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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: I am always fascinated by polytopes of higher dimensions, particularly the regular polytopes of the fourth dimension. In 4-space, six regular polytopes exist. The 5-cell corresponds to the tetrahedron of 3-space; the 8-cell, to the cube; the 16-cell, to the octahedron; the 120 cell, to the dodecahedron; and the 600-cell, to the icosahedron. But in 4-space there also exists a polytope known as the 24-cell, which would appear to have no corresponding figure in 3-space. This appearance is not totally true. The 24-cell has no corresponding regular figure in 3-space, but it does have a similarly constructed figure -- the rhombic dodecahedron, which consists of 12 rhombic faces, 24 edges and 14 vertices. Curiously, the rhombic dodecahedron is a space-filling figure. In tomorrow's discussion, we shall construct a 24-cell polytope from the ground up, but before then, we have today's fractal to examine. Today's image is named "Negative is Positive". I gave it this name for two reasons -- first: many of my images result from combining negative powers of Z, which is a positive thing, and second: I received a junk e-mail this morning from a spammer, which is nothing unusual. But this letter was intended to sell an anti-spam program to me. It began by saying "If you had our product, you would not have received this letter." My mental reply was, "It's always best to start the day with a laugh." Then, after a moment's consideration, I deleted the e-mail and named today's image. Though its name was given in jest, today's image has perhaps a bit more of the negative than the positive about it. The rating of 4 is quite a comedown from yesterday's, but then, we can't be above average on every day. And yes, the image was created by combining two negative powers of Z, in this case -3 and -5. The parent of today's scene is a rather comical figure resembling a lobster sitting on its blunted tail. Today's image is located at the tip of an irregular filament near the point where the lobster's right claw joins its body. The image renders from the attached parameter file in just over 3 minutes. The GIF file of the completed image downloads in even less time. The download may be found at: With a temperature of 25F -4C, Wednesday was very cold and windy here at F.C. Occasional flurries of snow made things even worse. The F.C. cats spent the entire day indoors, checking the outside conditions several times, but turning around as soon as their sensitive noses felt the chill. Today is starting just as cold, and more snow is threatening. The cats will have another unsatisfactory day. My day will be satisfactory if I find a perfect fractal, which is unlikely. Until next time, take care, and remember that no one is perfect. Jim Muth jamth@mindspring.com jimmuth@aol.com |
START 20.0 PAR-FORMULA FILE================================
NegativeIsPositive { ; time=0:03:15.26--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=MandelbrotMix4 function=recip passes=1
center-mag=+1.23970545929533/-1.269509193680854/9.\
148861e+009/1/170/7.41674007990883455e-005
params=1/-3/0.5/-5/0/0 float=y maxiter=1200
inside=0 logmap=60 periodicity=10
colors=000OCVPCYQAaT9gV7jW5iX3n`2qd1ph1nl0mp0kt0ix\
0hz0fz0ez0dx0ct2aq5`m8_iAYfCYcGX_IWWMVTOTPSSMVQJXQ\
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zLYzV`zcczmezwazq_zlXzhVzcSzYPzVSzWVzWXzW_zXazXdzX\
fzXizYlzYozYqz_tz_wz_zz_xzYxzYxzXxzXwzWwzWwzWwzVwz\
VvzTCz5FzAGzDHzHJzMLzPMzT }
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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