🚀 go-pugleaf

In article <cFNkLc1w165w@west.darkside.com>, 
  grip@west.darkside.com (grip) writes:
|> I've been away for a bit and missed a few (hundreds) discussions. I'm 
|> curious about a few items:
|> 
|> 1) Has the interior of the M-set been measured? If so what is it?

Physicist: "Oh, the cardioid particle - let's see, its area is 3*pi/8."

Applied Physicist: "We should do this more carefully, what about that
circle attached.  To second order the area is 3*pi/8 + pi/16 = 7*pi/16."

Applied Mathematician: "But there are two more attached circles, and a 
cardioid at -1.75.  There are more but by counting these it's at least
1.4309892040732159237629752361947403823682576266903366670965787878249048546101"

Mathematician: "Its area is a waste of time to evaluate."

Engineer: "It is less than 4*pi, pi/2 is close enough for me."

Mandelmaniac: "Counting 200,000 conponents, it's at least 1.50612."

Fractal Poet:

            My Mandelbrot Flower

  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.

|> 2) Are there points in the M-set that are NOT limit points of some open 
|> subset of the M-set? 

Simple answer: No.  Even the orbits of chaotic points approach zero.

|> Thanks to anyone who knows the answers or knows if they are still open 
|> questions.
|> grip

Warmly,
Jay "Not tonight honey, I'm still counting petals" Hill

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main(){for(;e<3920;){u=(256*(s=(r=.0325*(m=e%80)-2)*r+(i=.047*(e/80)-1.128)*
i)-96)*s+32*r<3?28:16+32*r+16*s<1?35:0;if(u==(s=f=0))do g=s*s-f*f+r;while((f
=2*s*f+i)*f+(s=g)*g<4&&++u<29);putchar(_[++e>3840&&m<36?35-m:m>78?36:u]);}}