🚀 go-pugleaf

Looks like I'm a day late on posting this, but it still has some useful
information in it.  I just read Brian Hunt's posting with some very
interesting information concerning the fractal foam equations, so some of
this may be redundant.  Anyway, here's what I've come up with.

The 8 end points of the attractors are:
   A =  ( (-1 - 3*a^2)/4 , a/2 )   <-- consistent with Brian Hunt
   B = map(A)
   C = ((3*a^2 + 2*sqrt(3)*a + 1)/8 , (3*sqrt(3)*a^2 - 2*a + sqrt(3)/8)
   D = map(C)
   E = map(D)
   F = ((3*a^2 - 2*sqrt(3)*a + 1)/8 , (-3*sqrt(3)*a^2 - 2*a - sqrt(3)/8 )
   G = map(F)
   H = map(G)

To find the value of 'a' I set
    E = map(H) and H = map(E)
so that the gap between the attractors would be zero.  Using the equation
solver on my calculator, I found that there are four values of 'a' for
which this is true.

   a = +-1.02871376822  and  +-1.29099444874

The first pair of values is consistent with the article and with the
polynomial in Brain Hunt's post.

BH>77 - 33 a^2 - 9 a^4 - 27 a^6 = 0,

The second pair does not produce any inside basin, but does make an
interesting fractal when using a coloring scheme based on the number of
iterations before |z| gets bigger than some value (like the Mandelbrot
Set).

Happy Fractaling,
-Wes
==============================================================
  Wesley Loewer
  internet: loewer@star.harc.edu     telephone: (713)292-3449