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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Today's image is that of a midget that lies on the grossly extended and distorted stem of a larger midget that lies on one of the lightning-like filaments extending northward from one of the Z^(1.23)+C Mandeloids. (There are an infinity of such Mandeloids.) I named the image "Typical Sub-midget" to point out that low-order midgets such as today's all start to look the same after a while, featuring converging branch cuts and long broken filaments. This sameness runs throughout the world of fractals. Back in the early days, when the only fractal available in the program I used was the Z^2+C Mandelbrot set, I wondered how much richer the higher-order fractals must be. If Z^2 produced such wonders, how much more wonderful must the Z^100 or Z^1000000 fractals be? But when I finally found a program that let me examine these fractals, I suffered a fractal let-down. I discovered that as the exponent grows larger, the fractal does not grow richer, it grows boring because it all starts to look the same. My next hope was that the lower-order fractals between 1 and 2 would hold the extra richness I sought. When Fractint finally added the ability to draw the fractals with fractional exponents, I got my chance to see what was down there. Here, I was less disappointed. There are some interesting scenes to be found in these realms, but once again, the farther from Z^2 I ventured, the less interesting the fractals became. I needed to prove to myself that the Z^2 of the Mandelbrot set is truly the apex of exponents. The same thing happened when I found the MandelbrotMix4 formula. This formula does actually draw images that have features not found in the M-set, and has become my most-often-used formula. But once again, I became ambitious. If combining two powers of Z draws such a varied collection of fractals, how much more varied would the fractals be when 6, 12, or 24 powers of Z are combined, I wondered. I wrote several formulas that did just this, but I used the formulas very little, because all the fractals started to look the same, and were not worth the time required to do the exploring. In the world of fractals, as in this world, simplicity is often the best solution. Individual fractals are infinitely varied, but they are infinitely varied within limits. The secret is to explore near those limits. The rating of 6 is based on the intentionally strong colors I gave the scene. Without the coloring, the underlying image would be worth only a 5 or so. The render time of today's fiery image is a modest 10-1/2 minutes on my overworked and under-powered Pentium 200mhz machine. If this is too slow, the completed GIF image may be found posted to Paul's web site at: The fractal weather Saturday turned out sunny enough, but the fractal cats still complained about the brisk breeze and temperature of 55F 13C, which chilled their ears when they ventured outdoors. Having had little success at taking it easy yesterday, I'm ready to try again today. Tomorrow, as always, I'll return with another fractal. Until then, take care, and give every fractal a fair chance. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Typical_Sub-midget { ; time=0:10:36.36--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=MandelbrotBC1 function=floor passes=1
center-mag=-0.80131730206810180/+3.261171920365556\
00/2.549835e+007/1/67.5/9.95332073081001134e-008
params=1.23/0/0/0 float=y maxiter=125000 inside=0
logmap=247 colors=000J00K00L00M00N00O00P00Q00R00S0\
0T00U00V00W00X00Y00Z00_00`00a00b20c30d43e53e63f73f\
83g93gA3gB3hC3hD3iE3iF3jG3jH3jI3kJ3kK3lL3lM3lN3mO3\
mP3nQ3nR3oS3oT3oU3pV3pW3qX3qY3rZ1qY3pY5pX7oX9oXBnW\
DmWEmWGlVIlVKkVMjUOjUQiURiTThTVgTXgSZfS`bI_dN`fSah\
Wbj`cledaxygpnmidsfQrdTrbVr`XrZZrY`rWbrUerSgrRirPk\
rNmrLowLwrKqnJkjIefI_bHUZGOVGJSKMQOOOSRMWTK_VHcYFg\
_DkbBodAvi9tg9sf8re8qd8pc7oa7n`6m_6kZ6jY5iW5hV5gU4\
fT4eS3dQ3cP3aO2`N2_M1ZK1YJ1XI0WH0VG0UF1WI1XK1YM1ZO\
2_R2`T2bV2cX3d_3ea3fc3ge8dfDafIZfNWfSTfXRgaOgfLgkI\
gpFgvCitDgrDfqDdoDcnDalD`jDZiDYgDXfDVdDUbDSaDR_DPZ\
DOXDNVDLUDKSDIRDHPDFNDEMDCKDBJDAHD8FD7ED5CD4BD29D1\
9D09D08D0EF3KG5QI7VJA`LCfMEnMDlNFkNGjNIiNJhNKgNMfO\
NeOOdOQcORbOTaOU`PV_PXZPYYPZXP`WPaWRaVPbVOcVMdVLeV\
JfVIgVGhVFiVDjVCkVAlT5nV9mXClZFl`IkaMjcPjeSigVhhWh\
iXhjMckNbkNbkNbkNbc3Wh2_f3Yd3Xb4W`4V
}
frm:MandelbrotBC1 { ; by several Fractint users
e=p1, a=imag(p2)+100
p=real(p2)+PI
q=2*PI*fn1(p/(2*PI))
r=real(p2)-q
Z=C=Pixel:
Z=log(Z)
IF(imag(Z) > r)
Z=Z+flip(2*PI)
ENDIF
Z=exp(e*(Z+flip(q)))+C
|Z| < a }
END 20.0 PAR-FORMULA FILE==================================
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times.
