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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: The formula -5Z^(-1.5)-0.5Z^(-15)+(1/C) when calculated with a bailout radius of 900 produces a north-south oriented double-lobed fractal with one east valley separating the two large lobes, and two smaller west valleys separating the lobes. Today's scene lies deep in the east valley, on the shore of a Mandelbrot bud that seems to have no reason for being in the otherwise non-critical appearing valley. I could have given the picture any of a number of names. The name I decided on is "Four Fractal Spirals", which is little more than a description. Actually, the more interesting part of the image lies inside the four spirals, closer to the central midget, where the brilliant leaf-like features surround the midget. After some thought, I rated the image a slightly above-average 6. With more time, I probably could have done better on the coloring by trying a logmap of -888 on the slow image, or by turning off the logmap entirely. The parameter file takes over one hour to render on a Pentium 200mhz machine. The fastest machines can likely finish the image in less than half that time. For convenience, the GIF file of the finished image will soon be posted to: The fractal weather today here at Fractal Central featured sunshine and cold temperatures. The 35F (1.5C) was too cold for the delicate fractal cats to venture into. Instead, they passed the afternoon shifting between heating themselves at the radiator, sitting in the window and wishing it were warmer outside, and worrying for food. This afternoon, while browsing through the February issue of "Scientific American" magazine, which is still my favorite 'pop' science publication, I came upon an article by Max Tegmark and John Wheeler, covering the 100 year history of quantum theory. The article contains nothing that hasn't been told many times in various popularized books on the subject, but a short section near the end of the article appeared particularly interesting. Here is a brief excerpt from that section: "The ultimate goal of physics is to find what is jocularly referred to as a theory of everything, from which all else can be derived. If such a theory exists, it would take top spot in the family tree [of theories], indicating that both general relativity and quantum field theory could be derived from it.... "A theory of everything would probably have to contain no concepts [such as sub-atomic particles] at all. Otherwise one would very likely seek an explanation of the concepts in terms of a still more fundamental theory, and so on in an infinite regress. In other words, the theory would have to be purely mathematical, with no explanations or postulates. Rather an infinitely intelligent mathematician should be able to derive the entire theory tree from the equations alone, by deriving the properties of the universe that they describe and the properties of its inhabitants and their perceptions of the world." Basically, what the excerpt states is that in the authors' opinion, a theory of everything will necessarily show that the ultimate reality, if and when it is found, will likely consist of pure mathematics. The universe, and every thinking being in it, will be shown to be composed of numbers. This claim set my philosophical muse into action. Numbers, especially such numbers as quaternions, and hypercomplex numbers, etc. are human-invented concepts themselves. The implication of the article is that human minds have invented the things of which the real world is composed. This is but another way of stating that the world is a creation of its observers, who with their minds have created the abstractions (numbers) of which the world is composed, and that without the minds of its observers there could be no world. This statement is the basic idea of Idealism. As an idealist, I can fit this idea into my belief system with no effort, but if I were a materialist, I would be struggling for a way to reconcile it with my system of beliefs, much as I once struggled to reconcile my religious beliefs with the discoveries of science. With modern science going the way it is, being an idealist is easier than being a materialist. This is why I am an idealist. The magazine article also describes Everett's many-worlds interpretation of quantum theory, and considers it to be one of the more likely interpretations. In this interpretation, reality consists of an infinity of parallel alternate worlds, each world as real as this world. And though the other worlds are not exactly theorized to be spirit worlds, the materialistic viewpoint would be a lot more secure if such ideas had not appeared in respectable science. With scientists hypothesizing such things as other worlds, how easy it is to remember the teachings of religion, and believe that when one dies, their self-awareness (spirit) flips into an alternate version of themselves, a version which still is alive in another world, (heaven?). For those who wish for immortality, the alternate world would be one where self-awareness has somehow been freed from dependence on a mortal brain. Of course, all this is pure fantasizing on my part. None of my speculations have been scientifically proved, but neither have they been disproved. A visit to the New-Age section of any book store will show how seriously the general public takes such a possibly unjustifiable misrepresentation of quantum theory. Well, I'm worn out, the cats are well fed, and it's time to shut down the fractal corner for another night. Until tomorrow around this same time, take care. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
FourFractalSpirals { ; time=1:04:00.01 -- SF5 on a P200
reset=2001 type=formula formulafile=critical.frm
formulaname=MandelbrotMix4 function=recip passes=t
center-mag=+0.3081096921170355/-0.0035040231105116/3\
.448459e+008/1/172.5 params=-5/-1.5/-0.5/-15/0/800
float=y maxiter=5000 inside=0 logmap=888 periodicity=10
colors=000E0cBNo8JmKOoYTqUSbNPeHEbBJimHaZGeKGhca5MRR\
YZDOSPEM`7UGSZ7MfIGYSAPa8YQ7SY6MdUMihMnWKmIIlOf0ETOc\
N`MJfWPrIKnF`RAQa2gIbm42`Z3Uc4NgsfEbYQMP`LQ9sMgUJiQB\
bQgxt63<2>IE`1NWgG7OGSR_hJTiCMjeIpNHmagM6Lr<2>5HlSTo\
<2>AJld4XIMF<3>8XH5_I3aI<3>4z3<2>QWR<3>Yad_chadkcfoe\
grgYehPU<3>clgbrjXji<3>DIf<3>`kUfrRlyOrNY<3>cHL_FHXE\
ETCBQB8<3>j`XogbtmhysnljP`a0<3>spUws`okd<3>NJq<3>HMj\
GNiFOgEOfPmm<3>NHs<3>TG`UGXJXY9mY<3>dFOk7MYAVJDc77Ne\
leJA_<2>8FheZUK3r<2>8DlWYwzwmhrrDWi7LM6IZgX6PQ5<3>hb\
5mc4rT7zIAr7C<2>cUeZKGUQgPNhKJjbyZ<2>DRhDbcw8eWChqLq\
TInlLX<2>GHhDkrA`o7Zm4cy<2>5rn6wn_zRPzYFzdHz4EzFFzQK\
z`PzQUzcZzaczfhzsmzorz`wzfzzD<2>zzczzBzzUzzA<2>zzbzz\
M<2>zzezzA<2>zzbzzt
}
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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