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Feb 26, 1994, 9:12:26 AM2/26/94

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The phenomenon you discussed in your previous post is documented in

_Chaos and Fractals: New Frontiers in Science_ by Heinz-Otto Peitgen, Harmut

Jurgens, and Dietmar Saupe (c) 1992 Springer-Verlag. The pi/mset relation

is on page 859, and is attributed to Dave Boll, who made a USENET posting

in 1991, in <1992Feb26.2...@yuma.acns.colostate.edu>.

A mathematical "suggestion," rather than a rigorous proof, is only

provided; the text notes "Although this is not a rigorous proof for the

observed phenomenon, it provides a supporting argument for it" (pg. 862).

_Chaos and Fractals: New Frontiers in Science_ by Heinz-Otto Peitgen, Harmut

Jurgens, and Dietmar Saupe (c) 1992 Springer-Verlag. The pi/mset relation

is on page 859, and is attributed to Dave Boll, who made a USENET posting

in 1991, in <1992Feb26.2...@yuma.acns.colostate.edu>.

A mathematical "suggestion," rather than a rigorous proof, is only

provided; the text notes "Although this is not a rigorous proof for the

observed phenomenon, it provides a supporting argument for it" (pg. 862).

If you have evidence that your friend's discovery predates that

mentioned in _Chaos and Fractals_, you may want to inform the authors.

Peter Wang

Feb 25, 1994, 2:52:33 PM2/25/94

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There was a discussion in 1992 (concidentally with the same title,

"Pi and the Mandelbrot Set") in alt.fractals (as it was then called)

and sci.math. It began with an observation much like the one

in the previous article by Dave Boll. Unfortunately, I am not aware of any

place these discussions are archived. In this case, a book author

was lurking, and included some of the material in his book

(with reverence to Usenet, of course):

Peitgen, Jurgens, & Saupe, FRACTALS FOR THE CLASSROOM, Part two,

p. 431--434.

Explanations for the point -3/4 and the cusp at +1/4 were obtained.

A somewhat heuristic approach was found in the text of Guckenheimer

& Holmes, NONLINEAR OSCILLATIONS..., sec. 6.8.

--

Gerald A. Edgar Internet: ed...@math.ohio-state.edu

Department of Mathematics Bitnet: EDGAR@OHSTPY

The Ohio State University telephone: 614-292-0395 (Office)

Columbus, OH 43210 -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)

Feb 25, 1994, 2:14:01 PM2/25/94

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Several years ago a dear friend (now deceased) Burton C. Gray found an

interesting empirical relationship between Pi and the Mandelbrot set

(MSET) which I haven't seen mentioned before (please correct me if I'm

wrong).

interesting empirical relationship between Pi and the Mandelbrot set

(MSET) which I haven't seen mentioned before (please correct me if I'm

wrong).

Let I(c) be the escape iteration count for the initial point c (a complex

number); that is, it counts the number of iterations through the familiar

formula z = z^2 + c, before the point can be shown to escape from MSET

using a rule such as abs(z) > 2.

Consider the points around MSET on the vertical line x=-3/4 (the dividing

line between the cardiod and the circle to its left), and the product

I(-3/4 + i*y) * y as y approaches zero. The iteration count increases to

infinity (because there's only one point on that line actually in MSET,

y=0, and the closer you get the higher the iteration count). The product

of one number increasing to infinity and another number decreasing to

zero is ...?

Empirically, it appears that the product is Pi, which we've confirmed to

nine significant places. That is, more mathematically

lim I(-3/4 + i*y) * y = Pi.

y->0+

There is another related formula concerning the line y=0 at the cusp of

the cardiod (1/4, 0):

lim I(x) * sqrt (x - 1/4) = Pi.

x->1/4+

I want to emphasize that these formulae are not supported by mathematical

proof, only empirical evidence.

1. Is this already known?

2. Are there other such formulae?

3. Can anyone prove this?

--

Bob Smith -- bo...@access.digex.net

Feb 28, 1994, 11:15:02 PM2/28/94

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>Peter Wang

Thanks for your response. My notes indicate that I was told of this

relationship on 14 Aug 85 and that I called John Hubbard at Cornell about

it in early Sep 85.

Mar 1, 1994, 8:35:08 AM3/1/94

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>A somewhat heuristic approach was found in the text of Guckenheimer

>& Holmes, NONLINEAR OSCILLATIONS..., sec. 6.8.

>& Holmes, NONLINEAR OSCILLATIONS..., sec. 6.8.

That date is 1983, so it precedes both Boll and Smith's unnamed friend.

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