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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: The non-fractal work is once again under control here at Fractal Central, leaving more time for a proper fractal search. My search today was quickly rewarded with an above average and striking image, well worth a look. The mathematical expression Z^(1.1)-Z^(-1.1)+(1/C) is purely whimsical. As far as I know, it has no special significance except that when iterated by the MandelbrotMix4 formula it produces the expected fractal. The fractal consists of a twisted Mandelbrot set centered at the top edge of the default screen. Today's scene lies in the intricate features along the northwest edge of this set, near the point where a gently curving filament makes an unexpected acute angle and doubles back on itself. When the time to give a name to the picture arrived, I had no hint of what to call it. The name I settled on, "Boiling Cauldron", was more a choice of desperation than a logical choice. I may already have given the same name to an earlier FOTD. I have no handy list available to check which names have already been used. And what harm would come from having a duplicate. Perhaps I should start naming the FOTD's with an Opus number, like is done with the works of composers. Even that would be a task however, for I also have no readily available count of the 3-1/2 years' of FOTD's. But if I started numbering without actually counting, I would start around 1250. Today's image is above average -- neither much above nor slightly above, but rather average above average, which translates to an honest rating of 7. The parameter file renders in 6 minutes -- slow though not overly frustrating. For those who would rather not go through the hassle of running the parameter file, the GIF image file has been posted to the usual two sites: The fractal weather today was partly cloudy with a temperature of 79F (26C), perfect for the cats, who chased each other around the yard for 1/2 hour before running out of energy and settling down for a snooze on the porch. The slow-down also allowed more time to ponder. The topic of my pondering was the nature of reality. I am sometimes asked why I as well as the real philosophers would concern ourselves with a problem that has such an easy and obvious answer. The answer would appear to be that reality is the world around us, in which we exist and which we accurately though incompletely observe with our senses. But things are not so simple. If they were, philosophers would never have needed to address the question of reality. As I pondered this afternoon, I remembered a sunrise I had observed from a cliff overlooking the ocean on the coast of Maine. It was a lovely sunrise, with a sky tinted every color imaginable. According to physiology, the colors which I had observed in that sky were a creation of my mind, which through my retinas, optic nerves and visual cortex had translated various frequencies of electromagnetic radiation into the subjective mental sensations known as red, blue, violet, etc. No colors existed in the actual 'out-there' sky. Then I imagined that the real sky out there must be without color, existing as merely different shades of gray, like a black-and-white photograph. But light and dark are also mental sensations that exist only in the mind, so the 'real' sky has no shades of gray either. It has only various frequencies and numbers of photons, moving through space in various directions at the speed of light. And then I remembered that photons when unobserved are a homogeneous mass of fuzzy spread-out waves, which conveniently collapse into particles only when we look for particles. And I remembered that we are aware of the apparent space in which we exist only by observing photons, which we create by the act of observing them. By this time I was doubting whether space or anything at all exists that might be called reality. "It truly is an act of faith to declare the physical universe real," I thought. Next, I stopped pondering and turned my energies to the far less profound mysteries of fractals. Well, the cats have long since ended their snooze, and are once again thinking of food. I guess it's time to shut down the fractal shoppe and put the fractal cats' minds to rest by filling their food dishes with their favorite snacks. Until tomorrow, when I'll return with more thoughts about reality, take care, and see you then. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Boiling_Cauldron { ; time=0:06:08.06 -- SF5 on a P200
reset=2001 type=formula formulafile=critical.frm
formulaname=MandelbrotMix4 function=recip passes=1
center-mag=-0.38580613488445680/+2.076060642776320/2\
.459343e+008/1/62.5 params=1/1.1/-1/-1.1/0/0 float=y
maxiter=750 inside=0 logmap=115 periodicity=10
colors=000RlzTkzWmzWizVezVdzV_zUZzUVzUUvTQsSOqRKmRIi\
QEgPCdP8`P5Z0ib0`V0RO0HG0AA222400800C00000N0IRHSjKiv\
NtzTzzSzzRzuOskOkbNd`PZ<2>VOOTNKRLERLBPK4NK0<3>FJ0GJ\
0EN0DR0BV09X08`07d07e05i03m02q00s00v00z00z00s02k44dE\
5XP8QZAMkDGvFAzH3zODzVKzaTzi_zogvspszvtzzqzzlzzizzZz\
pQzeHzS6zG0z20z00<2>z00z00x20v41u65uA9sDDqFGoGJoKNmM\
QkOTkPXbP_WPbPPgGOk7Nq0Mu0Mz0Mz0Oz0Qv0Ps0Ro0Tk6TiAVe\
DWbJWZMXVQXV<2>GVeCUi8Sm4Rq1Qv0Qz0Pz0Nz0Oz0Mz0Mz0Oz0\
Nz0Qz0Pz6Pz9RzFSvJSuMUqRUmVWkZXg`XddYcgYakZad__XXXUX\
WMXVHZV9ZV1`V0`V0bV0bV0Yb0Yg9YoHWuPXzYXzdWzkUzedzYju\
SsiKuYFuNHqIFkCGk6Gk0MkARkKWkTbkbhkilnorrxuszqszlqzf\
qzcpzYpzSozNozJozdpzvrzsszmpziqzcqxYtxUwxPzxJzwGzwAz\
w6zwCzfHzQLzANz0<3>pz0<2>uz0wz0wz0xz0zz0<2>zz0qz0gzE\
ZzTNzeCzo<2>Yzy<2>uzzuzzuzzszzszzszzszzqzzqzzHzyqzzq\
zzqzzNzmPzsRzz
}
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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