Summary:
In this video we provide a tool to approximate the boundary of the Mandelbrot set.
Content:
0:00 Describing Contour lines
1:25 Objective
2:00 The Mandelbrot set - Introduction
4:00 Zoom in on the imaginary unit
4:40 The generating polynomials
5:35 Mathematica two-liner
6:10 Why Two?
7:28 Two nearby points
8:00 The computation of the contours
10:19 Taylor series and inverse function
13:30 Towards the boundary
14:48 Outro: 100 arrow dance
16:14 Morphing
References:
The presentation is based on ideas investigated in the paper:
1.) Jungreis, Irwin. “The uniformization of the complement of the Mandelbrot set.” Duke Mathematical Journal 52 (1985): 935-938.
2.) Ewing, John H. and Glenn Schober. “The area of the Mandelbrot set.” Numerische Mathematik 61 (1992): 59-72.
Especially in the second paper you can find the recursion relation for the computation of the coefficients of the series expansion of the boundary.
Here is also a jypiter notebook, where the algorithm is implemented
gist.github.com/Gro-Tsen/4c5434b23c5a7e95e9c1cab345cdbf5f
I was introduced to this topic by Paul Abbott:
community.wolfram.com/groups/-/m/t/2684255?p_p_auth=vLAFPJ7D
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