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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Today's striking fractal was created by a rather unusual formula -0.002*(Z^1000)+0.02*(Z^(-2))+(1/C). The parameters were chosen entirely at random, as are the parameters for nearly all my fractals. I named the image "Ionized Mandelbrot" because of the fuzzy, purplish-blue aura around the midget, which reminds me of the ionized glow in a vacuum tube. Since the picture is better than the past few days' efforts, I rated it an above-average 7 on my 0-to-10 scale of fractal worth. A day or so ago, a message was posted to the Fractint list, asking what was necessary for a fractal to have midgets. The truth is that after working with the MandelbrotMix4 formula for over a year, I just don't know. To begin, it is difficult to define exactly what a midget is. Many fractals are filled with holes that I do not consider midgets. By a 'midget' I usually mean a tiny, often distorted, classic Mandelbrot shape embedded in a larger fractal. These tiny Mandelbrot shapes are sometimes called 'atoms'. The easiest answer would be that midgets are present in all fractals of the form Z^n+C, where 'n' is greater than 1 and Z is initialized at the critical point of the formula. But the midgets are almost impossible to find in fractals where 'n' is less than 1.2. As 'n' is increased to higher values, the midgets become easier to find, but ever less interesting, eventually degenerating into boring lopsided circles. The M-Mix4 formula automatically initializes Z to a critical point of the iterated formula, but not all formulae have midgets, and in many other fractals only certain limited parts have midgets. I have much more to discover about when and where midgets I'll return to the topic of midgets in a day or two. For now it's time to give a reminder that today's image takes 8 minutes to render from the parameter file, and only a minute or so to download from: The fractal weather today was perfect, with sunny skies and a temperature of 68F (20C), which lured the fractal cats onto the porch and then into the yard, where they sampled the cat-mint plant. The philosophy is still sleeping, but I've got a few hours to go yet in my day. It's just enough time to find a fractal and catch up on the Fractal-Art list, which has suddenly come alive. Until tomorrow, take care, and do fractals exist as real objects, or are they mere thoughts -- or are they both? Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Ionized_Mandelbrot { ; time=0:08:08.58 -- SF5 on a P200
reset=2001 type=formula formulafile=critical.frm
formulaname=MandelbrotMix4 function=recip passes=1
center-mag=+7.625829827008553/+0.00937269972207/3.58\
9962e+011/1/129.992/0.007 params=1/1000/-10/-2/-1.00\
2/5000 float=y maxiter=1400 inside=0
logmap=168 periodicity=10
colors=000805A07C08D0CF0DG0FI0IJ0JL0L<5>L0eL0hL0k<3>\
L1yL2zL2zL4zLOzDUzKYz0az0hz0oz0pz0ps1np<2>FehJceO`a<\
3>dPQhMOkJLnIOoGQpGTrFWsDZuDaxCdyCgzAiz8l<2>z7uy4lr1\
e<3>T0AM01G00I00J00J02L04M07M0AO0CO0FP0IQ0JQ0MS0PS0Q\
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cy0Wy0Px0Ox<3>0Qx0Vr<3>0ha0lY0oT0sO0xJ0zF<2>0z10z00z\
41zA4zGAzM<2>OydSviWuh`xg<3>pnauk`yhZzF0<2>n1Mi0Ve0a\
c0hh0en4csC`xJYzQVzYSzdQ<2>zhOziOzkM<2>yoLvpMsrOrsPo\
uQlvSkxThyTgzVdzWazY`zZYz`Wz`szGzz0zz1zz5zzAzzFzzI<2\
>yzVvzYuzarzepziozlrzh<3>xzSyzOzzJ<3>zz14z05z17z2
}
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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