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: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> 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=========================================