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Fractal of the Day
by Jim Muth
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Fractal enthusiasts and visionaries: Today's fractal rates a 7 -- the second above average fractal with such a rating in a week. But before getting to the image, I must finish what I started yesterday. In yesterday's FOTD I mentioned the surface of double revolution, while neglecting to tell what it is. That oversight will now be corrected. A surface of double revolution is a curved two-dimensional surface, which exists in the three-dimensional boundary of a four-dimensional hypersphere. It consists of the points lying at a given distance from a given great circle of the hypersphere, and so at the complementary distance from the polar, or completely perpendicular, great circle. A surface of double revolution is also the path traced by points of the hypersphere when it is subjected to a double revolution. A double revolution is a type of motion possible only in spaces of four or more dimensions. It is comprised of two simple but completely perpendicular rotations applied simultaneously to a single object, each rotation taking place in two separate dimensions. The surface of double revolution is of importance in the theory of functions with two complex variables. Nested surfaces of double revolution could serve to measure the latitude of a four-dimensional hyperplanet. In this case, they would start at the equatorial great circle of the planet, where they would resemble the torus or doughnut-shape of three-dimensional space. But they would undergo an impossible-to-visualise transformation before they converged, as inside-out doughnuts, on the polar great circle of the planet. It is the shape of this surface of double revolution when it measures 45 degrees of latitude that has me puzzled. But there is no puzzle about today's FOTD, which turned out unexpectedly good. The formula behind the image is 0.3Z^(-1.05)+Z^(1.05)+(1/C). The parent fractal is another of those strange figures that appear when Z and 1/Z, or values close to this are combined in various ways. Adding 1/C instead of C merely adds to the interest. I named the picture "A Midget in Blue" when I noticed the deep blue haze surrounding the central midget. The colors of today's picture blend together unusually smoothly. In fact, the image must be examined closely to be sure that it is not in true-color. The parameter file is unusually slow, taking almost 1/2 hour to render on an average Pentium. But as always, the GIF image has been posted to the Usenet binary group: The fractal weather today was absolutely perfect, with deep blue skies, puffy clouds, low humidity, a gentle north wind, and a temperature of 80F (26.5C). The cats must have approved, as they spent the better part of the day sleeping on the porch and keeping out of the blue jay's way. The fractal philosophy is still on hold, and likely will remain so for several weeks yet. But fear not, for the four-dimensional muse is once again stirring, giving me much to think and write about. And the impossible things that are possible in four dimensions make the illusions of the world's greatest magicians look like nothing. For now, the time to shut down the fractal shoppe has arrived. The cats need to be evicted and fed, and I need to settle in front of the TV to doze off trying to watch a junky old movie. But I'll be back in 24 hours, so until then, take care, and don't let those higher dimensions make you hyper. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
A Midget_in_Blue { ; time=0:27:15.63 -- SF5 on a P200
; Version 2000 Patchlevel 9
reset=2000 type=formula formulafile=critical.frm
formulaname=MandelbrotMix4 function=recip passes=1
center-mag=-4.919970039353122/+1.649879877881105/1.8\
42739e+010/1/-34.997 params=0.3/-1.05/1/1.05/0/0
float=y maxiter=3600 inside=0 logmap=511
colors=000P9T<3>SAQTBPUBO<19>mL_nM`oMa<3>rOcrPdsPdsQ\
esRd<38>tkitkitli<3>tnisohtni<35>uimuhmuhm<2>uhmuhmv\
jn<22>_N_ZMZYLY<3>UHXTGXSFWREWQDW<3>MHXLIYKJZJK_KJZK\
IYKHXKGUKFSLEQLCPLAMLBQLCPMDN<17>PPHPPHQQH<3>QTGPRH<\
20>hqBirBjsB<3>mxAkxBjyB<3>ezBdzBczB<3>ZzBQz`<11>bzT
}
frm:MandelbrotMix4 {; Jim Muth
a=real(p1), b=imag(p1), d=real(p2), f=imag(p2),
g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j,
k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel):
z=k*((a*(z^b))+(d*(z^f)))+c,
|z| < l
}
END 20.0 PAR-FORMULA FILE==================================
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