Pi and The Mandelbrot Set

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Peter T. Wang

Feb 26, 1994, 9:12:26 AM2/26/94
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).

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

Gerald Edgar

Feb 25, 1994, 2:52:33 PM2/25/94

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)

Bob Smith

Feb 25, 1994, 2:14:01 PM2/25/94
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

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.

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.

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

Bob Smith

Feb 28, 1994, 11:15:02 PM2/28/94

>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.

Gerald Edgar

Mar 1, 1994, 8:35:08 AM3/1/94
>A somewhat heuristic approach was found in the text of Guckenheimer
>& Holmes, NONLINEAR OSCILLATIONS..., sec. 6.8.

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

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