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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Midgets are found everywhere around the border and along the filaments of the Mandelbrot set. Some of the best midgets are found in East Valley, sometimes known as Elephant Valley, and in the two symmetrical valleys on the west side of the lake, known together as Seahorse Valley. Today's image pictures a typical though rather deep midget lying in Seahorse Valley. So far, so good. The image is filled with the seahorse tails typical of this area, and all the expected features surround the midget in the 2,4,8... series, without being cut off, just as they would be expected to do. But things go awry as we approach the actual midget at the center. Where there should be a minibrot, we find a roughly triangular hole, which has little resemblance to either a midget M-set or to a hole in a Julia set. Something strange indeed is going on. At first glance, one might suspect that the image is part of a perturbed Mandelbrot set, and that the fully-formed midget will appear if only Z (p3) were initialized to 0,0. Unfortunately, doing this results in a screen filled with diagonal lines, with no midget anywhere in sight. So what is going on? True, the image is perturbed, but it is not a Mandelbrot set. The answer lies elsewhere. The Mandelbrot sets and Julia sets are perpendicularly oriented slices of an abstract four-dimensional figure known as the Julibrot, only parts of which can be seen in our three-dimensional space. And four-dimensional objects can be sliced by two-dimensional planes in six mutually perpendicular directions. Today's picture shows a midget sliced in one of the odd perpendicular directions. Mandelbrot sets display the C plane, Julia sets display the Z plane. Today's image displays the plane determined by imag(C) and imag(Z). I have named this orientation 'Rectangular', because the slice in this direction through the origin of the Julibrot shows a roughly rectangular lake. If one wishes to call these slices 'Rectangular sets', very well. Though don't expect the midgets to be shaped like rectangles unless the slice cuts them exactly through the point that corresponds to the origin of the Mandelbrot set, and they happen to lie in the same orientation. In addition to the Rectangular direction, there are three other new directions, as well as an unlimited number of oblique directions in which to slice the Julibrot. And every new direction shows a new and unique aspect of the Julibrot. During the upcoming month of July, I'll be exploring these new directions. I named today's image "A New Seahorse" because though it is a scene in Seahorse Valley and is filled with sea horses, the midget at the center is unlike anything seen in the standard Mandelbrot or Julia views of this area. I rated the image at a 7, mostly because of the striking coloring. The parameter file takes only 3-3/4 minutes to render on a Pentium 200mhz machine. The already-rendered GIF image downloads in even less time. That GIF image may be found posted to: The fractal weather today (June 28) brought the first really hot day of the season, with a temperature of 95F (35C), and all the haze and humidity that goes with such heat. The fractal cats had no complaints, though they showed little activity in the sweltering conditions. Tomorrow (today, June 29) promises to be just as hot, so I'm going to take it easy and let the CPU do all the work. Until next time, take care, and when the going gets hot, get going on a hot fractal. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
A_New_Seahorse { ; time=0:03:42.07--SF5 on a p200
reset=2001 type=formula formulafile=allinone.frm
formulaname=multirot-XY-ZW-new function=flip/ident
passes=1 center-mag=+0.00000000006028511/-0.000000\
00000678208/1.223635e+009/0.2406/6.418/71.294
params=0/90/2/0/-0.7692602987818452/0.109559945816\
0586/-0.7692602987818452/0.1095599458160586 float=y
maxiter=2000 inside=0 logmap=388 periodicity=10
colors=000VzeYzfZzf_zfazfczfezffzfgzfizgkzgmzgnzgo\
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zYzzZzz_zyazybzyczyezyfzwfzwfzwfzwfzwfzvfzvfzvfzvf\
zvfzvfzvfzvfzvfzvfzvfzvfz
}
frm:multirot-XY-ZW-new {; draws 6 planes and rotations
;when fn1-2=i,f, then p1 0,0=M, 0,90=O, 90,0=E, 90,90=J
;when fn1-2=f,i, then p1 0,0=M, 0,90=R, 90,0=P, 90,90=J
a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994,
z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p3,
c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p4:
z=z^(p2)+c,
|z| <= 36 }
END 20.0 PAR-FORMULA FILE==================================
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