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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: In the world of science, things that are long assumed to be true sometimes prove to be untrue. Why should fractals be any different? Take for example the often stated assumption that there are many Julia sets but only one Mandelbrot set. Even when the perturbed Mandelbrot sets are ignored, this assumption is untrue. It is untrue because there are at least two 'true' Mandelbrot sets, with all their parts intact. The familiar M-set cuts through the four-dimensional Julibrot along the plane where Z equals zero. But there is a second slice that cuts through the Julibrot in a perfect M-set -- the plane where the values of Z and C are equal. This second M-set, which I call the shadow set, is oriented exactly halfway through the double rotation between the Mandelbrot orientation and the Julia orientation. It lies at an absolute angle of 45 degrees to the classic M-set, every line in the shadow set lying at an angle of 45 degrees to every line in the classic set. This shadow set appears exactly like the classic set, though its linear dimensions are 1.4 times those of the classic set and its associated perturbed sets are strangely different. And yes, it is possible to enlarge its features to 1.4 times the size of the identical features in the classic set before reaching the limit of math precision. Is this second M-set the only oblique Julibrot slice that gives a spurious Mandelbrot set? I think so, but I'm not sure. One of these days I'll do some serious investigating, perhaps along the surface where Z equals the square root of C, but for the present I've got a rather exceptional FOTD to talk about. The math expression that drew the image is a totally whimsical journey into the land of fractional negative exponents. A quick check of the parameters will show the exact values. Try as I might, even after spending 1/2 hour coloring the picture, I could think of no suitable name for it, so I settled on the generic description "A Nice Fractal Midget". A second look convinced me that the rating of an exceptional 8 is fully justified. The parent fractal is the latest in my series of 'parent' images, which are characterized by a prominent 'fan' and lots of rings. The scene of today's image is located at the edge of the fan. The parameter file takes 9-1/2 minutes to render, making a download of the GIF image file the wise choice. The image file has been posted to the Usenet binary group: The fractal weather today was less perfect than yesterday, but still perfect enough to keep the cats happy. The partly cloudy skies and temperature of 73F (23C) lured them onto the porch, from where they bounded down into the grass to chase butterflies. The philosophy is smouldering, and could burst into flames at any time. I'm currently pondering the relation between fractals and the mind. Something may come of my thoughts, or perhaps nothing will. Regardless, I'll be here again tomorrow with another fractal and more gossip. Until then, take care, and be happy in your fractaling. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
A_NiceFractlMidget { ; time=0:09:36.66 -- SF5 on a P200
reset=2001 type=formula formulafile=critical.frm
formulaname=MandelbrotMix4 function=recip passes=1
center-mag=+4.39976788244362300/+0.25116878814085030\
/80669.34/1/7.5 params=-1.11/-11.1/-11.1/-1.11/0/300
float=y maxiter=2000 inside=0 logmap=125 periodicity=9
colors=000`0Ta0T<7>f7Yf8Yg9ZgAZ<2>iD_iF_iF_jGajGajHa\
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>jixjixlix<3>qixoixEgxCgxCgxBgxBgx9gx<2>8fxFgxMgxThx\
Yhx<4>Mgx
}
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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