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