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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Today's image is another view of yesterday's midget. Of course, yesterday's image was not a picture of a midget, it was a slice through a hole in a four-dimensional object known as the Julibrot. Such slices are known as midgets only when the slices are oriented in the direction determined by real C and imaginary C, and the starting point of Z is 0,0. Under these conditions the holes appear as miniature versions of the entire Mandelbrot set. When the orientation of a slice is changed, or the starting point of Z is set to a value other than 0,0, a slice will not cut a hole in a miniature M-set. The slice of the hole will appear as a Julia set or as an irregular open area. True, some of these slices come very close to being intact mini-M-sets, but there is always some tiny imperfection when one searches deeply enough. When cut in an odd direction, (other than Mandelbrot or Julia), a hole often presents an irregular triangular shape, as did the hole in yesterday's image. But in yesterday's image the pattern around the hole was intact. In today's slice of the same hole, the pattern has been distorted and stretched in three directions until it is barely recognizable. It is apparently the same three-way distortion that shaped yesterday's hole, but this time it affects the entire surrounding area. This three-way stretching appears quite often around holes when they are sliced in odd directions. There must be some geometric feature of the Julibrot causing the effect, which would be quite apparent if we could visualize four-dimensional forms. In today's scene the hole itself has been reduced to a narrow slit. This also happens quite often in the odd slices, and is due to the unusual shape of the actual hole, which has two extended Z dimensions and two much smaller C dimensions. I named today's image "Head of Zardoz". Anyone who knows the movie, which features a flying head, knows why I chose the name. I rated the picture at only a 4. The twisting, stretching effect is really rather unattractive, and I chose it only to show what could be done by rotating the direction of the slice, while keeping it centered in the hole. I also kept the same palette of colors as an aid in identifying the same features. Taking an even 4 minutes to render, the image is mercifully convenient to render from the parameter file. Downloading the GIF image from: The fractal weather today continued sultry. The temperature of 90F (32C) and occasional distant thunder kept the cats alert. But all went well. No storm broke and the cats finally relaxed. And it's about time for me to relax. It's 10am on a Saturday morning, with other stuff to do. But I'll be here again, right on time with the next FOTD. Until then, take care, and though we can speak the words about the fourth dimension, we cannot picture what we are speaking of. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Head_of_Zardoz { ; time=0:04:00.58--SF5 on a P200
reset=2001 type=formula formulafile=allinone.frm
formulaname=multirot-XZ-YW-new passes=1
center-mag=-0.00000000262628144/-0.000000000366346\
96/1.493935e+009/0.02482/-8.307/87.302 params=91/1\
74/2/0/-0.7692602987818452/0.1095599458160586/-0.7\
692602987818452/0.1095599458160586 float=y
maxiter=2000 inside=0 logmap=389 periodicity=10
colors=000VzeYzfZzf_zfazfczfezffzfgzfizgkzgmzgnzgo\
zgqzgszguzgvziwziyzizzizzizzizzizzizzjzzkzzmzznzzo\
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zYzzZzz_zyazybzyczyezyfzwfzwfzwfzwfzwfzvfzvfzvfzvf\
zvfzvfzvfzvfzvfzvfzvfzvfz
}
frm:multirot-XZ-YW-new {; Jim Muth
; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect
e=exp(flip(real(p1*.01745329251994))),
f=exp(flip(imag(p1*.01745329251994))),
z=f*real(pixel)+p3, c=e*imag(pixel)+p4:
z=z^(p2)+c,
|z| <= 36 }
END 20.0 PAR-FORMULA FILE==================================
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times.
