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Topic: Findding points within epsilon := 1/10^100 of the boundary of the
Mandelbrot set: easy, or hard?

Replies: 6   Last Post: Oct 11, 2013 12:42 PM

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David Bernier

Posts: 3,892
Registered: 12/13/04
Findding points within epsilon := 1/10^100 of the boundary of the
Mandelbrot set: easy, or hard?

Posted: Oct 7, 2013 7:31 AM
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After looking at a video zooming in by 10^275 on the Mandelbrot set,
it's a weird set.

< > .

So, suppose A is a point "close enough" (say within 0.05 roughly)
of the centroid of the Mandelbrot set.

Let theta be a "random" angle in [0, 2pi[.

The L_{theta} be the ray from A at angle theta to the Real axis, going
from radius= 0 to radius = 100. It is a closed set; L_{theta} is
a compact set in R^2, or C.

If M is the Mandelbrot set, M is compact. So is L_{theta} /\ M
is also compact, and the distance function d:C -> R
given by d(X) = | X - A | is continuous on C.

Therefore, d() attains its maximum at a point P_{theta}.
The point P_{theta} is both in M and in L_{theta}.
Any open neighborhood of P_{theta} will contain a point in
L_{theta} that is not in M. Therefore, P_{theta} is on
the boundary of M.

epsilon := 1/10^100

For a pseudo-random theta, such as log(5) radians, how easy or hard is
it to determine a point Y in C such [alternatively: M]
such that | Y - P_{theta} | < epsilon ?

If epsilon = 1/10^100 is too easy, then we can ask same
for epsilon = 1/10^10000 or even epsilon = 1/10^1000000 .

David Bernier

Let us all be paranoid. More so than no such agence, Bolon Yokte K'uh

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