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GIPPP#3 and other Mandelbrot Set area calculations


          My Mandelbrot Flower
          (C) by Jay Hill, 1993

  She loves me, she loves me not
    Will it be no or will it be yes?
  From my flower of Mandelbrot
    The answer will be anyone's guess.

  It's the Fractal Set of Mandelbrot,
    With quadratic, cubic and quartic petal
  Whose total area is dearly sought.
    But each component is transcendental.

  The area in period one, 1.17809724509617, is simple.
    Since for a cardioid it is exactly 3pi/8, you see.
  So is 0.19634954084936, period two's bounding circle.
    But for the area, 0.05654241812768, in period three

  Just sum Giarrusso's form, it's analytical.
    The area, 0.023275137697, in period four
  Is found by integrating Stephenson's polynomial.
    Find the fifth area, 0.01311506748, as before.

  Now roots seem chaotic, I can't get a fix!
    The petals are tiny and hard to integrate.
  Keith Briggs computed the area in period six.
    Thank you, 0.00893479 is his estimate.

  With roots all a tangle and sums so tiny,
    Who can know the area beyond period eight?
  Pixel counters and distance estimators a plenty,
    With Richardson's method try to extrapolate.

  Thanks to Bill Broadley and dozens on internet
    Who helped Yuval Fisher and Jay Hill show,
  The bounds on the area of the Mandelbrot Set
    Are 1.5613027 from above and 1.5031197 from below.

  Will the net of computers, each plucking a petal,
    Someday circumscribe the Mandelbrot infinity?
  How near Robert Manufo's 1.506598 will it settle?
   We'll all know when the clock strikes eternity.


I obtained the area for period three by integrating
formula provide by Dante Giarrusso and Yuval Fisher,
and periods 4 and 5 by integrating polynomials due
to John Stephenson.  Keith Briggs has integrated
Stephenson's polynomials through period six. Keith's
value for period six is shown here. The bounds on the
area of the Mandelbrot Set, determined using distance
estimators and quad dividing to depth 17. Bill Broadley,
thank you, used our GIPPP#3 program to get this result.
The GIPPP#3 project is evaluating the bounds to depth
19 with results still pending.  Robert Manufo has used
the pixel counting method to estimate the interior.
To get our paper, email to mset@kepler.ucsd.edu the
word 'paper'.

Jay 'Not tonight honey, I have a hyperbolic component' 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<3;n++);
printf("%17.15lf\n",1/a);} /* 9 billion digits of pi, anyone? */