Fractal visionaries and enthusiasts: 

While yesterday's fractal was a non-critical loser, today's fractal, which is critical, is a notable improvement.  It is a scene in the fractal that is generated when yesterday's fractal is re-calculated with Z initialized to the first of the three critical values of the iterated expression.  This value appears as the real(p5) parameter of today's image.

At first glance, today's parent fractal, which is critical, appears little changed from yesterday's image.  In its depths however it is totally different, with intact minibrots lurking where least expected.  Today's image, which I named "Spiraling Onward", lies on the west side of the north branch of what could be called a kind of Seahorse Valley.

The math value remains a superb 8, while the artistic value has risen from an abysmal 3 to a very respectable 7.

The three critical values of initial Z in yesterday's loser image were found by entering the same parameters into the CriticalPoints formula, which appears in today's parameter file.  This calculates the Parabolic plane of the iterated expression and creates an image that is a rough graph of the expression.  All that is then necessary is to find the 'Y' coordinates of the tips of the three obvious though distorted parabolic curves.  I always choose the place where the default red adjoins light blue.  The tedious part is continually stretching the curves horizontally to determine the exact 'Y' coordinates.  The second image, named "Points Revealed", shows the resulting image-graph that must be analyzed.

The method is tricky, especially with negative and fractional exponents, and sometimes does not work, but it is still better than trying to write a formula to do the job or finding the points by hand calculation.

The calculation time of 2-2/3 minutes is somewhat slow, and I'm not sure the result is worth the wait.  Luckily, the web sites can speed up the viewing.

A mix of sun and clouds, and a temperature of 88F 31C made today quite summer-like here at Fractal Central.  The fractal cat was undecided whether to stretch out or curl up, and finally settled half stretched in my favorite chair.  The fractal people passed the day getting things done.

The next FOTD will soon be posted.  To see how soon, check back here frequently, and before you know what happened, the next fractal will be here.  Until whenever the fates decide, take care, and let's admit it -- according to science, objective reality is probably an illusion, which means that our mighty scientific method is founded on nothing more than an illusion.

Jim Muth
jimmuth@earthlink.net


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

Spiraling_Onward   { ; time=0:02:40.00 SF5 at 2000MHZ
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=MandelbrotMix3a function=ident passes=t
  center-mag=-0.511568295107038/+0.05469492019194518\
  /1.23e+010/1/140/0 params=0.1/2/1/3/0.5/6/0.01/-1/\
  0.2246790860663939/0 float=y maxiter=1750 inside=0
  logmap=138 periodicity=6 mathtolerance=0.05/1
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  KEcGBcC7c83c4xchhmXUmMFmBwmcdmlczhczdcz`czXmzUmzQm\
  zMmzImzFmzBmz7mz3mzSmznzzizzdzzmzzmzzmzzmzzmzzmzzm\
  zzmzzmzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
  zzzzzzzzzzzzzzzzzzzzzzzzz }

frm:MandelbrotMix3a {; Jim Muth
z=real(p5), c=fn1(pixel), a=real(p1), b=imag(p1),
d=real(p2), f=imag(p2), g=real(p3), h=imag(p3),
j=real(p4), k=imag(p4), l=imag(p5)+100:
z=(a*(z^b))+(d*(z^f))+(g*(z^h))+(j*(z^k))+c,
|z| <=l }

Points_Revealed    { ; time=0:00:20.00 SF5 at 2000MHZ
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=CriticalPoints function=ident
  passes=1 center-mag=0/0/0.6666667
  params=0.1/2/1/3/0.5/6/0.01/-1 float=y
  maxiter=500 inside=0 periodicity=6
  colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzz\
  zLzzz000555888BBBEEEHHHKKKOOOSSSWWW___ccchhhmmmsss\
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  OESSESSEOSELSEHSEESHESLESOESSEOSELSEHSEESEESHESLES\
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  G8AG88G88GA8GC8GE8GG8EG8CG8AGBBGCBGDBGFBGGBGGBFGBD\
  GBCGBBGCBGDBGFBGGBFGBDGBCGBBGBBGCBGDBGFBGGBFGBDGBC\
  G000000000000000000000000 }

frm:CriticalPoints  {; Jim Muth
z=imag(pixel), c=fn1(real(pixel))
a=real(p1), b=imag(p1), d=real(p2), f=imag(p2),
g=real(p3), h=imag(p3), j=real(p4), k=imag(p4):
z=(a*(z^b))+(d*(z^f))+(g*(z^h))+(j*(z^k))+c,
|z| < 10000 }

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