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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: In a recent FOTD I wrote that almost all midgets in fractals created by the MandelbrotMix* series of formulas (formulae for the over-educated) are variations of the quadratic Mandelbrot set. Note that word 'almost'. Today's image shows a midget that most definitely is not quadratic, but cubic. And to top off the strangeness, the midget does not appear to be fully defined, almost as though it were cut in a non-critical plane. The pattern around the midget is perfectly defined however. This appearance of an imperfect cubic midget is far from the strangest scene I have stumbled upon in my adventures in fractal land. I have actually found perfect cubic midgets side-by-side in the same frame with midgets of the 6th order. A FOTD that appeared several years ago illustrates the scene. At the moment, the particular image is lost, but I would like to find it for further investigation. For some reason, the real(p2) parameter of today's image, which determines the amount of Z^5 to be subtracted from Z^3, is extremely critical, as it is in all fractals that I have so far found which contain higher-order midgets. I named the image "Paddles". The three prominent arms remind me of the paddle blades of an old-fashioned windmill. (Old-fashioned always seems to be more picturesque.) I rated the image at a 6, which is above average, but not outstanding. When the image's short render time is considered, its overall value comes in at a hefty 357. In a recent letter to the Fractint list it was asked what I mean by a plane oriented halfway between the Mandelbrot and Julia orientations in the Z^2+C Julibrot. I am really not sure. The more I think about it, the less certain I become as to what I mean. I'll have more to say about this in the next FOTD that uses the Julibrot formula. I also recently was asked whether there is such a thing as twin fractals. I am not sure what the writer means by twin fractals, but there certainly are twin midgets. A pair of them will appear as tomorrow's FOTD. Though today's image is a fast one, the minor annoyance of running the included parameter file can be eliminated by downloading the completed image from Paul's web site at: The fractal cats, Thomas the lazy oversized orange and Tippy the thin wiry tabby, had one of their best days ever on Monday. It was one of those days when everything that could go right in the world of cats did go right. The weather was sunny, with a temperature of 63F 17C, but the big excitement came when a squirrel came too near Tippy, who appeared to be asleep. Tippy suddenly lunged and for a moment got his jaws on the squirrel's tail. Luckily for both the squirrel and Tippy, the squirrel pulled free and scampered up a holly tree, where he scolded loudly before going back to his hole. Thomas actually became alert during the scuffle. When they came indoors, I treated them to chicken, which they enjoy even more than tuna. But today is starting cold and rainy, and the two fractal cats are already turning sulky. We'll see how it goes. For me, it looks to be an average day. And if all goes well, the fractal I find will be above average. Jim Muth jamth@mindspring.com jimmuth@aol.com |
START 20.0 PAR-FORMULA FILE================================
Paddles { ; time=0:01:40.79--SF5 on a P200
reset=2003 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=ident passes=1
center-mag=-0.4615028914331138/0/9778.83/1/-90/3.8\
857805861880479e-016 params=1/3/-0.21/5/0/0 float=y
maxiter=2000 inside=0 logmap=25 periodicity=10
colors=000h0hh0hh0hh5hhAhhFhhKhhPggUffZeecddhc`k4V\
f9PbDJYIESMTLLZCKYHPYLTYPXXU`XYdXahWglWlpWqtYqrZrq\
_qoaznbulcpkdpj16l28m3An4Co5Ep5Gp6Iq7Kr8Ms9Ot9QtAS\
uBUvCWwCYwGTrKOnOJjREfV9bZ4Za0VZ5SWAPTFNQKKNPIKUFH\
YDFRNEKXCDeB6oA0xH6vNCtUIr_OpfUnl_lsejyjhvgiseiqci\
n`ilZiiXigViiWjjWjkWjlXkmXkoXkpYlqYlrYlsZmuZmvZmw_\
nx_ny_nz_nuakqbimcfiedefbag_YhYUjWQkTMlRImPEZRBKT8\
6VAARBENCIJELGFPCGT8HW5G_4Fb4Ee3Dh3Cl2Bo2Ar19u1Ct2\
Fs3Ir4Lr5Oq6Rp7Uo8Wo8VmAVlCVlDVmFVnGUoIUpJUqLUrNUs\
OTtQTuRTvTTwUTxWTyXdzdpzlkzmgznczo_zpWzqSzrOzsKztG\
zuOzdWzOcz7bzAbzDbzGazJazMazP`zS`zU`zX_z__zb_zeZzh\
ZzkZzmPz5Pz7Pz9PzBPzDPzEPzGPzIPzKPzMPzNPzPPzRPzTPz\
VPzWSzXUzYXzZZz_az`cz``zZYzYVzWSzVPzUMzSJzRGzQWzaj\
zmhzhgzdez`dzWczSazO`zJZzFYzBXz7Uz6Sz5Pz5Nz4Lz4Iz3\
Gz3Ez2Bz29z17z17z17z17z16z16z16z16z16z16z16z15z0Zz\
hbzkeznizqlztpzwszzrzyqzx }
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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