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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: I should have realized it. I see from the Fractint list that the 'bug' in the program is not a bug at all, but actually a result of the multi-valued nature of the complex log function. It turns out that the mirror images that result from switching the (p1) and (p2) parameters in the MandelbrotMix formulas are both correct. In fact, an infinity of other 'correct' images could be drawn from the same basic formula. I have a formula in my collection named MandelbrotBC1, (BC stands for branch cuts), which takes advantage of the multi-valued complex log function. I have posted quite a few FOTD images created by this formula. I also have a formula hidden somewhere which raises a complex exponent of Z to a power, then extracts the same root. Because of the two multi-valued steps, the final image has little resemblance to the starting image. Today's image is what I call a 'dry-lake', a term invented by Lee Skinner, who first drew my attention to these features. The symmetry is there, but the midget at the center is not. What has happened is that the slice of today's image has passed very close to a midget, but has not cut through it. By changing the (p4) parameter, the slice could be moved until it cuts the now-invisible midget. This is the reason I added the extra parameter. But finding the skimmed midget is not as easy as it seems. The C value changes along with the Z value, and, when the (p4) parameter is changed, the center of the dry lake must be chased as it moves off the screen. Today's image lies in the north branch of the East Valley area of the prominent Mandeloid feature located to the northeast of the 'fan' of the parent fractal. The elements in this area are very skeletal, consisting largely of inside points, so to produce a good picture, I set the inside to <atan> and let the iterations add up. The resulting image rates a 6 -- not bad, but with lots of room for improvement. The render time of 48 minutes could certainly be improved, and this improvement may be gained by visiting Paul's FOTD web site at: and downloading the image from there. The forecast rain never appeared here at Fractal Central on Thursday, and the temperature reached 75F 24C. As a result, the dynamic duo had an exceptional day in the outdoors. They even managed to stay out of trouble, (which isn't very difficult for near-13-year-old cats). Today is actually starting sunny, with a forecast of sun all day. It should be a good day for the cats. For me, it will be a good day when the work is finished. And in order to finish, I must start. So until next time, take care, and beware of multiple values. Jim Muth jamth@mindspring.com jimmuth@aol.com |
START 20.0 PAR-FORMULA FILE================================
The_Dried-Out_Lake { ; time=0:48:07.99--SF5 on a P200
; Version 2002 Patchlevel 5
reset=2002 type=formula formulafile=allinone.frm
formulaname=mandelbrotmix2 function=recip passes=1
center-mag=+1.28250208122812600/-3.009422644531135\
00/4914106/1/95.0004599045420264/1.230095661228919\
33e-005 params=-10/-1.1/-1/-11/0/525/0/0 float=y
maxiter=1200 inside=atan periodicity=10
colors=000JTSMYXQc`TgcYmg`rlcvojjrp_uvPxzDzz1zz0zz\
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0zx1zz3zz3zz6zz9zzCzzFzzIyzLxzOvzQsyTrxXpu_osamrdl\
pgjoPmg6p`0sT3ua9FGCJLGPP }
frm:MandelbrotMix2 {; 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 20.0 PAR-FORMULA FILE==================================
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