Editing Mandelbrot set (section)

===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}}