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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Try as I did, I could not think of a name for today's image. But that created no problem, as I simply named the image "A Nameless Fractal". I'll let the mathematically over-educated ponder whether today's FOTD actually has a name. Name or no name, today's image is busy. It has more colors than a galaxy has stars. In fact, it has too many colors -- a redundancy that drops it from a rating of 7 to one of a 6. But it's still a fun image to look at, and well worth the 16 minutes required to run the parameter file or the small effort required to go online and download the GIF image from: The formula that did the deed is Z^4+2Z^3+C, an apparently simple formula that creates a surprisingly complex fractal. In the vicinity of today's image can be found quadratic midgets, cubic midgets, quartic midgets, quintic midgets and hexic midgets (if that's what Z^6 midgets are called). There are also midgets of fantastic shape, which combine the features of midgets of several degrees. The most obvious appearance of these odd-order midgets in today's image is near the left edge, about 1/3 down from the top edge, where a cubic midget lies. But do not try to resolve the cubic midget. It lies beyond resolution. The mystery of the day's weather however can always be resolved -- that is if one waits until the next day. On Monday, partly cloudy skies and a temperature of 86F 30C made things here at Fractal Central perfect for cats. It was so perfect in fact that I had a hard time cajoling them to return indoors when the time came. Since writing yesterday's FOTD, I have pondered the chances for my self-aware existence (or the existence of any conscious individual). Before I was born, I was undefined, rendering the problem virtually meaningless. But if forced to give odds, I would guess that the chance of my eventual existence was unity divided by the maximum number of self-awarenesses that could possibly exist, or 1 divided by infinity, which is an infinitesimal number, effectively zero. In other words, before my birth I was impossible. But despite that impossibility, I am here -- this fact is 100-percent certain. When I am gone, the chance of my re-emergence into the physical world, to hopefully resume my fractal exploration, will once again be extremely small, but it will not be as infinitesimally small as it was before I appeared the first time, for at that future time I already will have been defined. This line of thought leads me to conclude that reincarnation is extremely unlikely, but not as impossible as was incarnation the first time. And with these odds, and an infinity of time and space available for it to happen in, my re-appearance is virtually certain. At this time I'll not speculate on what it is that might reincarnate. And my use of myself as an example is no indication of an overblown ego. It is simply easier to write 'I' than to write 'any given individual'. Once again the time has come to get busy. The work is waiting and it's 10:30am. So until next FOTD, take care, and enjoy the fractals. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
A_Nameless_Fractal { ; time=0:16:14.98--SF5 on a P200
reset=2001 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=ident passes=1
center-mag=-0.41562595889429820/-0.000000209527199\
68/1.010792e+011/1/-12.498 params=1/4/2/3/0/0
float=y maxiter=2100 inside=0 logmap=343
colors=0003qz3qz4rz5rz6sz7sz8sz9tzAuzBvzCwzDwzFsqH\
nhJg_LZRMUINQ9JN10HGQBVR5hQeTUWRZNQcDPh4OMFq3aK7ZR\
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eRMaX_ZbmXlEOfSFQe7AsEFoKKlQPhXUebZahcZngWfdU_bST`\
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U_zXXz_TzbPzeMzhIzkEznBzmFzlIzlLzkOzkRzjUziXzi_zhb\
zhezedzbdz_dzXczUczhlz_nz
}
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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