; Date: Wed, 08 Dec 2004 11:45:53 -0500 ; From: Jim Muth ; Subject: [Fractint] FOTD 08-12-04 (Theme and Variations [5]) ; Id: <1.5.4.16.20041208114738.0d772384@pop.mindspring.com> ; --------- ; ; FOTD -- December 08, 2004 (Rating 5) ; ; Fractal visionaries and enthusiasts: ; ; After more than two months' travel, we have finally completed ; the tour of the built-in escape-time formulas. But we have not ; even dented the surface of the library of formulas that have ; been written by the Fractint users over the years. This ; collection is far too vast for me to explore before the 22nd ; century arrives. In fact, I doubt that I could explore even my ; own formulas in that time. ; ; We turn today to the HyperMandelbrot formula, which I wrote ; several years ago but have never intensively explored. My ; interest in this formula became re-kindled when I posted the ; FOTD created by the 'hypercomplex' formula of Fractint, and ; remembered the limitations of that formula. My formula adds ; four additional changeable parameters, and can draw far more ; interesting images. But everything it draws is based on the ; familiar Mandelbrot set that we know and love. ; ; When I first saw some of the images drawn by my formula, I ; wondered if they were mere artifacts. The image changes were ; too great for very small changes of the parameters. But then I ; remembered the Z^2+C Julia sets, and how very small parameter ; changes can sometimes result in very large image changes. When ; the questionable features proved stable under magnification, I ; decided they are true parts of the Hypercomplex Mandelbrot set. ; ; The default image drawn by the formula when all parameters are ; set to zero is the Mandelbrot set. Today's image shows what ; happens to the M-set when only two parameters are changed by a ; small amount. The set appears to be closing in on itself, the ; valleys joining into bridge complexes. I have named the image ; "Theme and Variations". The Mandelbrot set is the theme; the ; images posted in the next week or so will be the variations. ; ; To me, the most interesting part of today's image lies in the ; area of Seahorse Valley and the north bud. The Seahorse Valley ; has blossomed into an incredible complex of interlinked valleys, ; while the north bud has collapsed into a kind of M-set with two ; feathers in its tail. The inner details of both these areas are ; unlike anything to be found in the familiar M-set. ; ; I have rated the image at a 5, which equals average. It can be ; no more than average because I have done nothing with the parent ; fractal but enlagre it. Over the next few days I will post ; several additional curious slices of the hyper-M-set drawn by ; today's formula. These will likely be average also. Then I will ; search for the interesting inner details that might lie hidden in ; the images I have posted. These could well rate higher, as the ; 4-D M-set appears far richer than the familiar 2-D set. ; ; Because the periodicity must be turned off for the HyperMandel- ; brot formula to work correctly, the render times of images drawn ; by the formula are rather slow. But today's image, being of ; such a low magnitude, is quite fast. The render time of 37 ; seconds joins with the rating of 5 to give an overall worth of ; an outstanding 814. And the image can always be downloaded from ; the FOTD web site at: ; ; ; ; though this will take a bit more time than the rendering. ; ; Rain, fog, and a chilly temperature of 46F 8C kept the fractal ; cats indoors all day Tuesday. A good bit of tuna was expended ; in the evening as a result. This morning is milder and the sky ; is clearing, but the wind is up, and fractal cats do not like ; wind, which conceals the sound of intruding cats. Hopefully, if ; the wind abates this afternoon as scheduled, the cats will have ; enough time in the yard to keep them happy. ; ; For me its work before fractals. The next variation on the ; Mandelbrot set will appear in 24 hours. Until then, take care, ; and may the best happen. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE======================================= ThemeAndVariations { ; time=0:00:36.86--SF5 on a P200 reset=2004 type=formula formulafile=allinone.frm formulaname=HyperMandelbrot center-mag=-0.462673/4.44089e-016/0.9363296 params=0/0/0.25/0/0.01/0 float=y maxiter=4000 inside=0 logmap=3 symmetry=xaxis periodicity=0 colors=000ICTIDVIFYIHbIKeHMhHPjHSmHUpHVrHXsHZuE`wH\ cuKcsMaqPZoRWmUUkWSiZPgaOecMcfKahJ_kHYoBYmGWlKUjOS\ iTQgXOf`MeeKciIbmG`rE_vCWzCZzBayAcw9fv8it7ks6nq5os\ 4pp5qm5rj6rg6sd7tb7u_8uX8vU9wR9xOAyK7xMAxOCxQExSGx\ UIxVKwXMwZOw`QwbSwdUvfZweWxdUxcRybPyaNz`Kz_IzZFzYD\ zXBzW8zV6zV4zT9zSDzRHzPLyOPxNTxLXwK`vJdvIhrKeoMczO\ azP_zRYzTWzVTzWRzYPz_NzaLzbJz5Zz6_z7`z8az8bz9czAdz\ BezBezCfzDgzEhzEizFjzGkzGkzJjzLizOizQhzTgzVgzYfz_f\ zaezddzfdziczkbznbzpazrazu_zwYzt_zr`zpbznczldzifzg\ gzehzcjzakz_lzbezd_zfTziNzkGzmAzo4zp5zp6zp7zp7zp8z\ q9zq9zqAzqBzqBzoEzmGzlIzjKzhMzgOzeQzdSzbVz`Xz_ZzY`\ zWbzVdzTfzShzXizaizeizjjznjzsjzwjzugztezrbzq`zpZzn\ WzmUzlSzjPziNzgKzfIzeGzcDzbBza9zbnz`kz_izZgzYezWcz\ VazU_zTYzRWzQUzPSzOQzNOzRMzULzXKz`JzcIzfGzjFzmEzpD\ zsCzoDzlDziEzfEzbFz_FzXGzUGzQHzNHzKIzHIzDQzAYz7ezA\ fzCgzFhzHizKizMjzBLzDOzFQ } frm:HyperMandelbrot {; periodicity must be turned off a=(p1),b=(0,0): q=sqr(a)-sqr(b)+pixel, b=(p2+2)*a*b+p3, a=q, |a|+|b| <= 100 } ; END PARAMETER FILE========================================= ; ;