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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Today's FOTD, which is quite a come-down from yesterday, shows some elephants. I named the picture "The Elephants Walk" because that's what the elephants appear to be doing. For purposes of clarification, various parts of the Mandelbrot set have been given descriptive names such as Seahorse Valley, Elephant Valley, Scepter Valley, etc. The names refer to the shapes of the features in these valleys, and a bit of imagination is necessary of course to see the vague resemblance to the objects the valleys are named after. Perhaps the easiest things to find are the guardian elephants trotting out of East Valley, the valley that begins at 0.25 on the X-axis of the M-set. In this area, each elephant is aligned with one Mandelbrot bud, and stands guard over its bud, protecting the bud from whatever it is that attacks Mandelbrot buds. Today's fractal shows two complete though strangely distorted elephants, and the front part of a fragmentary third elephant. The elephants are nice, but we've all seen the East Valley elephants many times, so to make things more interesting, I have captured a view of the elephants from an entirely new direction -- the direction I call Oblate, which consists of the planes of the Julibrot defined by the imag(c) and real(z) axes. In today's tilted picture the narrow diagonal line is oriented in the imag(c) direction, with the positive direction toward the upper right, while the perpendicular direction is real(z). The elephants are obvious enough, but where are the buds they are guarding? The secret is that the one-bud-per-elephant arrangement is true only in the Mandelbrot orientation. Actually, one bud appears in today's picture, but it appears so distorted that it would never be recognized. Look carefully at the straight-edged features extending diagonally from the lower left to the upper right of today's image. Pay special attention to the hair-thin straight line. This line is what I call a bridge. Bridges appear all the time in the four odd planes of the Julibrot figure. This bridge is actually the tip of a Mandelbrot valley sliced from the side, and the diagonal empty space between the line and the main diagonal straight-edge is the open inside of one of the buds along the southern edge of East Valley. The reason the buds appear as straight lines is that they are actually four-dimensional hypercylinders in the 4-D Julibrot figure -- a shape impossible to visualize with mere 3-D minds. I find it most curious that these straight bridges continue their way wherever they are unobscured regardless of the low-iteration material surrounding them. In fact, as can be seen in several places in today's picture, the bridges actually attract the low-iteration material to themselves, and increase their mass by absorbing this material. The parameter file of today's image takes 5-1/2 minutes to render on a P200 machine. The GIF image file downloads in less than half that time from: The fractal weather today was partly cloudy and quite cool for July. The temperature of 79F (26C) must have suited the fractal cats perfectly, for they spent several hours outdoors, romping in the yard. The philosophy made little progress however. But tomorrow is another day, (it always is), and some surprise philosophy could appear then. Check in to see what happens. Until next time, take care, and don't lose your head in the fourth dimension. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
The_Elephants_Walk { ; time=0:05:20.22 -- SF5 on a P200
; Version 2000 Patchlevel 9
reset=2000 type=formula formulafile=multirot.frm
formulaname=multirot-XY-ZW function=ident/flip
passes=1 center-mag=+0.3181278040886614/-0.013000620\
65059610/275.3077/0.04602/0.659/80.235
params=0/90/0/0/0.285/0 float=y maxiter=2500
inside=0 logmap=14 periodicity=0
colors=000YGU<3>`JXaKYbLZcM_dO`eRagUb<3>nfhpijrklsmm\
qli<2>hd_daX`YUYVRTSN<3>FGABD78A4571<3>PPNUTSYXX<3>M\
RRKQPHPOENMBML8RJ6TI7YH8cH9iHAmHDoG<4>RXEUTDXQD<3>gB\
B<7>VOVTQYRS_QTbOVdNWf<3>F_sD`vC`v<3>A_vA_vA_v<6>Ccs\
CcrCdr<2>DeqDfpGfkDfp<6>FimFjmFjl<3>Glk5hi<3>8Ta9P_9\
O_AN`<3>CXdDZeDafEcgFehKciPajU_mZYl<2>kSl<3>iKliIlhG\
lhElhClgBl<58>KfkKgkJhk<3>IjkHjnHkqGktGmwMmz<3>JmrIm\
pHmnGmlImj<3>HmaHm_HmYHmW<4>HmXHmXHmXHmYHmYHmY<2>HmZ\
HmZImZJm_<3>KmW
}
frm:multirot-XY-ZW {; draws 6 planes and many 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))+p2,
c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p3:
z=sqr(z)+c,
|z| <= 36 }
END 20.0 PAR-FORMULA FILE==================================
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