|
Fractal of the Day
by Jim Muth
|
|
Fractal visionaries and enthusiasts: There is the original classic Mandelbrot set, which we all are familiar with. And then there are the countless sets which surround the M-set, and are almost exactly like it, but with strange and interesting differences. These sets, which I like to think of as supernumerary, are the sets with exponents of Z such as 2.05 and 1.95. Unlike the classic M-set, these supernumerary sets cannot be displayed on a single screen. In fact they cannot be displayed in their entirety at all, because they are infinite in surface extent. Their actual shape cannot be visualized, since it is hyperdimensional, but it can be approximated by an infinite spiral, somewhat like the blade of a post-hole digger. The closer the exponent of Z lies to the integer 2, the more slowly these supernumerary fractals change as the calculated section of the spiral is removed from the ground level. When the exponent is exactly 2, there is no change at all. In today's fractal, the exponent of Z is 2.05, and the view has been moved 10 -- let's call it revolutions -- up the spiral. The overall shape of the parent fractal closely resembles the classic M-set rotated 45 degrees clockwise, but the distortions are immediately apparent. The main bud, which at this new orientation points northwest, is broken by a spurious valley on its northeast shore. This valley is filled with closely spaced branch cuts and sinuous filaments, on one of which lies today's midget. With a generating formula of Z^(2.05)+C, the midget has *almost* 8 sinuous elements surrounding it, though some of these elements are broken and unevenly spaced. Considering the exponent of 2.05, I named the image "Hyper-Two", and rated it an all-too-easy-to-achieve 5. The parameter file's render time in the 3-minute range makes it not too inconvenient to run, but as always, the ultimate in convenience can be found on the internet at: The fractal weather today (Saturday Oct 6) started with a squall of wind and rain, which caused a brief power outage. By midday however things settled into a partly cloudy, breezy pattern, with a temperature around 60F 15.5C. The fractal cats dislike wind, which covers the signs of approaching danger. They spent the afternoon on their shelf by the window, watching the world go by. I'm still in the process of digesting the current issue of 'Skeptical Inquirer' magazine, which addresses the topic of science and religion. As expected, it is heavy on Scientism (empirical science taken as a source of religious truth) and Atheism, but with a good bit of worthwhile information mixed in with the dogma about a meaningless universe. The message of the magazine appears to be: believe what the evidence says you must believe, not what is necessary to make you feel good. This is solid advice, but unfortunately, there are many opinions of the evidence and what it says must be believed. I'll have much more to say in a few days when I finish reading. And I'll also find a way to keep things on the fractal topic. Well, it's once again Sunday morning here at Fractal Central. And Sunday is my day of rest. I think I'll rest today by taking a trip to the land of fractals to see what I can find there. If I find anything of interest, it will appear as the next FOTD. Until the next FOTD, take care, and believe what you will. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Hyper-Two { ; time=0:03:21.69--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=MandelbrotBC1 function=floor passes=1
center-mag=-0.26165001846746670/+0.849300872259936\
60/2.095812e+008/1/-114.999999882607526/-2.1637172\
242794378e-007 params=2.05/0/10/0 float=y
maxiter=1200 inside=0 logmap=92 sound=off
colors=000wXutWtrWtoVtmVsjUshUsWQiKM`LQcMTfNXhO_kP\
bnQfpRisRluP_kONaLIPJEDQFJWGOaHUgIZhOUhTPhZKicFiiA\
in5is1gf8eUFcHMa5SiLXq_`xndvhctccsZcqUcoPcnKcYNWHQ\
O0SH9UMIWRRYWZ__5CTDINLOITUD`_8cRAgICj9Em0Gl1Dk2Aj\
37i44i52l7Ro9omCjlEfkHbjJYiMUhOQgQMbLTYH_UDfP9mL5s\
1dcAbbJabS_a_ZahX`qW`yV`rWYkXVdYSYZQR_NK`KEaIMWEUQ\
BWRGXRLYRPZRU_RZ`RbaRgbRkeSegT`iUWlVQnWLpXGrXBqaGp\
eLoiQnnVmr_mvdip_ekVbfRZaMVXHSSD`JFiAHr2JcAOPHSBOW\
DPSEPOFPKGQGHQCIQ9qcxhWr`OmSHhK9cC2ZN9cXFhgMmqSqnP\
kkMfhJ`eGWbDQ`ALUpAZiJbbSgW`kPipIrtBzyecqf`jgYbhVW\
iSPjPLiMHiJDhGAhE8lF6pG4tHHvKTxNdyQVnJLdDCV7LO5TH4\
aA3i32zykovWdtGbUbiUgpUlwUqmUrdUsVUtMUuNUoOUjPVdQW\
_RXUSYPQ_VOa_NcdLeiJgnIisMkoQmkUogXqc`s_duWhwSkyOn\
zSqzVszZvzaxzd_zdCzdIzYNzSTzMYzGbzAXzHSzONzVHzaCzh\
7zoCzjGzfLzaPzYTzUYzPazLezHbzE`zCZz9Xz7Vz4Tz2QzFNz\
RKzcIzoLzfOzYKzRhzpmzrrzt
}
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==================================
|
times.
