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Fractal of the Day
by Jim Muth
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Fractal enthusiasts: What does a hypersphere look like? Of all the higher-dimensional figures, the hypersphere is the simplest. Any n-dimensional slice of a higher-dimensional hypersphere is a hyper-sphere of n dimensions. For example, any 3-D slice of a 4-D hypersphere is a sphere. Any 2-D slice is a circle, and any 2-D slice of a hypersphere of any number of dimensions is a circle. Today's fractal is a revelry of circles, which I assume, though I have not checked, are slices of 4-D hyperspheres. The three-dimensional effect is due entirely to the coloring, to which I devoted an unusual, (for me), amount of time an effort. The image resembles nothing as much as a scene in orbit around a giant planet in some far away galaxy. The entire picture is composed of circles colored to appear as spheres. These circles are arranged into interesting sheets and tendrils, which need investigating. Anyone out there who would like to investigate further, go ahead. I've got too many more fractals waiting, to have time to spend more than one day on a given image. I named the picture "Hyper-Spherical" more out of hope than fact. I hope that these circles are slices of 4-D hyperspheres, but I have a strong suspicion that they are not. The GIF, (no JPEG with pictures with sharp edges), file has been posted in all its planetary glory to: The weather at fractal central has not changed. Partly cloudy skies with occasional sprinkles of rain and a temperature of 57F 14C made it a chilly spring day, but a perfect day for finding fractals. The original purpose of the Philofractal list was to discuss science, religion and fractals. So today, let's knock religion around a bit. I got a real kick this morning from Morgan's list of the different types of Atheists. I was a cross between the adolescent and campus types. But I can't be blamed. Having been educated in a Catholic elementary school where one hour I was told that God created the world 6,000 years ago, and the next hour was told that dinosaurs roamed the earth 100-million years ago, I naturally got confused. When I asked the sister to explain the discrepancy, she told me that when I study religion the world is 6,000 years old, and when I study science, it is 4-billion years old. When I asked which date was correct, she told me that both dates are correct. The next time I opened my mouth I was told to sit down and shut up. The next year, in a public school, I was a smart-mouthed Atheist. Actually, religion is a necessary part of the human psyche, a part that is not going to go away in the foreseeable future. My main complaint nowadays is that there are so many different religions teaching so many conflicting things. Religion runs into trouble when it starts taking the countless myths and allegories of the worlds various cultures as literal fact. I would like to strip away the excess baggage and get down to basics. The two things that almost every religion teaches are: 1-- A supreme power exists, which created and sustains the universe. 2-- A single human lifetime on earth represents but a tiny part of that individual's total existence. It frustrates me to see the religions disputing trite details of dogma when they have failed to demonstrate their two most basic principles. Having relieved a bit of my frustration, I shall close down the fractal shoppe for another evening, and call it a day. But I'll return tomorrow with more fractal news and some more fractosophy. Until then, take care, and keep the faith. Jim Muth jamth@mindspring.com |
START FORMULA=============================================
MandNewt12 {; Jim Muth
z=g=pixel, a=real(p1), b=imag(p1), c=real(p2),
d=imag(p2), e=real(p3), f=imag(p3):
h=z^a+(g-1)*(z-b)
j=c*z^d+g
z=z-e*h/(f*j),
.0000000000000000000000000000005 <= |h|
}
END FORMULA===============================================
START PARAMETER FILE======================================
Hyper-Spherical { ; 68min on a 486-100, 640x480
reset=1960 type=formula formulafile=mandnewt.frm
formulaname=MandNewt12 passes=1
center-mag=+0.789705601237999/-0.02843030167377993/4\
77729.2/1/30 params=-4.3808/-0.5139/-1.2595/1.0881/\
-0.81/2.519 float=y maxiter=250 bailout=25
inside=bof60 logmap=yes symmetry=xaxis periodicity=0
colors=000yxYyxY<16>gfHfeGecFdaE<2>ZWCYUCWTBVRBTPA\
<11>A3596899A6656666660008WT8WT8WT8WT<10>7yq<16>xPy\
<4>k8L<18>OP3<36>xWp<12>foJ<6>rtQstRrqS<22>\
Y2SY0SZ1R<28>r6K<12>IDC<19>YcdZdc_dcwTg<2>wQg
}
END PARAMETER FILE========================================
START 19.6 PARAMETER-FORMULA FILE=========================
Hyper-Spherical { ; 68min on a 486-100, 640x480
reset=1960 type=formula formulafile=mandnewt.frm
formulaname=MandNewt12 passes=1
center-mag=+0.789705601237999/-0.02843030167377993/4\
77729.2/1/30 params=-4.3808/-0.5139/-1.2595/1.0881/\
-0.81/2.519 float=y maxiter=250 bailout=25
inside=bof60 logmap=yes symmetry=xaxis periodicity=0
colors=000yxYyxY<16>gfHfeGecFdaE<2>ZWCYUCWTBVRBTPA\
<11>A3596899A6656666660008WT8WT8WT8WT<10>7yq<16>xPy\
<4>k8L<18>OP3<36>xWp<12>foJ<6>rtQstRrqS<22>\
Y2SY0SZ1R<28>r6K<12>IDC<19>YcdZdc_dcwTg<2>wQg
}
frm:MandNewt12 {; Jim Muth
z=g=pixel, a=real(p1), b=imag(p1), c=real(p2),
d=imag(p2), e=real(p3), f=imag(p3):
h=z^a+(g-1)*(z-b)
j=c*z^d+g
z=z-e*h/(f*j),
.0000000000000000000000000000005 <= |h|
}
END 19.6 PARAMETER-FORMULA FILE===========================
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times.
