; Date: Tue, 10 Dec 2002 09:47:04 -0500 ; From: Jim Muth ; Subject: [Fractint] FOTD 10-12-02 (Hypercomplex Bud [5]) ; Id: <1.5.4.16.20021210094921.2aff7a80@pop.mindspring.com> ; --------- ; FOTD -- December 10, 2002 (Rating 5) ; ; Fractal visionaries and enthusiasts: ; ; I see that the existence of triternions has been noticed. I had ; not seen them mentioned until Tim mentioned them in his recent ; letter, though I had assumed that they did exist and were less ; well-known mainly because they are less useful than quaternions. ; ; Three-part numbers would draw a three-dimensional Mandelbrot set ; and a six-dimensional Julibrot. But I wonder whether they would ; actually create new Mandelbrot material. The Quaternions and ; Hypercomplex numbers draw interesting Julia sets, but merely ; expand and distort the Mandelbrot set, introducing little if any ; actual new material. This might be due to the nature of the ; M-sets or it might be due to the way they are implemented in the ; program. I question the implementation because the Julia ; versions of the hyper-sets have 6 parameters, while the Mandel ; versions have only 2 working parameters. ; ; The quaternions draw a figure resembling the 3-D figure that ; would appear if the M-set were rotated around its X-axis. This ; figure, which basically consists of circles, can then be sliced ; through any of its parts, but the result is always a stretched ; and distorted version of the same Mandelbrot set that we know ; and love. The Hypercomplex M-set appears as two separate sets ; that overlap. Some interesting things happen where the two sets ; intersect, but once again, no new material is created. ; ; Perhaps the triternions would create a M-set with new material ; to explore, perhaps not. I rather suspect the latter, though it ; would be most interesting to actually try it and find out. ; ; Several years ago, bored with the sameness, I wrote a generaliza- ; tion of the Hypercomplex formula that does create new material. ; I have never been convinced that this new material is real ; rather than an artifact of mathematical imprecision, but it ; exists in a consistent way that would imply that it is indeed ; real. ; ; Today's image, which was created with this formula, and rather ; resembles a skewed gothic window with converging bars before it, ; has been named "Hypercomplex Bud". I gave it this name because ; it is a bud in the Hypercomplex Mandelbrot set unlike anything ; that appears in the familiar complex M-set. Not only is the ; image unique on the surface, but its inner details are surpris- ; ingly complex -- more intricate than anything that appears in ; the traditional M-set. I will show some of this incredible ; inner richness in tomorrow's FOTD. ; ; I had to rate today's image at an average 5. Its mathematical ; interest is above average, but its artistic merit is below. My ; alibi, as always, is lack of time. If I had more time, I could ; have found a better color palette. ; ; With a render time of 2/3 hour on my machine, the parameter file ; is a slow one. A far better way of viewing the scene is to ; download the finished GIF file from the internet and Paul's site ; at: ; ; ; ; or from Scott's site at: ; ; ; ; Monday was sunny but cold here at F.C. With snow on the ground ; and a temperature that never reached freezing, (it stopped at ; 28F -2C), the dynamic duo of fractal cats spent the day huddled ; by the heat. Late in the afternoon, a treat of turkey eased ; their distress, and consequently my distress also. ; ; As is usually the case, I find myself with several non-fractal ; tasks to accomplish before setting out on my fractal journey of ; the day. This is a certain sign that I must now get busy. So ; until next time in 24 hours, take care, and when you walk with ; fractals, don't expect much conversation. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE================================ Hypercomplex_Bud { ; time=0:39:51.09--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=HyperMandelbrot passes=1 center-mag=-0.04037558685446038/+0.084551148225468\ 65/122.6667 params=0/0/10/0/1e-014/0 float=y maxiter=12000 inside=255 logmap=177 periodicity=0 colors=000fRKgSKhSKiTKjTKkUKlUKmVKnVKoWKpWKqXLrXLs\ YLtYMsXNrWNrWOqVOqVPpUPoUQoTQnTQnSRmRRmRSlQSkQTkPT\ jPUjOUiOUiNVhMVgMWgLWfLXfKXeKYdJYdJYcIZcHZbH_bG_aG\ ``F``Fa_Ea_EaZDbZCbYCcXBcXBdWAdWAeV9eU8fV9eV9eV9eV\ 9eV9dV9dV9dV9dV9dV9cVAcVAcVAcWAcWAbWAbWAbWAbWAbWAa\ WAaWBaWBaWBaWB`WB`WB`XB`XB_XB_XB_XB_XC_XCZXCZXCZXC\ ZXCZXCYXCYXCYYCYYDYYDXYDXYDXYDXYDXYDWYDWYDWYDWYDVY\ EVZEVZEVZEVZEUZEUZEUZEUZEUZETZETZFTZFTZFTZFS_FS_FS\ _FS_FS_FR_FR_GR_GR_GQ_GQ_GQ_GQ_GQ_GP`GP`GP`GP`HP`H\ O`HO`HO`HO`HO`HN`HN`HN`HN`HNaIMaIMaJMaJMaKMaKLaKLb\ LLbLLbMLbMLbMKbNKbNKcOKcOKcPJcPJcPJcQJcQJdRJdRIdRI\ dSIdSIdTIdTIeUHeUHeUHeVHeVHeWGeWGeWGfXGfXGfYGfYFfZ\ FfZFfZFg_Fg_Eg`Eg`Eg`EgaEgaEhbDhbDhcDhcDhcDhdDhdCi\ eCieCieCifCifBigBigBjhBjhBjhBjiAjiAjjAjjAjjAkk9kk9\ kl9kl9km9km9km8ln8ln8lo8lo8lo8lp7lp7mq7mq7mr7mr6mr\ 6ms6ms6nt6nt6nt5nu5nu5TVn } 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================================== ; ;