; April 26, 1997: Halebopp Falls to Earth
;
; halebopp
;
; 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
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}

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

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