)|)^2 + c</math>.{{Citation needed|date=March 2025}}