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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Most cyclones consist of a lot of wind and rain, but today's fractal cyclone was created by numbers. In recognition of this fact, I named the image "Cyclone of Numbers". It's a rather striking and brilliant image, one that, with its bronze disk around the midget, honestly earns its exalted rating of 8. The formula that produces the image subtracts some Z^(0.6) from some Z^(2.3) and adds the standard C. The overly critical values of real(p1) and real(p2) show that the parent fractal, which is well worth a look, was found with the evolver feature of Fractint, a feature I often use when I need a fractal and time is short. Unfortunately, the parameter file is slow. It takes almost 27 minutes to render on my creaky old Pentium running at 200mhz. But as always, salvation is at hand in the form of the rendered image, which is available on the W.W.Web at: The fractal weather today turned out to be much more to the cats' liking. The sunny skies, light winds, and temperature of 61F 16C made conditions on the porch ideal for sleeping. This is what the fractal cats did all afternoon, while in some amazing manner continuing to guard Fractal Central like two fearless watchcats. (When an intruder threatens, the cats' attack is quite effective. The draft behind them as they dash for cover knocks the intruder flat.) But even such intrepid cats would be lost on a 4-dimensional hyperplanet without a longitude-altitude-latitude grid to fix their position. To actually fix a position on such a planet, we can start at a defined zero-zero-zero point, just as we do on earth, where the zero-zero point lies off the west coast of Africa. From this point we can lay out the great circle of the equator, which, just as on earth, circles the planet at its broadest point. By specifying a point on this equatorial circle (longitude), we define a great sphere of the hyperspherical planet. We can then discard the rest of the equatorial circle, and let it slip into hyperspace. We are left with a normal 3-D sphere, which we can fully visualize. This sphere can be taken as a longitude surface of the hypersphere. Our location on this sphere can then be taken as one of the sphere's poles. The equator of the entire hypersphere still cuts through the sphere at this polar point, though the remainder of the equator is no longer visible. And now, for the first time, we can see the entire polar circle of the hypersphere, which appears 90 degrees away as the apparent equator of the newly-defined sphere, which in this case is a cold zone. From the pole of the sphere it is a simple matter to mark a direction, which defines a circle of the sphere and the altitude of the hypersphere, and a distance, which fixes a point on the circle of the sphere and the latitude on the hypersphere. By discarding one dimension halfway through, we have reduced the locating process to one that can be visualized. If we attempt to remain in hyperspace all the way, things become more difficult. We start with the equator of the hypersphere, just as before, but instead of marking a point on the equator (longitude) and a direction from the equator (altitude), we skip directly to the distance from the equator (latitude). Since the surface of a hypersphere is three-dimensional, we have an entire 360-degree circle of directions in which to depart from the equator. We have defined a shape much like a doughnut. Very near to the equator, this doughnut is very thin, like a hula hoop. The surface of this doughnut marks a latitude surface of the hypersphere. As we move farther from the equator, the doughnut grows fatter, but it no longer fits into 3-D space. In some manner beyond our power of visualization the latitude doughnut reaches 45 degrees toward the polar circle, at which point it surrounds both the polar circle and equator of the hyperplanet in the same manner at a mutual distance of 45 degrees. From here on, the latitude doughnut approaches the polar circle, growing thinner as the latitude increases, until at last it forms a hula hoop around the polar circle. . . . Well, I'll have to finish tomorrow. I seem to have gotten carried away in hyperspace, and now find myself with certain tasks that need to be done. The way to get the tasks finished is to get them started, so until next time, take care, and then do even more. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Cyclone_of_Numbers { ; time=0:26:40.37--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=ident passes=1
center-mag=-0.94163953656643470/+0.130898140289193\
70/6.093502e+009/1/-25.0000100100820291/0.00017467\
5444399419177 params=-0.40736/0.6/0.573185/2.3/0/0
float=y maxiter=3000 inside=0 logmap=560
colors=000zXfzXmzXsoduVluAsv0zvTzczzMzyMvuMrpMolMj\
gMfaMcYM_TMVPMSLMOFMJAMG6MC1M70M40M60L70J90IA0GC0F\
D0DF0DG0CI0AJ19L17M36O36PLCQTIQZOUfTcn_mocmlVceOUZ\
FGV7DO09A047004003000000000000C04Q19c6DrAIzCJzCLzC\
LzDMzDOyDOxDPvFQvFQuFSsFTrGTrGVpGXoGXoL_fP`_TcSXdL\
`gCdi4il0lm0fo0ap1Yp6TrCPsILsMGuSCvY7va3xg0xl1rm4m\
o7ip9crC_sFVuGQvDYjAc`9jQ6pF3x41z04z06x07v0As0Cr0D\
o0Fm0Ij0Jg0Lf0Oc0Pa0Q_0SY0PY7OYDLYJJYPIYVFYaDYgAYm\
9Ys7YyC_zG_zL`zP`zTazYazaazfczjczodzsdzxdzs`zpYzmT\
ziQzfOzcJz_GyXCxT9vP6uM1sJ0rG0pL0lO0iQ1fT3cX4`_6Ya\
9VdASgCPjDMmFJpGGsIDuMAvP9xS7yX4z_3za1zd0zi0zl0zo0\
zr0zp0zo3zm7zlCzjIyiMygQxfVvd`ucdsair`mr_rg`mYaiRh\
jKhm9ipCgM3lM0pM0uMGmFT69TCITGPV_JWcRXgVMgb0gl4gvK\
gzVgzifzpgzxizrjvllgamzcozdpvfrpJs_OuaQvdVxgYyjazm\
dzp9zsDzvIzyMzzQzzVzz_zzczzzzzzzzyzzvzzszzpzzmzzjz\
zzzzzzzyzzszzvzzxzzzzzzzz
}
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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times.
