Fractal visionaries:

Today's fractal doesn't make much of a picture, but it's one of the strangest I've ever uncovered.  It doesn't even look like a proper fractal; it looks more like the greatly enlarged surface of a CD-ROM disk.  Or it could be a mysterious encoded message picked up by my computer and sent by a race of space aliens plotting to invade Earth.

Actually it's none of these.  It's just a Newton's fractal.  In Newton's method fractals no two areas ever actually touch.  The areas are always separated by an infinitely divided area of fractal chaos.  This fractal area of chaos between different roots is immediately obvious, but it also exists in the bays of points converging on the same root, where it separates the iteration bands.

Normally this chaotic area is too small to find, but by making the cutoff value sufficiently small the chaotic area can be made visible.  As one explores ever deeper into this chaotic area, one incredible pattern after another emerges, as the edges in one pattern when magnified break up into the next pattern.  And the patterns never stop.

But now, I must stop.  There's another alien message coming into my computer.  ;-)

Jim Muth
jamth@mindspring.com


START PAR FILE FOR 19.6=========================================

Alien_Message     { ; time=0:00:03.57-SF5 on P4-2000
  reset=1960 type=formula formulafile=basicer.frm
  formulaname=NewtonTest26 passes=t periodicity=10
  center-mag=-1.08216744913928/+0.634312891440502/\
  8.60215e+007 params=1/3/3/1/1/2.4e-032 float=y
  maxiter=90 inside=0 logmap=yes
  colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzz\
  zLzzz000555888BBBEEEHHHKKKOOOSSSWWW___ccchhhmmmsss\
  zzz00zG0zW0zj0zz0zz0jz0Wz0Gz00zG0zW0zj0zz0jz0Wz0Gz\
  00z00zG0zW0zj0zz0jz0Wz0GzVVzbVzjVzrVzzVzzVrzVjzVbz\
  VVzbVzjVzrVzzVrzVjzVbzVVzVVzbVzjVzrVzzVrzVjzVbzhhz\
  mhzqhzvhzzhzzhvzhqzhmzhhzmhzqhzvhzzhvzhqzhmzhhzhhz\
  mhzqhzvhzzhuzhqzhlz00S70SE0SL0SS0SS0LS0ES07S00S70S\
  E0SL0SS0LS0ES07S00S00S70SE0SL0SS0LS0ES07SEESIESLES\
  PESSESSEPSELSEISEESIESLESPESSEPSELSEISEESEESIESLES\
  PESSEOSELSEHSKKSMKSOKSQKSSKSSKQSKOSKMSKKSMKSOKSQKS\
  SKQSKOSKMSKKSKKSMKSOKSQKSSKQSKOSKMS00G40G80GC0GG0G\
  G0CG08G04G00G40G80GC0GG0CG08G04G00G00G40G80GC0GG0C\
  G08G04G88GA8GC8GE8GG8GG8EG8CG8AG88GA8GC8GE8GG8EG8C\
  G8AG88G88GA8GC8GE8GG8EG8CG8AGBBGCBGEBGFBGGBGGBFGBD\
  GBCGBBGCBGEBGFBGGBFGBDGBCGBBGBBGCBGEBGFBGGBFGBDGBC\
  G000000000000000000000000 }

frm:NewtonTest26 {; Jim Muth
a=real(p1), b=imag(p1), c=real(p2),
d=imag(p2), k=real(p3), f=imag(p3),
z=(pixel):
zx=z^b-a
zy=c*z*z
z=z-(d*zx/zy)^k,
|zx| >= f }

END PAR FILE FOR 19.6===========================================