🚀 go-pugleaf

Attention Mandelbrot zoomers, ever wanted to zoom in on the island
Mandelbroties in the spike toward -2?  Ever need a road map into
the deep zoom zone?  Here is a quicky C program for approximate
coordinates of islands corresponding to period k.

#include <stdio.h>
#include <math.h>
void main(){double y=1.025;int k;for(k=0;k<21;++k){
printf("%3d %20.16lf\n",k,-2+1/y); y=4*y+0.5*k; } }

It gets more accurate with increasing k, but it better since the 
islands get smaller.

The Grand Internet Parallel Processing Project to compute the area
of the Mandelbrot set (GIPPP#3) using the distance estimator method 
has finished and here are some preliminary results including a pixel 
counting area estimate: A = 1.506644118 which includes 0.000615624 
from unescaped non-periodic pixels.  We used a maximum iteration limit 
of 65536.  A larger limit leaves the distance estimator results unchanged 
but does affect the pixel counting result.  Some part of the unescaped 
non-periodic area could have been escapers, so A should be considered 
high.  The Mandelbrot set was effectively partitioned into 524288x524288
cells.  To obtain a preprint of our paper (a postscript file) email
the word 'paper' without quotes to mset@kepler.ucsd.edu.

Areas were totaled as a function of period so a correction to the
distance estimation results can be made for areas which are known
'exactly'.  These include period 1 (3pi/8) and period 2 (pi/16).
Areas for periods up to 7 are also known (K. Briggs and J. Hill have
evaluated them as posted earlier).  The following table shows the
calculated and 'exact' lower bound areas through period 7.

Period       Lower bound
 k        area in period k,       'exact' value
 1      1.178089947490357000   1.17809724509617242
 2      0.196348107510587178   0.19634954084936207
 3      0.056424484007997877   0.056542418127676
 4      0.023205824115790110   0.023275137696299
 5      0.013068361060828690   0.0131154247261598
 6      0.008898768326736384   0.008934790
 7      0.005035557418435822   0.005057927

There are differences in the 10^-5 digit, indicating the accuracy for
each period.  Since these are lower bouunds, we expect all to be SMALLER 
than the exact value.  The result is the lower bound is 1.502771 and the 
upper bound is 1.554063.  The Mandelbrot set boundary was divided into 119 
cells.  Sixty net users tried to join the project with the first 41 
contributing to the calculation.  The list will be posted a little later.

Warmly, Jay  (GIPPP#3 author with Yuval Fisher).

Ref. Keith Briggs <kbriggs@spam.maths.adelaide.edu.au>
Yuval Fisher <fisher@inls1.ucsd.edu>
-- 
{ hilljr@jupiter.saic.com } begin writeln(3*ln(640320)/sqrt(163):17:15) end.
void main(){double sqrt(), y=1/sqrt(2.), a=.5, m=1,z; int n=0;
for(;m*=2,z=sqrt(1-y*y),y=(1-z)/(1+z),a=a*(1+y)*(1+y)-m*y,n<3;n++);
printf("%17.15lf\n",1/a);}