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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Today's image, which is from the same area of the same parent fractal as yesterday's, is impossible. It's incredible. It has no right existing. But here it is. I rated it a 9-1/5 because in the mathematical sense, and with no exaggeration, it is the most unusual fractal I have ever seen. (Artistically however it is not all that much.) I have long noticed that the midgets I find in the fractals created with the MandelbrotMix4 formula are all of the quadratic variety. That is, they are all shaped like miniature Mandelbrot sets. At the same time, I have heard rumors that midgets of other higher orders are possible if only one knew enough math to find them. Until today, I have found higher-order midgets only in formulas such as Z^(2.99)+Z^(3.01)+C, which doesn't count, because such a formula is basically 2(Z^3)+C, and the fractal it draws is the expected cubic. Today however I finally found an elusive higher-order midget. But this is not all. Not only did I find a cubic midget, but right beside it and in the same frame I found an order-6 midget. The cubic is not all that unexpected, since the generating formula combines portions of Z^3 and Z^4, but the order-6 midget is a total surprise. Yesterday's image, which showed a quadratic midget with 6-way symmetry, hinted at something very unusual in the area, but not until I stumbled upon today's scene did I realize how unusual. The more mathematically erudite may see what is happening and understand how a scene such as today's is possible. For myself, I enjoy mysteries, and I'll consider today's image a mystery. But I'm going to do a lot of exploring around the area of today's image to see what other wonders I can uncover. I have named today's image "Simply Amazing". That is how I felt when I saw it, so that's what I named it. The parameter file renders in just under 5 minutes, making it a toss-up whether to run the parameter file or download the finished GIF image from: The fractal weather today was more typical summer stuff, with hot sun and a temperature of 88F 31C. The cats, who enjoy such conditions, were kept happy. The clock over my desk tells me that it's already past the time when I should get to the waiting work. So until the next FOTD, take care, and who knows what strangeness lurks deep in the world of fractals. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Simply_Amazing { ; time=0:04:53.85--SF5 on a P200
reset=2001 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=ident passes=t
center-mag=+0.44864363812428920/+0.000055024630770\
93/72716.29/1/72.5 params=1/3/-0.35/4/0/0
float=y maxiter=3000 inside=0 logmap=72
colors=000SGUSGUTOZUWbVcfWkkXsoXzs`qodhlg_ikRfoIcr\
9`zFSi`cUvn`rcfoTmkIsh7rc8q_8pW9oS9nOAmKAlGBkCBcaF\
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znwzowzpxzqxzrxzrxzsyztyzuyzvazRczUezWgzZiz`kzcmze\
nzhpzjrzmtzovzrxzt5zu7zv8zvAzvBzvDzvEzvGzvHzvJzvKz\
vMzvNzvPzvQzvSzvJzpNzsQzu
}
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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