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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: The well-known Mandelbrot set is actually a slice through the center of a less-known four-dimensional object known to Fractint users as the 'Julibrot'. Since the Julibrot is four-dimensional, the M-set does not cut the Julibrot in half, but it does slice a two-dimensional hole all the way through it. All the 'perturbed' M-sets are parallel to the 'true' set, and taken as a single unit, form the Julibrot. The many Julia sets are also slices through the Julibrot. The 2-D Julia sets also cut the 4-D Julibrot in parallel slices, and when taken as a single unit, form the same Julibrot. The difference between the Julia sets and the Mandelbrot sets is the direction in which they cut the 4-D Julibrot. The Julia sets are oriented in a direction completely perpendicular to the Mandelbrot sets. This relation is not the same as that of perpendicular planes in normal 3-space, such as a wall and ceiling, which must always intersect in a line and have common sets of parallel lines. In the 4-D Julibrot, M-sets and J-sets intersect in only a single point, and every line in every M-set is perpendicular to every line in its intersecting J-set. Just as M-sets and J-sets are slices in different directions through the same four-dimensional Julibrot object, there are many other directions in which this 4-D object may be sliced. Today's 7-rated image, which I have named "New Eastern Angle" shows East Valley of the M-set sliced in one of these new directions. Since the slice is in a new direction, there are no Mandelbrot elephants, buds, or brain-like Julia sets to be seen. What is visible is totally unlike any M-set or J-set, and must be seen to be appreciated. The image can be seen by running the attached parameter file or by picking it up from the W.W.Web. The parameter file runs in exactly 7 minutes on my worn-out Pentium 200mhz machine with a noisy fan. Most of the 7 minute render time is spent on the bottom 1/2 inch of the image. The pre-rendered GIF image downloads in even less time from the W.W.Web at: The fractal weather today here at Fractal Central was once again perfect, with sunny skies and a temperature of 71F 21.5C. The fractal cats agreed, and showed their pleasure by spending most of the afternoon lurking in the yard, watching the squirrels, who are busy gathering their nuts for the coming Winter. Seeing no signs of life yet from fractalus.com, I once again filed away the day's philosophy until and if fractalus has a resurrection. It's now time to file away the fractals and turn to more profitable but less enjoyable activities. Until tomorrow, take care, and be sharp. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
New_Eastern_Angle { ; time=0:07:00.73--SF5 on a p200
reset=2002 type=formula formulafile=allinone.frm
formulaname=multirot-XZ-YW-new passes=1
center-mag=+0.60570814332461560/+0.250416550291830\
40/8155.282/0.0003492 params=0/57/2/0/0/0/0/1e-005
float=y maxiter=45000 inside=0 logmap=135
colors=000SUhTUiUUjVUkWUlXUmYUnZUo_Up`UqaUrbUsbUtb\
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}
frm:multirot-XZ-YW-new {; Jim Muth
; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect
e=exp(flip(real(p1*.01745329251994))),
f=exp(flip(imag(p1*.01745329251994))),
z=f*real(pixel)+p3, c=e*imag(pixel)+p4:
z=z^(p2)+c,
|z| <= 36 }
END 20.0 PAR-FORMULA FILE==================================
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