FOTD -- October 20, 2004 (Rating 6) Fractal visionaries and enthusiasts: Today's FOTD takes us to the hypercomplex formula of Fractint, which draws a four-dimensional version of the Mandelbrot set. At one time I used this formula quite extensively, but since then I have written my own formula, the HyperMandelbrot, which draws all the same images plus many new images that the hyper- complex formula does not. The hypercomplex formula is not to be cast aside as outmoded however. It still draws some very interesting images, and it draws them faster than my own formula. In a case like today's, where the image takes over one hour to render, speed becomes quite important. In addition, the hypercomplex formula is the only one in which the various functions can be applied. Being four-dimensional, the overall shape of the hypercomplex Mandelbrot set is impossible to visualize, but when the function is 'sqr', it can be taken as two familiar Mandelbrot sets partly overlapping. The position of the overlapping sets is controlled by the Z parameter. The other functions produce different figures, but the figures behave in pretty much the same way. The most interesting parts of the fractals drawn by the hyper- complex formula are the places where different parts of the two overlapping M-sets intersect. One such place is seen in today's image, which shows an intersection of Elephant Valley and Seahorse Valley. When two elements intersect, the element with the lower itera- tion count always obliterates the higher. In today's image, Seahorse Valley appears as the brownish shaft shooting up the center of the frame, while the rest of the scene is a part of Elephant valley. Examine the brownish elephant-trunk features just above the center of the frame. The elephant spirals do not go on indefinitely, as would be expected. Instead, as their iteration count increases, they grow more diffuse as they grow smaller, until their iteration count finally becomes higher than the count of Seahorse Valley, which until now they have been obscuring, and they vanish entirely. The moral of all this is, 'once a point escapes, it is not coming back'. I named the image "Elephant-Seahorse", because it is a combina- tion of both valleys. I rated it at a 6, more for its mathema- tical interest than its artistic worth. The unusually slow render time of 1 hour 18 minutes reduces the overall worth to a pathetic 7.6. But that abysmal worth can be ignored by downloading the file of the completed GIF image from the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> A chilly rain here at Fractal Central on Tuesday, with a temper- ature of 52F 11C, kept the dynamic duo of fractal cats indoors all day. They accepted their confinement stoically, though an extra treat of tuna was needed to help them find their stoicism. Today is starting chilly and cloudy, but at least it is dry. What the cats will make of this is anybody's guess. My guess is that the next FOTD will appear in 24 hours, and it is a pretty fair guess that the image will be memorable. Until that glorious moment arrives, take care, and search for the hidden meaning in fractals. Jim Muth jamth@mindspring.com jimmuth@aol.com START PARAMETER FILE======================================= Elephant-SeaHorse { ; time=1:18:35.73--SF5 on a P200 reset=2003 type=hypercomplex function=sqr passes=1 center-mag=-0.22062180436286390/-0.029775589430454\ 16/607.5435 params=0/0/0/0.53 float=y maxiter=27000 bailout=9 inside=0 logmap=25 periodicity=10 colors=000OSVPTVQUWRVWSWWUXWVYXWZXX_XY`XZaY_bYacYb\ dYceZdfZegZfhZgi_ij_jk_kl_lm`mn`no`opapqaop`nn_lmZ\ kkYjjXiiWggVffUedUccTbaSa`R`_QZYPYXOXVNVUMUSMTRLSQ\ KQOJPNIOLHMKGLIFKHEIFDJGEKHELHFMIFNIGOJGPKHQKHRLIS\ LITMJUNJVNKWOKXOKYPLZQL_QM`RMaRNbSNcTOdTOeUPfUPgVQ\ hWQiWRjXRkYRkZRj_9i`BhaCgbEfcFedGceIafJ_gLYhMWiNUj\ PSkQQlSQmRRnRSnRToQUoQVoQXpPYpPZpP_pP`qOaqObqOdrNe\ rNfrNgsMhsMisMktNjsMjsMjsMjsMjsMjsMjrMjrMirLirLirL\ irLiqLiqLiqLiqLhqKhqKhpKhpKhpKhpKhpKhpKhrIhpKhoMhm\ OhlQhjShiUhgVhfXhdZhc`habh`dhZehYghWihVkhTmhSohRpg\ SofTofToeUneUndVncVncWmbWmbXmaYmaYl`Zl_YmZXnYWoXVp\ WUqVTrUSsTRtSQuRPvQOwPNxOMyPLzPKzQJzQIzQHzRGzRGzSH\ zSIzTIzTJzTKzUKzULzVMzVMzWNzWOzWOzXPzXQyYQyYRyZSxZ\ SxZTx_Ux_Uw`Vw`Ww`Ww_VuZVtYUsXUrWTqVToUSnTSmTRlSRk\ RQiQQhPPgOPfNOeMOcLNbLNaKM`JM_ILYHLXGKWFKVEJUCISEJ\ TFKTGLTHMUINUJOUKPUMQVNRV } END PARAMETER FILE=========================================