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Fractal of the Day
by Jim Muth
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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 generalization 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 surprisingly 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: 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 20.0 PAR-FORMULA 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\
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9eV9dV9dV9dV9dV9dV9cVAcVAcVAcWAcWAbWAbWAbWAbWAbWAa\
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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\
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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 20.0 PAR-FORMULA FILE==================================
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