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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Does two have a square roof? According to yesterday's C-FOTD it does. It also has a square root, but then so does a tree growing in a square pot. But regardless of whether the roof or root is square, the image is the same. This morning's letter from Andrew Coppin that mentions the square roof also mentions spooky things that sometimes happen in the world of fractals, where ghosts of things that no longer exist are sometimes visible. I can't comment on the fractal that Andrew was referring to because I have not seen it, but I have done lots of experimenting with fractal ghosts in my 13 years of fractaling. These ghosts are quite common, needing only the proper color scheme to make them evident. They exist in all planes and all orientations of four-dimensional Julibrots except the Julia set. Today's picture shows some very obvious ghost-buds in one of the perturbed Mandelbrot sets. The ghosts are indeed shadows of things that exist in other planes. In today's picture, the brilliant bud-like outlines in the background are shadows of the buds that exist at the same coordinates in the unperturbed Mandelbrot set. To see the buds unobscured, simply change the Z parameters of today's image to 0,0. The reason the ghosts exist can be explained by the geometry of the four-dimensional Julibrot. I'll not go into the involved explanation at this time, other than to say that the Mandelbrot shape exists as an interrupted 4-D hypercylinder in the Julibrot. The coloring of today's image was done with the under-utilized ranges and banding features of Fractint. A total of only 13 colors are used in the image, though the impression is one of far more colors. I named the picture "Spooky Radicals" after the ghost-like outlines of the buds. (A radical consists of a bud plus all its associated filaments and midgets.) I rated the picture a 4, which may be a bit modest. With a render time of just under 5 minutes, running the parameter file risks impatience. The remedy is to download the GIF image file from: The fractal weather today was partly sunny, with a temperature of 57F (14C), which was mild enough for the fractal cats to enjoy several hours out of doors. The philosophy is still stifled by the ridiculous political situation here in the U.S., which is giving me no end of frustration. I suppose we'll have no philosophy until we have a president. Right now, I'm going to ignore a couple pressing tasks and have myself an enjoyable evening before the TV. I'll return tomorrow with fractal and blather. Until then, take care, but not so much care that it stops being fun. Jim Muth jamth@mindspring.com |
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