# π IN THE MANDELBROT SET

@article{Klebanoff2001IT,
title={$\pi$ IN THE MANDELBROT SET},
author={A. Klebanoff},
journal={Fractals},
year={2001},
volume={09},
pages={393-402}
}
The Mandelbrot set is arguably one of the most beautiful sets in mathematics. In 1991, Dave Boll discovered a surprising occurrence of the number π while exploring a seemingly unrelated property of the Mandelbrot set.1 Boll's finding is easy to describe and understand, and yet it is not widely known — possibly because the result has not been rigorously shown. The purpose of this paper is to present and prove Boll's result.
Is the Mandelbrot set computable?
• P. Hertling
• Computer Science, Mathematics
• Math. Log. Q.
• 2005
It is concluded that the two-sided distance function of the Mandelbrot set is computable if the famous hyperbolicity conjecture is true. Expand
Hypercomputing the Mandelbrot Set?
The Mandelbrot set is surveyed and two models of decidability over the reals are considered, two over the computable reals (the Russian school and hypercomputation) and a model over the rationals. Expand
Principles for the Mandelbrot set Karsten Keller
New insights into the combinatorial structure of the the Mandelbrot set are given by ‘Correspondence’ and ‘Translation’ Principles both conjectured and partially proved by E. Lau and D. Schleicher.Expand
Local properties of the Mandelbrot set at parabolic points
We formulate the technique of parabolic implosion into an easy-touse result: Orbit correspondence, and apply it to show that for c0 a primitive parabolic point, the Mandelbrot set M outside the wakeExpand
Families of Homeomorphic Subsets of the Mandelbrot Set
• 1999
The 1=3-limb of the Mandelbrot set M is considered as a graph, where the vertices are given by certain Misiurewicz points. The edges are described as a union of building blocks that are calledExpand
Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account
A key point in Douady and Hubbard's study of the Mandelbrot set $M$ is the theorem that every parabolic point $c\ne 1/4$ in $M$ is the landing point for exactly two external rays with angle which areExpand
Equivalence between subshrubs and chaotic bands in the Mandelbrot set
• Mathematics
• 2006
We study in depth the equivalence between subshrubs and chaotic bands in the Mandelbrot set. In order to do so, we introduce the rules for chaotic bands and the rules for subshrubs, as well as theExpand
Invariant Peano curves of expanding Thurston maps
We consider Thurston maps, i.e., branched covering maps f:S2 → S2 that are post-critically finite. In addition, we assume that f is expanding in a suitable sense. It is shown that each sufficientlyExpand
Chaotic bands in the Mandelbrot set
• Mathematics, Computer Science
• Comput. Graph.
• 2004
This paper has focused on the ordering of the hyperbolic components and Misiurewicz points located at the shrub0 of any shrub of the Mandelbrot set. Expand
External arguments of Douady cauliflowers in the Mandelbrot set
• Mathematics, Computer Science
• Comput. Graph.
• 2004
This work studies the Douady cauliflowers, giving a binary tree model to calculate the binary expansions of the external arguments of both the main cardioid of each minute Mandelbrot set and Misiurewicz points, and creating a horsetail model that explains the arrangement of its corresponding external rays. Expand