Thread View: sci.fractals
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Started by Jay R. Hill
Mon, 11 Oct 1993 20:03
Re: MS Area, Oct. 11, 1993
Author: Jay R. Hill
Date: Mon, 11 Oct 1993 20:03
Date: Mon, 11 Oct 1993 20:03
51 lines
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2978 bytes
On the list FRAC-L <FRAC-L%GITVM1.BITNET@uga.cc.uga.edu> Fausto Barbuto
wrote:
> Rio de Janeiro, BRAZIL, October 11, 1993:
>
>
> ====================================================
> 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
> ====================================================
>
> Period detection by Jay Hill, Mar/Apr 1993.
> Computer code by Fausto Barbuto & Jay Hill.
> System IBM Risc/600 Power 530H.
> Execution time: not computed.
Looks like a considerable execution time, Fausto, very impressive.
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. Also Robert Manufo calculated 1.50659530 +- 0.00000209
by averaging 40 runs with grids approximately 8192x4096. Fausto's
result is about 95 standard deviations below Robert's result.
I hope we can settle this someday, since in theory exact areas of
components can be calculated. The total through period eight is
1.48584014237616. That is the sum of 236 components. To get the
area to accuracy comparable to that claimed by pixel counters will
require sums to beyond period 64, more than 18446744073709551616
components ;-]. Of course most of these are ignoreable. Since
computing machines are increasing in speeds, doubling in speed every
2 years, I should finish this sometime in 2034. ;-) Stay tuned.
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);}
Re: MS Area, Oct. 11, 1993
Author: Jay R. Hill
Date: Tue, 12 Oct 1993 21:42
Date: Tue, 12 Oct 1993 21:42
60 lines
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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);}
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