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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: A few days ago I received an e-mail from a fractal fan, asking me to tell more about fractals. Since I had no idea of where 'more' started, I didn't know quite what to tell. But once I sat at my keyboard this morning to reply to the e-mail, the philosophical muses came to life, and the words flowed almost of their own accord. The following four paragraphs are a slight revision of the letter I sent earlier today. Perhaps the philosophy is once again ready to arise. My interest in fractals is mathematical, artistic, and philosophical. Mathematically, fractals are simply graphs of iterated mathematical functions. Artistically, fractals offer a means of expressing one's artistic aspirations, though I consider the importance of this aspect to be somewhat exaggerated. My main interest lies in the philosophical aspect of fractals. Perhaps the question most often asked about fractals is, "what are they?" In addition, I often wonder, "are fractals real?" The answer can only be, "fractals are the things numbers do, and numbers are pure abstractions". The Mandelbrot set does not exist in the sense that a tree does. No one will ever find a 'real' Mandelbrot set; they will find only pictures of it. The M-set exists only because human beings evolved with the sense of vision, and to better understand the workings of math functions, find it helpful to turn the functions into pictures. In essence, the Mandelbrot set exists only because we created it with our minds and sustain it with our computers. Much is also made of the fractal nature of the real world. We hear about the fractal nature of trees, ferns, clouds, coastlines, etc. These things do indeed have a fractal surface appearance, but they are not true fractals in the mathematical sense. A true mathematical fractal continues unchanged to infinity regardless of how much it is magnified. The 'real world' fractal objects such as trees and clouds ultimately break down into individual cells and water droplets, and finally into atoms, which no one shall ever observe directly. But according to quantum theory, atoms also are nothing more than convenient pictures, models created in human minds from mathematical functions. And I have heard it said that numbers themselves are creations of the human mind. So is the 'real world' the world's greatest fractal? The answer to this challenging question is what I am currently seeking. Another challenging question is why I named today's C-FOTD "Challenging Midget". I'm sure I had a good reason for such a provocative name, but the reason totally eludes me. Regardless, it's a fine fractal, which renders from the parameter file in under 3 minutes. The iterated expression, -5.5Z^(-0.5)-5.5Z^(0.5)+(1/C), as are so many of mine, is totally whimsical. The fractal it draws at first appears to be a total loss, but a close examination soon reveals tiny areas of promise. Today's midget lies in one of these promising areas. It's a lively little midget, centered as it is in its square of chaos, with gracefully curving arms radiating from the four corners before an electric violet background. The midget is so lively in fact that I rated the image a lofty 8, a level rarely reached on my conservative scale of fractal worth. Since the parameter file takes only 2 minutes to complete, running the file is most likely the fastest way to view the image. But for those who find running parameter files too much of a hassle, the GIF image will soon be posted to: The Usenet postings will continue to be delayed by 12+ hours until I solve my technical difficulties, a task that I have no intention of tackling in the near future. The fractal weather today here at Fractal Central was once again comfortable, though not nearly as mild as yesterday. The partly sunny skies and temperature of 42F (5.5C) lured the fractal cats onto the porch and into the yard, but once in the yard they quickly decided it was a bit too chilly, and soon returned to their radiators. And this leaves me with nothing to do but shut down the fractal shoppe and call it a night. I'll watch a junky old movie if I can stay awake. Until tomorrow, take care, and beware of the fractal witch. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Challenging_Midget { ; time=0:02:42.96 -- SF5 on a p200
reset=2001 type=formula formulafile=critical.frm
formulaname=MandelbrotMix4 function=recip passes=1
center-mag=+0.05445382910798185/+0.01773957995824642\
/8.333333e+007/1/4.999 params=1/-0.5/1/0.5/-6.5/300
float=y maxiter=1300 inside=0 logmap=45 periodicity=9
colors=000bFz<2>cKz`FzYDzWCzT9wP8tO6oL3kJ1hG0eD0bC0Y\
<2>40O10L00I10L10M10O11P34R39T3CV3FW4JZ4M`4Rb4Vc6Ye6\
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LzVIzWFzWCzY9zZ6zZ3z`0zb<2>0zc0z`0zZ0zW0zV0zR`ztbzw
}
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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