Pi & the mandelbrot set, also DiffEQ question

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dave boll

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Apr 24, 1991, 11:02:10 AM4/24/91
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I posted this to alt.fractals about a month ago, and it occurred to me
that readers of this newsgroup would probably be interested also.

I was trying to show that the 'neck' of the mandelbrot set at (-.75,0) is
actually of zero thickness. Accordingly, I wrote a quickie program and
started checking the number of iterations that points of the form
(-.75,dy) went thru before blowing up (with dy being a small number).
Here's a table of results:

dy # of iterations
.1 33
.01 315
.001 3143
.0001 31417
.00001 314160

Notice anything peculiar about the # of iterations? How about the product
of the # of iterations with dy?

I tried again at the 'butt' of the set at (.25,0), using points of the
form (.25+dx,0). Another table:

dx # of iterations
.01 30
.001 97
.0001 312
.00001 991
.000001 3140
.0000001 9933
.00000001 31414
.000000001 99344

This shows the same type of relationship, with (sqrt dx)*(# of iterations)
equal to pi.

Note: if anyone tries to verify these results, use double precision in
your program.

I gave a half-hearted attempt at showing this mathematically, but got
stumped trying to figure out the 2**n coefficients of (c^2 + c)**n
in a non-recursive form.
The first couple are easy, the last n are Catalan numbers, and after that
is gets ugly.

Now on to a DiffEQ question that has always bugged me: separation of
variables in polar coordinates. The assumption is that F(r,theta) can
be expressed as A(r)*B(theta). Well, my question is how can you assume
the function is separable when a constraint you must have is that
B(theta) goes to a constant as r goes to zero? Yes, I know, the answer
works out that way, but I'm still not satisfied assuming it before
we see the answer.

If anyone has any comments or insight on any of this, e-mail me or
post.
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Dave Boll bo...@handel.cs.colostate.edu
"Things are more like they are now than they ever have been before."
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jo...@ohstpy.mps.ohio-state.edu

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Apr 24, 1991, 3:03:25 PM4/24/91
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In article <14...@ccncsu.ColoState.EDU>,
bo...@handel.cs.colostate.edu (dave boll) writes:

> I was trying to show that the 'neck' of the mandelbrot set at (-.75,0) is
> actually of zero thickness. Accordingly, I wrote a quickie program and
> started checking the number of iterations that points of the form
> (-.75,dy) went thru before blowing up (with dy being a small number).
> Here's a table of results:
>
> dy # of iterations
> .1 33
> .01 315
> .001 3143
> .0001 31417
> .00001 314160

This is very interesting. Try dy = 0.00345 or anything besides 10^{-n}.
What do you see then? That is, what is dy * (#_of_iterations)?
--
O /
-------------------------------- X --- cut here -----------------------------
bob jones O \

Disclaimer: "I just said what?"

internet: jo...@ohstpy.mps.ohio-state.edu
US mail: robert jones, POBox 3194, Columbus, OH, 43210
telephone: (614)-447-0214 (home) and (614)-292-1648 (school)
pi: 3.14159265358979323846264338327950288419716939937510582097494459230781...
e: 2.71828182845904523536028747135266249775724709369995957496696762772407...
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