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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Today's fractal is in two parts. The first part rates a 4; the second, a 5. I bunched the two images together and named them "Before and After". The two images show the result of a rotation in four dimensions. The "Before" image shows a very typical midget in the Mandelbrot set, with some Julia-like decorations surrounding it. This midget is situated deep in the East Valley area of the most prominent midget on the west filament of the North Radical. The "Before" picture is a pleasant enough scene, colored with a rather Halloween-like palette. But it's nothing outstanding -- we've seen dozens of similar images. Because such images have become hackneyed, I have been able to rate it at only a somewhat below average 4. The Mandelbrot set however is only one slice of a four-dimensional object known as the Julibrot, and Mandel midgets are actually four-dimensional holes scattered through the Julibrot like holes in a hyperswiss cheese. If we slice this hypercheese in an absolutely perpendicular direction, we get the various Julia sets, but what do we get when we slice through a midget in some remote oblique direction? We get what I call oblique midgets, which can take any shape whatever. The "After" image is centered at exactly the same C-coordinates as the "Before" image. The only difference is that the direction of the slice has been double-rotated 60 degrees through the fourth dimension. The color scheme is exactly the same, so that the same parts of the fractal appear in the same colors in both images. Instead of a nice neat Mandel midget, we now have an object that resembles the sun about 1/4 risen above the horizon. And the nice nearly circular array of Julia-like features is still there, but it has been transformed into a vaguely triangular mass of misplaced elements. Closer in toward the central hole, the features become distorted beyond recognition. This distortion is mild compared to the distortion that appears when the midget is sliced at some other angles. I might show some of these even more distorted scenes in the near future, but for now I feel we've given this little midget a hard-enough time for one day. After all, no one enjoys having their very worst aspects made public. The two fractals in the parameter file each render in less than one minute. With such a fast render time, running the parameter file might be more efficient than downloading the two separate images. But for those who would prefer to download, the images may be found at: The fractal weather today was absolutely perfect, with a blue sky decorated with angel-hair cirrus, and a temperature of 80F (26.5C) that brought out the best in the fractal cats. The cats spent several hours outdoors, enjoying the sunshine in their middle years. The philosophy still sleeps, but eventually it will awaken. And before long, I'll awaken and return with more wonders from the world of fractals. Until then, take care, and don't go into the fourth dimension -- you may never get back out. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
1-Before { ; time=0:00:48.22 -- SF5 on a p200
; Version 2000 Patchlevel 14
reset=2000 type=formula formulafile=slices.frm
formulaname=Mandelbrot passes=1
center-mag=-0.00000004627230047/-0.00000004133611691\
/1082251 params=0/0/-0.153481762634/1.03017708079
float=y maxiter=900 inside=0 logmap=88 periodicity=10
colors=00060Q60Q<3>60Y61_85a<3>CLiDPkETk<3>JhmKlmLpn\
MtnNxn<4>MffMbdMZb<3>MLXPHVUEUbASfMRmCUuJPxOMzQKxN8q\
L6<2>I33E74NA5<3>fM8kP9oS9<3>cm6`r5Yw5<8>SkOSiQRhS<3\
>Pc_<2>c1TdMOdmKduGevF<3>eiCefBecB<5>qWKrVLtUMvTOxRP\
zQQ<3>yMW<3>bI`XHaSGc<2>BEf<2>8U`<3>FYJHZEI_AK`5La1<\
4>JR5JO5JM6<3>IE8<3>FUUFYZEac<3>Cpx<3>Wpf_padpY<3>wp\
G<3>aqJWrKRrLLrMGrM<3>LpdMoiNonOnsPnwcjoqggzd_zb`<4>\
bVbYTcTSc<3>AMe<3>WCPa9Lf7H<2>v05<5>XIVSLZOOb<3>7Zs<\
3>b8Jj2Al3G<3>t4`v4ex4kz4pz4u<2>z9fzIhxRjw_l<3>xeUyg\
PzhL<2>zl7<5>zmQzmUzmX<3>zmhzmhzmhzmfzmL<3>zmN
}
2-After { ; time=0:00:47.29 -- SF5 on a p200
; Version 2000 Patchlevel 14
reset=2000 type=formula formulafile=multirot.frm
formulaname=multirot-XZ-YW passes=1
center-mag=-0.00000080034637373/-0.00000045133115239\
/485574.3/0.5971/179.913/62.877
params=-60/60/-0.153481762634/1.03017708079/-0.15348\
1762634/1.03017708079 float=y maxiter=900 inside=0
logmap=88 periodicity=10
colors=00060Q60Q<3>60Y61_85a<3>CLiDPkETk<3>JhmKlmLpn\
MtnNxn<4>MffMbdMZb<3>MLXPHVUEUbASfMRmCUuJPxOMzQKxN8q\
L6<2>I33E74NA5<3>fM8kP9oS9<3>cm6`r5Yw5<8>SkOSiQRhS<3\
>Pc_<2>c1TdMOdmKduGevF<3>eiCefBecB<5>qWKrVLtUMvTOxRP\
zQQ<3>yMW<3>bI`XHaSGc<2>BEf<2>8U`<3>FYJHZEI_AK`5La1<\
4>JR5JO5JM6<3>IE8<3>FUUFYZEac<3>Cpx<3>Wpf_padpY<3>wp\
G<3>aqJWrKRrLLrMGrM<3>LpdMoiNonOnsPnwcjoqggzd_zb`<4>\
bVbYTcTSc<3>AMe<3>WCPa9Lf7H<2>v05<5>XIVSLZOOb<3>7Zs<\
3>b8Jj2Al3G<3>t4`v4ex4kz4pz4u<2>z9fzIhxRjw_l<3>xeUyg\
PzhL<2>zl7<5>zmQzmUzmX<3>zmhzmhzmhzmfzmL<3>zmN
}
frm:Mandelbrot {; Jim Muth real(c),imag(c)
z=p1, c=pixel+p2:
z=sqr(z)+c,
|z| <= 16
}
frm:multirot-XZ-YW {; 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)+p2, c=e*imag(pixel)+p3:
z=sqr(z)+c,
|z| <= 36 }
END 20.0 PAR-FORMULA FILE==================================
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