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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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Apr 24, 1991, 3:03:25 PM4/24/91

to

In article <14...@ccncsu.ColoState.EDU>,

bo...@handel.cs.colostate.edu (dave boll) writes:

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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