Jim, Thanks for the test image. It's calculating away in Richard's Fractint for Windows beta 5 (now called Iterated Dynamics) - not for general release. And, here's Jim's image and some variations: http://www.emarketingiseasy.com/TESTS/FOTD/2016/F160609.jpg http://www.emarketingiseasy.com/TESTS/FOTD/2016/F160609A.jpg http://www.emarketingiseasy.com/TESTS/FOTD/2016/F160609Z.jpg http://www.emarketingiseasy.com/TESTS/FOTD/2016/F160609Y.jpg http://www.emarketingiseasy.com/TESTS/FOTD/2016/F160609X.jpg http://www.emarketingiseasy.com/TESTS/FOTD/2016/F160609W.jpg
I am now trying to visualize a 4-D object called the Clifford Torus... Wikipedia https://en.wikipedia.org/wiki/Clifford_torus
has a nice animation of a 4D Clifford Torus rotating through 3-space and further states: "Although having a different geometry than the standard embedding of a torus in three-dimensional Euclidean space, the square torus can also be embedded into three-dimensional space, by the Nash embedding theorem; one possible embedding modifies the standard torus by a //// fractal //// set of ripples running in two perpendicular directions along the surface.[1]" The article has a link ([1] above) to this paper: http://www.pnas.org/content/109/19/7218.full based on John Nash's (yes, I believe *that* John Nash) work that shows (if I understand the basic concept correctly) an embedding of a 4-space torus in 3-space using "fractal ripples." The article at the link above has interesting images that show toruses (tori) with fractal ripples in them. - Hal Lane ######################## # hallane@earthlink.net ######################## -----Original Message----- From: Fractint [mailto:fractint-bounces@mailman.xmission.com] On Behalf Of Jim Muth Sent: Monday, May 9, 2016 6:29 PM To: fractint@mailman.xmission.com Subject: [Fractint] FOT? test May 09, 2016 FOTD -- May 05, 2016 (Rating A-4,M-4) Fractal visionaries and enthusiasts: Today's image is a test I am doing to again become familiar with the key strokes of the DOS version of Fractint, which were near automatic for many years. The image has little artistic or math worth other than the fact that all fractals are unique. Actually, the key strokes are coming back to me faster when I let my fingers go their own way rather than trying to consciously recall them. The image might soon be posted to the FOTD web site at: <http://www.crosscanpuzzles.com/Archives.html> It might also soon be posted to other web sites if anyone cares to make an effort. Unfortunately, Paul Lee, who hosted the FOTD web site for so many years, is no longer with us. It was rather pleasant here at Fractal Central today, with morning sun giving way to afternoon clouds and a temperature of 70F 21C. The three fractal cats, Nicholas the black 13-year old guy, Jasmine the black 3-year old somewhat overweight gal, and Lida, the 2-year-old frisky lady tabby cat, are doing quite well, though the lovelorn local tomcats prowling the area are keeping all the fractal cats dashing from window to window. During my absence, I have produced about 75 pencil drawings in my own unique semi-surrealist style, mostly of imaginary scenes on distant planets. I also delved briefly into politics and conspiracy theories, but found most but not all of them too crazy to take seriously. My greatest distraction has been into meanings of the latest experiments in field of quantum theory, where I found rational materialistic skeptics, atheists, etc. almost going bonkers trying to defend the reality of the real physical world against the claims of many new-agers and even some quantum scientists that the real world is only some kind of matrix-like illusion. Such skeptical passion, mixed with typical skeptic ad-hominen attacks, must be based on something very serious and philosophically very important. (It must be that old quantum bugaboo, the measurement problem.) I have also not forgotten the fourth dimension. I must not yet have had enough trouble trying to visualize the 4-D cube, since I am now trying to visualize a 4-D object called the Clifford Torus, which is a 2-D curved surface lying in/on the curved 3-D surface-space of the 4-D hypersphere. This strange thing, which divides the 4-D hypersphere into polar and equatorial halves, is a kind of torus in which the inside is also the outside, the latitude lines are also the longitude lines, and the geometry is flat euclidian. More on all of this later. Until next time, take care, and the best is yet to come. 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