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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: On the surface, today's image resembles a piece of pop art. Actually it's "A Midget in a Quandary", which is the name I gave to the image. The midget's problem lies in its symmetry. Clearly, the midget is a quadratic one, but where is the 2,4,8,... symmetry that normally appears around such midgets? Instead of a 4-way and an 8-way symmetry, we have a very well defined 6-way symmetry, and this is not a pseudo-symmetry, it is a true 6-way symmetry. To make matters worse, the 6-way symmetry breaks down into a 36-way symmetry as it nears the midget. This would be expected in a fractal created by a formula such as Z^6+C, but today's fractal was created by a combination of Z^3 and Z^4, giving no reason at all for the 6-way symmetry. Features with 3-way symmetry also fill the image. The most prominent of these are located in the six radiating arms, though no midgets lie at the center of these 3-way features. And the area within the 36-way symmetry breaks down into the kind of non-symmetrical chaos that would be expected around the fragmentary midgets of a perturbed fractal. But a closer examination of today's midget shows that it is intact. All-in-all, today's image is a real mystery. If I were rating it according to its appearance alone, I would give it a 6. But when the puzzling mathematical aspect is considered, the picture is well worth its rating of 8. At a bit over 10 minutes, the parameter file renders reasonably quickly. And as always, the GIF image will soon be posted to the web sites of Paul and Scott at: The fractal weather today was of the pleasant uneventful summer variety, with hazy sun and a temperature of 81F 27C that pleased the cats. To please me however it will be necessary for the day's work to be behind me. And it's now time to set out on the road to pleasure. So until next time, take care, and start looking forward to philosophy of the fantastic kind. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
AMidgetInAQuandary { ; time=0:10:39.17--SF5 on a P200
reset=2001 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=ident passes=1
center-mag=+0.44736824626302050/-0.000845686608124\
88/1.476198e+007/1/-22.499 params=1/3/-0.35/4/0/0
float=y maxiter=2600 inside=0
logmap=154 periodicity=10
colors=00060020000000200600A0DE0HH0IL0LP0QU0VYbYan\
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n0qw0sz0nz0ez2UsEHiL6aU0Pe0Ek06w00z00z00z00w00n0Ai\
0Le0YY0iP0nH0zE0z65zsAknFkkKkhPnmUnrZnwe0zY6zPEzHL\
zAYz2ez0kz0wz0zz0zz0zz0zz0zz0zz0zz0zz0zz0zzAzenzLz\
z2zzAzzHzzPzzYzzezzkzzsnzzezzUzzHzz6zz0zz0zz2zz2zz\
6zz6zzAzzEzzEzzHzzHzzLzzLzzPzzUzhUssYanaLka6ie0ei0\
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zzzzzzzzszzezzUzzHzz6zz0zz0zz0zz0zz0zz0zz0zz0zzAzz\
HzzPzzYzzezzkzzszzzzzzzzP
}
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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times.
