Tale of a Lost Midget

Today's fractal has a little story behind it.  As you know if you live in the northern hemisphere, there is a comet now visible in the evening sky -- the brighest in many years.  The other evening as I sat comet gazing, I remembered a fractal comet I had discovered a year or so ago.
I recalled some work I had done in the ultra-low-exponent Mandeloids.  As is common knowledge, the larger the exponent of Z, the more and larger the midgets become, and the less interesting they become in the resulting fractal, until finally the midgets become vague, lopsided circles with a bit of fern- like detail in between.  The most interesting midgets appear in the classic Mandelbrot set, with an exponent of 2.

But what of the midgets in the Mandeloids with exponents less than 2?  If the higher-order midgets are less interesting, one would think the lower-order midgets would be even more interest ing.  To check this out, I tracked down a few midgets in the Z^(sqrt(2))+C figure.  In this range the midgets become ever harder to find, because they slip off the screen into imaginary planes, but I managed to find a few that were still visible.  The results were interesting, but not quite worth the effort.

Then I thought of the most obvious midget of all -- the one on the negative tail at -1.76.  What happens to this midget when the exponent drops below 2?  A quick check told me that it vanishes into some obscure imaginary space.  Then I remembered the cmplxmarksmand formula in Fractint, which splits and spreads the fractal along the negative tail.  What would happen to the buried tail midget if I tried the cmplxmarksmand trick on a Mandeloid with an exponent of 1.5 or so?

Surprise.  The additional term pulled the buried tail midget out from its hiding place onto the screen, where its distortions were clearly visible.  Now, by carefully adjusting the exponent and additional term, I had a means of keeping the midget in sight while I lowered the exponent to any arbitrary value.

I stopped at an exponent of 1.065, which has the midget resembling a comet, and is the image built by the attached formula and parameter file.  But 1.065 is by no means a lower limit.  I'm hoping to track this midget, which actually still is the main midget on what's left of the negative tail, down to an exponent of under 1.01.

The image takes 15 minutes to draw on a 486-100mhz, and of course is posted to ABPF.

Jim Muth
jamth@mindspring.com


START FORMULA===================================================

HaleBopp_Falls_to  { ; time=0:00:31.36-SF5 on P4-2000
  reset=1950 type=formula formulafile=basicer.frm
  formulaname=JimsCompMand passes=1 logmap=yes
  center-mag=-1.14547434438765100/+0.006011885318538\
  45/421.2545/1/-85/0 params=1.065/0/1.651/0/0/0
  float=y maxiter=860 inside=255 periodicity=10
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  S0XT0ZU0aV0dX0fY0iZ0gWzzz }

frm:JimsCompMand {; Jim Muth
z=c=pixel:
z=z^p1*(c^(p2-1))+c,
|z| <= p3+100 }

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