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Fractal of the Day
by Jim Muth
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Fractal enthusiasts and visionaries: Today became an irritating one here at Fractal Central, when one of the fractal cats, Tippy the tabby, had a dispute with a burdock bush. The bush won. When Tippy came inside, he had a clump of burrs the size of a tennis ball stuck to his side. I worked at the burrs almost an hour, during which time he protested loudly and constantly. But patient combing and cutting finally prevailed, and Tippy is almost as good as new. During Tippy's ordeal, Thomas, the other fractal cat, watched in bemused sympathy. The burr problem cut into my fractal-searching time, but I did manage to find an interesting scene in the Z^2+C Julibrot. In the Mandelbrot set, the mouths of some buds have simple, rational coordinates, such as -0.75, -1.25, 0.25+0.5i, and -1.0+0.25i. The final set of coordinates points to the mouth of a small period-4 bud on the north shore of the large bud centered at -1. Though the valleys here have some interesting scenes, I examined the area in one of the more remote oblique orientations -- halfway between the Parabolic and Elliptic directions. (I am never one to do things in the accustomed manner.) The resulting image is nothing at all like a Mandelbrot set. Nor is it anything like a Julia set. It is simply an oblique slice, slicing through the Julibrot figure at the point on the north shore of the period-2 bud. I have several times explained the nature of the parallel stripes that appear in the odd slices of the Julibrot; I will not repeat myself here other than to say that today's picture has a fine display of these stripes. I have named the picture "Wraiths" because it has a certain delicate, ethereal feeling about it. Despite the unusually high maxiter, the parameter file renders in less than 3 minutes. The JPEG'd file of the finished image has been posted as usual to Usenet on the binary newsgroup: The weather was perfect today, with sunshine and a temperature of 70F (21C) making conditions ideal for fractal hunting as well as cat cleaning. I have yet to give my solution to the puzzle of why I am myself instead of just another someone else. I have the (IMHO) solution ready, but due to the irritation caused by the fractal cat's misadventure, I just couldn't work up the concentration to put it into easy-to-understand words. But my solution is radical to say the least, so check again tomorrow, when with all seriousness, I shall offer my humble opinion to be considered, torn apart or ignored. Until then, take care and be happy. Jim Muth jamth@mindspring.com |
START FORMULA==============================================
multirot-XZ-YW {; Jim Muth
; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect
e=exp(flip(real(p1*.01745329251994))),
f=exp(flip(imag(p1*.01745329251994))),
z=f*real(pixel)+p2, c=e*imag(pixel)+p3:
z=sqr(z)+c,
|z| <= 36 }
END FORMULA================================================
START PARAMETER FILE=======================================
Wraiths { ; time=0:02:43.41 on a p233, SF5
reset=2000 type=formula formulafile=multirot.frm
formulaname=multirot-XZ-YW passes=1
center-mag=-0.16014146980676970/+0.0361142073067994\
0/7.139926/11.992/90 params=0/45/0/0/-1/0.25
float=y maxiter=120000 bailout=25 inside=0
logmap=10 symmetry=yaxis periodicity=10
colors=000S1E<8>`AN<3>dFReGSeHR<3>fINfJMgJL<3>gLIgL\
IhLIhLHhLHiMH<3>dMGcNGbNF<3>ZOEZOEXOE<11>OQAOQANRAN\
RAMR9<4>JS8IS8HS7<2>GT7FU6EV6EW6DX6<3>A`4Aa49b4<3>7\
c36c35c25b24b2<2>4`24`24Z2<2>5V45T55T66S76R86R96P9<\
33>DBbDBcEAd<2>E9gE9gFAf<46>RiQRjPRjP<2>SmOSmORnP<3\
>RrSQsSQtT<2>QwVQxVRyUSzUTzT<24>UzLUzLUzK<18>UzI
}
END PARAMETER FILE=========================================
START 19.6 PAR-FORMULA FILE================================
Wraiths { ; time=0:02:43.41 on a p233, SF5
reset=2000 type=formula formulafile=multirot.frm
formulaname=multirot-XZ-YW passes=1
center-mag=-0.16014146980676970/+0.0361142073067994\
0/7.139926/11.992/90 params=0/45/0/0/-1/0.25
float=y maxiter=120000 bailout=25 inside=0
logmap=10 symmetry=yaxis periodicity=10
colors=000S1E<8>`AN<3>dFReGSeHR<3>fINfJMgJL<3>gLIgL\
IhLIhLHhLHiMH<3>dMGcNGbNF<3>ZOEZOEXOE<11>OQAOQANRAN\
RAMR9<4>JS8IS8HS7<2>GT7FU6EV6EW6DX6<3>A`4Aa49b4<3>7\
c36c35c25b24b2<2>4`24`24Z2<2>5V45T55T66S76R86R96P9<\
33>DBbDBcEAd<2>E9gE9gFAf<46>RiQRjPRjP<2>SmOSmORnP<3\
>RrSQsSQtT<2>QwVQxVRyUSzUTzT<24>UzLUzLUzK<18>UzI
}
frm:multirot-XZ-YW {; Jim Muth
; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect
e=exp(flip(real(p1*.01745329251994))),
f=exp(flip(imag(p1*.01745329251994))),
z=f*real(pixel)+p2, c=e*imag(pixel)+p3:
z=sqr(z)+c,
|z| <= 36 }
END 19.6 PAR-FORMULA FILE==================================
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