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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: When I discovered the Mandelbrot set some 16 years ago, the most fascinating thing about it was the existence of countless midget sets, no two midgets exactly the same, (except for the mirror image symmetry around the X-axis). I have never really recovered from my fascination with these midgets. Probably nine of ten of my FOTD's feature a midget at the center, usually a midget of the quadratic variety. But midgets come in many varieties other than quadratic. Today's midget for example exists in a remote part of the fractal created by the formula Z^(sqrt(sqrt(2)))+C. Down in this low range there is no standard shape for the midgets, which come in all shapes. In this case, the colorful pattern surrounding the midget resembles a butterfly or moth. The midget itself, where its border is not defined by one of the discontinuities, has an ill-defined shoreline, with several tiny bays, which closer examination reveals to be totally chaotic. These chaotic shorelines exist in most midgets of a low order between 1 and 2, and are not fully revealed until the maxiter reaches a very high value. The maxiter of today's image is 75,000, and even this could be raised. I tried the image at a maxiter of 150,000, which sharpened the shoreline slightly, but it also made the render time inordinately long and buried too much of the color range right along the shoreline. I named the picture "4th root of Two" and rated it a 6. With a render time of almost 20 minutes, the parameter file is a slow one, making a download of the GIF image file by far the better choice. The image file will soon be posted to: The fractal weather today was cloudy and 74F (23C). The atmosphere was very unsettled all afternoon, with heavy thunderstorms and a tornado in the area. Fractal Central escaped the worst of it, though the heavy rain kept the cats well under cover. I now hear Tippy calling for his morning snack, so I guess it's time to turn my attention to more pressing matters and get some food into him. When Thomas sees Tippy eating, he'll turn hungry also, and rush to the dish to keep Tippy from getting all the food. The philosophy is simmering and could erupt at any moment, so stay tuned. But for now, and until next time, take care, and having a second cat is the best way of getting the first cat to eat his food. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
4th_root_of_2 { ; time=0:19:56.88--SF5 on a P200
reset=2001 type=formula formulafile=allinone.frm
formulaname=MandelbrotBC1 function=floor passes=t
center-mag=-1.311059032031873/+10.1354130773853/50\
793.51/1/114.999 params=1.189207115/0/240/3500
float=y maxiter=75000 inside=0
logmap=67 periodicity=10
colors=000A0AA0AA0AA1AA2AA3AA4AB5AE6AH7AJ8AM99PA8R\
B8UC7XC6_D5ZC6YC7XC8WC9VC9UCATBBSBCSBCRBCQBBPBAOA9\
NA8KA7GA6AA0070AA0PDAYFKdINkLPrKRqNQpPQoRPnTPnVOmX\
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RerJkuBqz1xx3vw4uv6tt7ss8rrApqBooDnnEmmFlkHjjIiiJh\
hLgfMfeOddPcbQbaSa`T`_U_YSYXRXWQVVOUUNTTMRSKQRJPQI\
NPGMOFKNEJMCILBGKAFJ9EMCGPEHSGIVJKXLL_NMbPNeSPhUQj\
WRmZTp`UsbVudWrfUohSljQilOfnMcpK`rIYtGVvETwDRvCQuC\
PtCOsCNsCMrCKqCJpCIoCHoCGnCFmCDlCCkCBkCAjC9iC8hC5n\
A7hC9cDAYECUFDUGFUHGUIIUJJUKIUQHUVGU_GUdFUjEUoDUtD\
UyBUvAUs9Up7Vm6Wj5Xh6Yg7Zf8_f9`eAadBbdBccCdbDebEfa\
Fg`Gh`Hi_HjZIkZJlYKmXLnXMoWNpVLqUNrVOsVQtWRuWTvXUw\
XVxYXyYYzZ_zZ`z_az_cz`dz`fzagzahzabzE`zG_zHYzJXzKW\
zMUzNTzPRzQQzSPzTNzVMzWKzYJzZIz`GzaFzcDzdCzfBzg9zi\
8zj7zkDzlIzmNznSzoXzpbzqgzrlzsqztvzuuzrtzotzlszj5z\
b6z`6zZ6zY7zW7zU8zS8zQ9zO
}
frm:MandelbrotBC1 { ; by several Fractint users
e=p1, a=imag(p2)+100
p=real(p2)+PI
q=2*PI*fn1(p/(2*PI))
r=real(p2)-q
Z=C=Pixel:
Z=log(Z)
IF(imag(Z) > r)
Z=Z+flip(2*PI)
ENDIF
Z=exp(e*(Z+flip(q)))+C
|Z| < a }
END 20.0 PAR-FORMULA FILE==================================
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times.
