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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: At first glance, today's fractal may appear disappointing, but don't give up hope too soon, for today we begin our 12-day journey into the fourth dimension, that impossible yet real abstraction where planes intersect in a single point and surfaces have three-dimensional volume. The object of investigation is the period-4 bud on the northeast shore of the main bay of the Mandelbrot set. This bud, along with many of its associated filaments and midgets, is clearly illustrated in today's FOTD. The bud is described as being of period-4 because the iterated points within it have four separate attractors. I have not given today's image a name or rating because it not a separate unit, but rather the first in a series of 12 images, which must be considered as the unit. The series of images will follow the bud in today's image as it moves and rotates through the fourth dimension and eventually becomes its own Julia set. On the way, it will become clear that the actual four-dimensional shape of the bud is that of a hypercylinder. As a matter of interest, the mouth of the bud is located at the C coordinates of 0.25+0.50i. The center lies at C=0.281+0.531i, and the mouth of the bud is tilted at an angle of exactly 45 degrees. It is a very familiar feature of the Mandelbrot set, so familiar in fact that little attention is usually given to it. The things I will do to the bud in the next 11 images however should shake most of the familiarity out of it, and explain the four-dimensional nature of the Mandelbrot set, Julia sets, and Julibrot as well. Today's image as well as the eleven following images will render from their parameter files in a few seconds, making the download superfluous. But for those who would still prefer the download, the GIF image file may be found on Usenet at: The fractal weather today was cold and windy. Winter is now setting in fast, and today's temperature of 42F (5.5C) along with howling winds made it feel like winter has fully arrived. The fractal cats dislike winter. They'll most likely spend the next several months as they spent today -- curled by the warmest radiator. I'll shut down the fractal shoppe at this time, but the 4-D story will continue tomorrow and for the following 10 days as well. Until next time, take care, and can an abstraction be real? Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
4D-01 { ; time=0:00:40.34 -- SF5 on a P200
reset=2001 type=formula formulafile=multirot.frm
formulaname=multirot-XY-ZW function=flip/ident
passes=t center-mag=-1.11022e-016/8.32667e-017/10.4\
1667 params=0/0/0/0/0.281/0.531 float=y maxiter=3600
inside=0 logmap=yes periodicity=10
colors=00040P<4>90UA0VB2W<3>RF_VI`ZLa<3>nXbr_bvbbteb\
vf`<3>ijWfkVcjV<8>MiVKiVIiV<3>AiV<5>WS`_PacMb<3>qAfu\
CgxGguKh<4>SccWZp<9>cm`dnZdpY<3>guS<3>aqa`qdZpf<2>Vm\
nUmpVlk<3>ZlU_lP`lK<2>bl7<9>ekOekPekR<2>fkWfkXhiY<3>\
ld`nbaoaa<2>rYcigKaq0hd8nTGoHOmKQ<3>PTXJWZDY`7_a<3>D\
cWFcVGdU<2>LgQMgPOfR<3>WqYXt_Zwa`xbbyd<3>izkkzmmzonz\
p<3>czNazGZz9Xz2Yz6ZzA_zE<3>kzSnzWqz_tzbwzczzczzdzzd\
<4>zzYzzXzzW<3>zzSzzKzzD<3>zzUzzYzz_<5>zzlzzozzq<2>z\
zxzzzzzzzzzzzzzzzzzyzzxzzz<30>zzz
}
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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