🚀 go-pugleaf

          My Mandelbrot Flower
        (C) by Jay Hill, August 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, 1.17809724509205, is simple.
    Since for a cardioid it is exactly 3pi/8, you see.
  So is 0.19634954084867, period two's bounding circle.
    But for the area, 0.05654241812767, in period three

  Just sum Giarrusso's form, it's analytical.
    The area, 0.02327513769624, in period four
  Is found by integrating Stephenson's polynomial.
    Find the fifth area, 0.01311506747875, 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.
    Thanks, 0.008934790525 is his estimate.

  Keith found 0.00505792907 in period seven,
    And I'm adding 0.00446801353778 for eight.
  Of the Mandelbrot area, some make an obsession,
    staying at the terminal until very late.

  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.



Verse 6 is new, announcing two more area estimates.
Note the acknowledgement of GIPPP#3 participants at 
the end of this article.

On bit.listserv.frac-l <SMTPUSER@GACVAX1.BITNET> asked
'Query:  What is the area of the Mandelbrot Set?' way
back on 18 Nov 89. Then on 30 Jun 1992 from S. Wukovitz 
<SXW103@PSUVM.PSU.EDU> the question went out again.  
Gerald Edgar's lengthy repley did not stop the inquiring 
minds.  On 10 Feb 1993 Jon Noring suggested that 5000+ idle 
computers on the net could compute the Mandelbrot set area 
coining it the Grand Internet Parallel Processing Project.  
Adam S. Nealis agreed, 'Anyone else game for this?'   

'Hurry, hurry, hurry. Get your computational pieces now.' 
shouted Yuval Fisher <fisher@INLS1.UCSD.EDU> on 17 Feb 
1993.  And GIPPP was off and running.

With my posting of an early version of this poem, I 
showed my secret wish for an analytical value for the
area.  I obtained the area for period three by integrating
formula provide by Dante Giarrusso and Yuval Fisher.
'That is three petals down, infinity to go',  I wrote
on 12 Apr 1993.  And after the poem I, with Yuval Fisher, 
announced GIPPP#3, a project to 'waste a great deal of 
time and effort, not simply by doing a pretty useless 
calculation locally, but by having many many other people 
all over the world waste their time too.'  ;-)

Then I integrated period 4 and 5 polynomials published
by John Stephenson.  Keith Briggs has also integrated
Stephenson's polynomials through period seven. Keith's
values for periods six and seven are shown here. High
degree polynomials for the Mandelbrot Set boundary
have been derived through period 7 (Dante Giarrusso)
and period 8 (John Stephenson).  John Stephenson has 
also tabulated the component centers through period 
eleven.  Using independent methods, Keith Briggs' and 
my values are now in agreement to about 11 digits for 
the areas through period 7.  

At least 3 GIPPPs have been run.  Area bounds for the 
Mandelbrot Set can be found with the Yuval Fisher/Jay 
Hill GIPPP#3 program.  It determines bounds using 
distance estimators and quad dividing.  A preprint is 
available.  Values of the bounds were found by Bill 
Broadley using our GIPPP#3 program to depth 17.   

The GIPPP#3 project evaluated the bounds to recursive 
depth 19 using about 50 machines by 40 netters with
preliminary results completed.  See netter list below.  
Since some machines may have had greater precision than 
others, or other errors may have crept in, there remains 
some doubt about the result.  A verification of the 
results will be attempted soon.

Robert Manufo used the pixel counting method to estimate 
the interior area.  His project is known as GIPPP#2.

The following contributed to the GIPPP#3 computation of 
the Mandelbrot set area.  Thank you all!

AGDP@gar.union.edu <DAVIS PHILIP, A G>
adam@lbs.lon.ac.uk <Adam Shaun Nealis>
albertel@cps.msu.edu  <Guy Albertelli>
broadley@neurocog.lrdc.pitt.edu <Bill Broadley>
cc_s525@kingston.ac.uk <Francis Bell>
davidb@mcil.comm.mot.com <David Bar-On>
emp@prs.hmc.psu.edu <Emmanuel M. Paul>
fisher@inls1.ucsd.edu <Yuval Fisher>
hagmanti@cps.msu.edu <Tim Hagman>
herman%athens.nosc.mil@nosc.mil <John W. Herman>
hilljr@Jupiter.SAIC.Com <Jay R Hill>
holliday@sioux.unl.edu <Dave Holliday>
jag@wfw.wtb.tue.nl <Bram de Jager>
jason@trg.saic.com <Jason S. Holloway>
jco2p@faraday.clas.virginia.edu <James Clifton Osborn>
jennings@abb.com <Thomas J. Jennings>
jewright@mathlab.mtu.edu <Jason Wright>
jkanerva@chance.ling.indiana.edu <Jonni M. Kanerva>
macleod@cu50.crl.aecl.ca
mcdavey@maths.tcd.ie <Matthew Davey>
meltsner@chipman.crd.ge.com <Kenneth J. Meltsner>
mitchell@mitch.larc.nasa.gov <Kerry Mitchell>
mlh@cygnus.ta52.lanl.gov <Michael L. Hall>
molson@bml4380.cpg.cdc.com <Mark Olson>
netcom.com!autodesk!autodesk.com!danielg <Daniel Green>
poe.austin.eds.com!austin.eds.com!jefftep <Jeffrey Grills>
poea0@elnw09.centel.com <Art Poe>
rmag@inforisc.cr
rnelson@uc.d.umn.edu <Ron Nelson>
rolfs@mcil.comm.mot.com <Rolf Salomon>
ros@apex.spectrospin.ch <Roland Stern>
s9329053@SANDCASTLE.COSC.BROCKU.CA <Darcy Boese>
smb3u@kiptron.psyc.virginia.edu <Steven M. Boker>
terescoj@tardis.union.edu <Jim Teresco>
thomas.riehm@medizin.uni-ulm.de <Thomas Riehm>
tim@ibmoto.com <Tim Olson>
tk_ounst@pat.uwe-bristol.ac.uk <Toby Ounsted>
tom@vlsivie.tuwien.ac.at <Thomas Winder>
uh@acm.Princeton.EDU <Uwe Hollerbach>
wgilbert@fatou.uwaterloo.ca <William Gilbert>
zordan@cribi1.bio.unipd.it <Zordan Mauro>

Warmly,
--Jay 'Not tonight honey, I'm connected to an 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<4;n++);
printf("%17.15lf\n",1/a);}