Archive for May, 2012

The Fractal Conical Sections

Tuesday, May 1st, 2012

Feed: Dr. Myron Evans
Posted on: Monday, April 30, 2012 7:17 AM
Author: metric345
Subject: The Fractal Conical Sections

I think it is very important to write up this discovery as a paper, probably UFT217. I have carried out a literature search and can find no previous work at all on the fractal conical sections:

r = alpha / (1 + epsilon cos (x theta))

Fractals are a highly developed subject in mathematics, computation and computational art, and so are the conical sections, and put the two together produces a major new subject. These graphs by Horst Eckardt show clearly that Y plotted against theta for three different values of x produce the same pattern but subdivided. These curves are for a constant alpha and epsilon. This is the basic fractal property – patterns repeated by subdivision. In the fractal conical sections the subdivision is caused by increasing the bland or ordinary looking x, in the case of the illustrations by Horst Eckardt here from x = 0.3 to 1.0 to 2.5. So every single very well known equation of the conical sections assumes a dramatically new importance. In physics, all of these fractal patterns are produced by the same universal force law, an amazing thing, and I do not often use hyperbole applied to the hyperbola. Fractals appear in the Book of Kells for example, repeated triskelions and were known to the ancients. The triskelion is made up of three spirals, and we know that the spiral can be obtained from the fractal conical section, and indeed the spiral is a fractal conical section, so the spiral is a fractal and back to the Book of Kells. A lot of very striking computer art can now be produced from the above simple looking equation. It is anything but simple, and is a profound discovery that makes Einsteinian general relativity instantly obsolete. The computer and computer art play a central role in it. Fractals were first highly developed by computer at IBM in the Mandelbrot Group in New York State, I think it was IBM Yorktown Heights. He was still there in 1986 – 1988 when I worked for IBM as an IBM professor in the Kingston, New York, environment of IBM Fellow Enrico Clementi. Fractal art generated by computer is now well known. We could share art skills at AIAS and produce a new book in colour for CISP based on such art: the three artists are Horst Eckardt, Ray Delaforce and last but not least, Robert Cheshire. I should think the fractal conical sections are related in some way to the Mandelbrot set that produces amazing patterns by computer.

217(1).pdf

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