🚀 go-pugleaf

On Mon, 11 Oct 1993 15:40:00 BSC 
"Fausto A. A. Barbuto (Totxo)" <BJ06@C53000.PETROBRAS.ANRJ.BR> wrote:

[...]
->              F I N A L   R E S U L T S
->  p-axis grid (NPIX) =       20000
->  q-axis grid (NPIY) =       10000
->  Iteration limit (ITERMAX) =   10000000.0000000000
->  Total number of points in the window =   200000000.000000000
->  Total number of points inside Mandelbrot =   53560708.0000000000
->  Ratio =  0.267803540000000007
->  Area of the window =   2.81250000000000000
->  --------------------------------------------------
->  MANDELBROT SET AREA =   1.50639491250000002
->  IN PERIODS 1, 2 =   1.43028393750000005
->  IN UNDECIDED =  0.325687499999999954E-04
->  DECIDED AREA =   1.50636234375000000

The line 'IN PERIODS 1, 2' likely should read 
  'IN PERIODS 1, 2 AND PERIOD 3 BUDS'
since that sum is 1.430582789 and I gave Fausto a test for these 
components.  My period 3 test checked for points in a circle 
completely contained in the peroid 3 bud.  The period 3 cardioid 
at -1.75 (area=0.0004064146471) was not included in the test.

hilljr@jupiter.saic.com (Jay R. Hill) writes to the list FRAC-L and in 
article <1993Oct11.200319.21531@ast.saic.com> 
[...]
|> This is to be compared to the calculations of Fisher/Hill/Broadley
|> who obtained bounds on the area 1.5613027 from above and 1.5031197
|> from below.

On Tue, 12 Oct 1993 14:18:13 EDT Gabor Csejtey said:
>(We need a theoterical proof that the the aera of the set is PI/2.)

On Tue, 12 Oct 1993 15:13:28 EDT Valdis Kletnieks <VALDIS@VTVM1.CC.VT.EDU>
wrote:
>Is there a good reason why we "need" one, given that I see the value
>of PI/2 being 1.570796... and we've gotten *upper* bounds on the area
>of the Mandelbrot set of 1.56....?
>
>Yes, it would be interesting if the area *were* exactly pi/2 - but first
>you'll have to find the error in some of the "upper bound" calculations
>that have been posted here..

The upper bound should be good to all digits shown, since we used a 
double precision calculation of the area of cells completely outside 
the Mandelbrot Set.  We used distance estimators to determine that the 
nearest point in the Set is outside each cell in question.  Also, if 
a cell was completely inside the Mandelbrot Set, as determined using 
a different distance estimator formulation, we included its area in
the lower bound.

Warmly, 
Jay 'Not tonight honey, I only need O(ln 2^n) more seconds' Hill
-- 
{ 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<4;n++);
printf("%17.15lf\n",1/a);}