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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Despite the sound of its name, today's image was not named because it resembles a very sick bird; it was named because it breaks the law. Certain rules exist in the world of fractals that cannot be broken. Well, today's fractal breaks a major rule, which makes it "Totally Illegal". And this is why it got its name. Considering the image's hasty coloring, the rating of 8 is high, but the rating is based not only on the artistic worth, but on the mathematical interest as well. What rule does the image break? It looks harmless enough. On the surface, it resembles nothing more than a large raspberry surrounded by smaller raspberries still on their stems. To find the crime we must look at the generating formula, which subtracts 0.425 parts of Z^4.5 from Z^3.5. These exponents are fractional, and because of the many-valued nature of the complex log function, fractional exponents must always draw fractals filled with discontinuities -- branch cuts as I now call them. Today's image however contains no visible branch cuts. It has therefore broken the law by doing something totally illegal. I have not yet checked to see whether the breaks exist at a deeper level, but I soon will. I suspect that the breaks are there, even while hoping that they are not. Regardless, the surrounding area holds many interesting midgets, some of which will surely be making their appearance as FOTD's in the days to come. A benefit of the shallow location of today's scene is the speed with which the image renders. Even on my tired old fractal computer, the image finishes in a matter of seconds. I have used the passes=b algorithm because not only is it the fastest, it is also the most fun to see in operation. And it draws the image with no loss of detail. Those who prefer not to render the image from the parameter file can still find it posted on the internet at the web sites: The weather today here at Fractal Central turned out to be another normal summer day with mostly sunny skies and a temperature of 87F 30.5C. The fractal cats enjoyed the weather, but got into a tussle when both wanted the same porch chair, which left them sulking at each other for over an hour. I'm sulking a bit because I've failed once again to find time to write that great philosophy of mine. But tomorrow is another FOTD, and I'll have more time before then to think and write. So check in 24 hours and see what happens. Until then, take care, and one person's great wisdom is another's foolish prattle. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Totally_Illegal { ; time=0:00:51.90--SF5 on a P200
reset=2001 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=ident passes=b
center-mag=+0.53136563368005390/0/149296.2
params=1/3.5/-0.425/4.5/0/0 float=y maxiter=400
inside=255 logmap=40 symmetry=xaxis
colors=0008Ac9Ac9AeAAgABiBCkBDmAEnAEoAFpAFqAFr9Gs9\
Gt9Hu9Hv9HwCLsEPpHTlJWiMaeOjbRrZTwWmzSszP_zMZqMYmM\
YiMXeMXaMWXMWSMVOMVKMUHMUGMWIOXMQYPSZTU_WV`_XabZbf\
`ciaagc_fdYeeWdfUbhSaiQ`jO_kNYlNWmNVnNToNRpMQqMOrM\
MsMLtMJuLHvLGwLExLCyLBzMEyNHxOJwPMwQOvRRuSTtTWtUZs\
V`rWcqXeqYhpZjo_mn`on_rr_tvaqpbojdldejZggTheNabRV`\
UPZXMTQJNKGHDDB7A61DCAFIIHORKUZM_gOeoQdlSciUcfWbdY\
ba_aZaaWc`Ue_Rg_OiZLkZJmYGoYDqXAsX8pY9mY9jZAgZAd_B\
b_B_`CX`CU`CRaDPaDMbEJbEGcFDcFBcFCcCCcAEdEGeIIfMJg\
PLhTNiXOj_OebOcdOcgOciOclOcnOcqOcsRcqTcpVcnXcmZck`\
cjbchdcgfgfdhgcigbjgajg`kgZlhYmhXmhWnhVohUohVlhWih\
WfhXchYch`chcchfchichlchochrchuchucgucgtcgtcftcfsc\
fscercercerddqgdqjdqldphbpeapb`o__oYZo_XnaWncVneUm\
gTmiRmkQlmPloOlqNksOjuPiwQhyRgzSfzTezUdzVczWbzXazY\
`zZ`zZXzcSzhOzmKzrGzvMzpRzkWzf`z`ezWjzRozMmzQlzUkz\
Yjzaizehzigzm7zU7zX8zZ000
}
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.
