April 26, 1997: Halebopp Falls to Earth | April 25 | April 28 | 1997 | FOTD Home |

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==============================================