Editing Mandelbrot set (section)

===Further results===
The [[Hausdorff dimension]] of the [[boundary (topology)|boundary]] of the Mandelbrot set equals 2 as determined by a result of [[Mitsuhiro Shishikura]].<ref name="shishikura"/> The fact that this is greater by a whole integer than its topological dimension, which is 1, reflects the extreme [[fractal]] nature of the Mandelbrot set boundary. Roughly speaking, Shishikura's result states that the Mandelbrot set boundary is so "wiggly" that it locally [[space-filling curve|fills space]] as efficiently as a two-dimensional planar region. Curves with Hausdorff dimension 2, despite being (topologically) 1-dimensional, are oftentimes capable of having nonzero area (more formally, a nonzero planar [[Lebesgue measure]]). Whether this is the case for the Mandelbrot set boundary is an unsolved problem.{{Citation needed|date=July 2023}}

It has been shown that the generalized Mandelbrot set in higher-dimensional hypercomplex number spaces (i.e. when the power <math>\alpha</math> of the iterated variable <math>z</math> tends to infinity) is convergent to the unit (<math>\alpha</math>−1)-sphere.<ref>{{cite journal|last1=Katunin|first1=Andrzej|last2=Fedio|first2=Kamil|title=On a Visualization of the Convergence of the Boundary of Generalized Mandelbrot Set to (n-1)-Sphere|url=https://reader.digitarium.pcss.pl/Content/295117/JAMCM_2015_1_6-Katunin_Fedio.pdf|access-date=18 May 2022|date=2015|journal=Journal of Applied Mathematics and Computational Mechanics|volume=14|issue=1|pages=63–69|doi=10.17512/jamcm.2015.1.06}}</ref>

In the [[Blum–Shub–Smale machine|Blum–Shub–Smale]] model of [[real computation]], the Mandelbrot set is not computable, but its complement is [[Recursively enumerable set|computably enumerable]]. Many simple objects (e.g., the graph of exponentiation) are also not computable in the BSS model. At present, it is unknown whether the Mandelbrot set is computable in models of real computation based on [[computable analysis]], which correspond more closely to the intuitive notion of "plotting the set by a computer". Hertling has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is true.{{Citation needed|date=July 2023}}