An Interesting Property of the Mandelbrot Set .... An Escape Limit
G. A. Edgar
fra...@archives.math.utk.edu wrote:
>Dear colleagues .....
>
>Someone recently sent out a reply to a question, in which he included
>something I had never seen before ... I regret that I have lost the actual
>message and response .
>
>The respondent was quoting a theorem in which complex values for C in the
>classic M-Set definition lieing on a vertical line through the point x = -.75
>share a sepcial relationship ... namely, the number of iterations needed for
>such points to escape a circle of radius 2 (at the origin) are related to PI as
>follows:
> C = -.75 +/- e i and n iterations are needed for escape
> then lim (as e goes to 0) of n times e approaches PI
>
>Two questions, please .....
>
> Has anyone seen a proof of this?
Some Usenet discussion in early 1992 in alt.fractals and sci.math,
under the title "Pi and the Mandelbrot Set".
See: Peitgen, et.al., _Chaos and Fractals : New Fontiers of Science_.
pages 559--862. Note the footnote references to Usenet articles.
But I am not aware of any place that keeps Usenet articles
on file indefinitely. I do not have it in machine-readable form,
just hard-copy.
>
>Does any such interesting (or even analagous) relationship apply to other
>bifurcation points along the negative-x axis?
>
Presumably, all of them. But I have seen it discussed only
for c = -3/4 and c = +1/4. At c = +1/4 we get something similar,
but we must use the number of iterations times the square-root
of e.
At the end of the Usenet discussion in 1992, I proposed looking at
c = -5/4 or c = -2, but no one did.
--
Gerald A. Edgar ed...@math.ohio-state.edu
Department of Mathematics telephone: 614-292-0395 (Office)
The Ohio State University 614-292-4975 (Math. Dept.)
Columbus, OH 43210 614-292-1479 (Dept. Fax)
david petry
ed...@math.ohio-state.edu (G. A. Edgar) writes:
>>
>>Does any such interesting (or even analagous) relationship apply to
other
>>bifurcation points along the negative-x axis?
[we're talking about the product of the number of iterations needed
to escape to infinity in the Mandelbrot formula, times the distance
from the x axis.]
>Presumably, all of them. But I have seen it discussed only
>for c = -3/4 and c = +1/4. At c = +1/4 we get something similar,
>but we must use the number of iterations times the square-root
>of e.
>At the end of the Usenet discussion in 1992, I proposed looking at
>c = -5/4 or c = -2, but no one did.
Well, I looked at it for -5/4, and I even posted the result at
the time. In fact, I posted a "C" program so that others could
investigate the result for themselves.
As I recall, I found that the product is always a half-integer
multiple of pi. I'm fully aware that it cannot always be a half-
integer multiple, just by continuity considerations. But the
empirical evidence was surprisingly strong.
G. A. Edgar
Sorry for the mis-type: It is in fact 859--862.
david petry
>>
>>Does any such interesting (or even analagous) relationship apply to
other
>>bifurcation points along the negative-x axis?
[we're talking about the product of the number of iterations needed
to escape to infinity in the Mandelbrot formula, times the distance
from the x axis.]
>Presumably, all of them. But I have seen it discussed only
>for c = -3/4 and c = +1/4. At c = +1/4 we get something similar,
>but we must use the number of iterations times the square-root
>of e.
I did a little investigation of this problem, and I think the results
are pretty interesting.
Let me go over the known result first. The point (-3/4, 0) is a
bifurcation point (or node) of the Mandelbrot set. That is, if that
point is removed, the set becomes disconnected, and also a vertical
line through that point intersects the Mandelbrot set only at that
one point. It was noticed that for points along that line, the number
of iterations of the Mandelbrot formula ( x(n+1) = x(n)^2 + P , x(0) = 0,
where P is the point being considered) to escape to infinity (actually
just past the circle of radius 2) times the distance from the x axis
is a good approximation of pi when the point is near the x axis. This
is surprising. The question is, what happens at the point (-5/4,0)
and other bifurcation points.
First of all, there are tiny fingers of the Mandelbrot set which extend
over the vertical line through (-5/4, 0), so the problem is not exactly
analogous to the former problem. However, the curve defined by
x = -5/4 - 3.3y^2 does not intersect the Mandelbrot set exept at
(-5/4, 0). Along this curve, N*y -> pi/2 as y -> 0, where N is
the number of iterations required to escape, and y is the distince
to the x axis. The number "3.3" in the equation of the curve was
determined empirically using FRACTINT, and the form of the equation is
just a guess.
Along the veritcal line through (-5/4, 0), the behavior is even more
interesting. The value of N*y jumps around wildly as the line is
traversed, and since it does intersect the Mandelbrot set, it must
be infinity at times. However, picking points at random along the
line seems to never give a point on the Mandelbrot set, and in fact
for any such point the product N*y is a multiple of pi/2. Table I
gives some actual values for a sampling of points on the line.
I investigated also the next bifurcation point, which occurs at
(-1.36809894, 0). The curve which never intersects the Mandlebrot
set is x = -1.36889894 - 15.3y^3, and N*y -> 0.702 as y->0 along
that curve. The number 0.702 doesn't seem to be related to pi in
any simple way.
TABLE I (notice that N*y*(2/pi) varies wildly for small changes in
in y, but it is always nearly an integer!)
y N*y*(2/pi) N
0.000001373 3.000026926 3432215
0.000001374 14.999995432 17148426
0.000001375 7.00001999 7996804
0.000001376 25.999887838 29680616
0.000001377 7.000033859 7985205
0.000001378 2.000039069 2279865
0.000001379 3.000051942 3417310
0.00000138 2.000016361 2276535
0.000001381 2.000014136 2274884
0.000001382 7.000088435 7956377
0.000001383 18.00004801 20444258
0.000001384 3.000026191 3404935
0.000001385 7.000029386 7939076
0.000001386 1.000036473 1133372
0.000001387 2.000015798 2265045
0.000001388 10.000044866 11317027
0.000001389 4.00008006 4523622
0.00000139 6.000017093 6780435
0.000001391 3.000026038 3387800
0.000001392 12.000012926 13541362
0.000001393 7.000054323 7893510
0.000001394 1.00003584 1126867
0.000001395 3.000073214 3378139
0.000001396 5.000089542 5626162
0.000001397 2.000056385 2248877
Here's the "C" program that produced the above table.
#include <stdio.h>
#define maxdepth 100000000
#define XI (-1.25) /* try also XI = -0.75 */
#define PI 3.1415926536
long int
depth(double xi, double yi) {
long int k=0;
double x = xi, y = yi, a=0.0, b=0.0;
while( a+b < 4.5 && k++ < maxdepth) {
a = x*x;
b = y*y;
y = 2.0 * x * y + yi;
x = a - b + xi; }
return k; }
main() {
double s= 0.000001373; /* start at s, increment by p */
double r,p = 0.000000001; /* for k iterations */
int k; /* note: I chose s to be some */
long int m; /* small number picked at random */
for(k=0;k<10;k++, s += p) {
m = depth(XI, s );
/* m = depth(XI - 3.3*s*s, s ); */ /* try this instead! */
r = (double) m * s;
printf("%11.9f %12.9f %ld\n", s, r*(2.0/PI), m); }
return 1; }
Bengt Mansson
>
>
>I did a little investigation of this problem, and I think the results
>are pretty interesting.
< Agreed, but I delete here since I just want to add a little remark to
the next case. >
>I investigated also the next bifurcation point, which occurs at
>(-1.36809894, 0). The curve which never intersects the Mandlebrot
>set is x = -1.36889894 - 15.3y^3, and N*y -> 0.702 as y->0 along
>that curve. The number 0.702 doesn't seem to be related to pi in
>any simple way.
Playing around for a while with my calculator I found this,
pi/(2*sqrt(5)) = 0.702481... . Can that be of some value?
< Deleted >
Regards, Bengt.
=======================================================
Bengt Mansson, Krondammsv 57, S-433 43 Partille, Sweden
e-mail: ben...@algonet.se, fax: +46 31 44 28 06
=======================================================