Fractal visionaries and enthusiasts:  

The expression behind today's image combines 1.96 negative parts of Z^(2.3) with 0.98 parts of Z^(4.6), then adds C on each iteration.  When written as a function, this expression has two critical points, zero and -1.  (I previously had this wrong.)  The MandAutoCritInZ formula automatically initializes Z to zero, but Z can be initialized to -1 by entering that value as the real(p4) parameter.

For some strange reason, the minibrots in the fractal that is produced when initial Z is set to zero are quadratic M-sets, while the minibrots in the fractal with initial Z set to -1 are of order 2.3.  This might be because the value of -1 is a point of inflection on the graph of the function, while the value of zero is the tip of a parabola.  In both cases the derivative is zero and the fractal will therefore be undistorted.  I'm not sure if this is true of all fractas created by expressions with multiple critical points.

Regardless of the correctness of the math, today's image lies in the fractal that results when Z is initialized to -1.  The minibrot at its center is of order 2.3 as expected.

The rating of a 7 is perhaps a bit conservative.  I did put a half hour or so of effort into the coloring.

The calculation time of 8 minutes is slow.  Much time may be saved by viewing the finished image on one or more of the FOTD web sites.

A typical late winter day passed uneventfully here at Fractal Central, with lots of clouds during the morning and a clearing in the afternoon.  The temperature of 37F +3C was right on the mark.  The fractal cat played with his catnip ball until the sun appeared.  Then he stretched out on his shelf in the southwest window.  The humans made it through the day with a minimum of annoyances other than the usual miseries of the flu.

The next FOTD will be posted when it is ready.  Until whenever, take care, and I hear from Old Fractal Central back in Bodymore Murderland, (as the locals there call Baltimore, Maryland), that the weather forecast there today is partly cloudy with a near 100-percent chance of scattered bullets.

Jim Muth
jimmuth@earthlink.net


START PARAMETER FILE=======================================

The_Test_of_Time   { ; time=0:08:00.00 SF5 at 2000MHZ
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=MandAutoCritInZ function=cosxx float=y
  center-mag=+0.102350608821/-0.60041985916/3.4e+008\
  /1/150/0 params=-1.96/2.3/0.98/4.6/0/0/-1/0
  maxiter=5400 inside=0 logmap=365 periodicity=6
  colors=000KMUKMUKMUKMUKMUJMUIMWGOXEQXCSXAUX8WX6X`B\
  YdG`gLbkQfnVjrZnuTrzNvzIxzCzz7pzIczTUmcUcmbWzHvMJn\
  zPNzPNzPNzQNzQNzRNzRNzRNzSNzSNzTNzTNzgQyfPxfPwePwe\
  PvePudPWSxBt9JoGQjNXeUc``jWgqRmrTkrVjrXhsZgs`esbds\
  dbteatg_tiZtkXumWuoUuqTurSsnUqkVphWneYlbZk__iXagUb\
  fRcdOebLfaIg_FiYCjX9k`WScY9o_KzIUzJTzJTzKSzKSzLSzM\
  RzMRzNQzOQzOQzPPzQPzQOzROzSOzSNzTNzdWzaTzZRzWPqqJp\
  pKooKonKnnKnmKmlKmlKlkKzjLziLziLzhLzgLzgLzfLzeLzdL\
  zdMzlJzrGzzCpzEmuGguIgvKfvMewOewQbsR`pSYmSWjTTfTRc\
  UP`UMYVKVWHRWFOXCLXAIY8FY7JX7NX7QW6UW6XW6`V5cV5gU5\
  jU4nU0zm0zmmzTmzQGSNTT_RVaQXbOZdNaeKdfKhhKizUpkUpl\
  Uomcnocmpcmqcqnhozjnpmmfmmcrm`vmZzmWzmQzmTzmWzkZzm\
  VzmQzmMzmIzmIzmIzmIzmJzcJzcJzcJzcJzcJzcJzcJzcJzcJz\
  cJzcKzcKzXKzXKzXKzTOzXKzWKzWKzWKzWKzWLzWLzWczWczWc\
  zVczVczVmzVmzVmzVmzVmzVzzVzzUzzUzzUzzUzzUzzUzzUzzU\
  zzUzzTzzTzzTzzTzzTzzTzzTz }

frm:MandAutoCritInZ {; Jim Muth
a=real(p1), b=imag(p1), d=real(p2), f=imag(p2),
g=1/f, h=1/d, j=1/(f-b), z=(((-a*b*g*h)^j)+(p4)),
k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel):
z=k*((a*(z^b))+(d*(z^f)))+c,
|z| < l }

END PARAMETER FILE=========================================