icon

    Orbit in three dimensions defined by:

      p = lambda + alpha * magnitude + beta * (x(n)*zreal - y(n)*zimag)
      x(n+1) = p * x(n) + gamma * zreal - omega * y(n)
      y(n+1) = p * y(n) - gamma * zimag + omega * x(n)

      (3D version uses magnitude for z)

    Parameters:  Lambda, Alpha, Beta, Gamma, Omega, and Degree


This fractal type was inspired by the book "Symmetry in Chaos"
by Michael Field and Martin Golubitsky (ISBN 0-19-853689-5, Oxford Press)

To quote from the book's jacket,

  "Field and Golubitsky describe how a chaotic process eventually can
  lead to symmetric patterns (in a river, for instance, photographs of
  the turbulent movement of eddies, taken over time, often reveal
  patterns on the average."

The Icon type implemented here maps the classic population logistic
map of bifurcation fractals onto the complex plane in Dn symmetry.

The initial points plotted are the more chaotic initial orbits, but
as you wait, delicate webs will begin to form as the orbits settle
into a more periodic pattern. Since pixels are colored by the number
of times they are hit, the more periodic paths will become clarified
with time. These fractals run continuously.

There are 6 parameters:  Lambda, Alpha, Beta, Gamma, Omega, and Degree

    Omega  0 = Dn, or dihedral (rotation + reflectional) symmetry
          !0 = Zn, or cyclic (rotational) symmetry

    Degree = n, or Degree of symmetry