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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: The previous image in the series of 4-D rotation images showed the period-4 northeast bud of the M-set as it appears when viewed from the side, with the Rectangular direction displayed. At this angle the displayed slice cuts through the length of the Mandelbrot cylinder, resulting in features with perfectly straight edges and narrow straight bands which I call bridges. In the previous 4D-05 image, the screen X-axis represents imag(Z), the Y-axis imag(C), and the Z-axis real(C). How then are we to rotate the image to bring the real(Z) axis, which is now the fourth dimension, onto the screen? To find the rotation, we must rotate the entire 4-D Julibrot around the image on the screen, until the image's Z axis represents real(Z). In such a rotation, the screen image will be the axis of rotation, and therefore will show no change as every point turns in place on itself, but the rest of the unseen 3-D slice will change dramatically. Now, as we rotate around the screen's X-axis, the screen's Y-axis will rotate toward real(Z) instead of real(C). We are rotating from the Rectangular direction toward the Julia direction. As the 3-D slice rotates through the fourth dimension in this manner, the shape of the cylinders themselves changes. The originally circular cylinders gradually elongate into ever more eccentric elliptic cylinders, and finally into flat, plate-like features of varying thickness. In today's image, the illustrated slice has been rotated 60 degrees from the Rectangular direction toward the Julia direction. The bridge, which appears as the brilliant feature in the background, is actually part of a 3-D plate. Since the slice of the image is cutting through this plate at an angle of only 30 degrees, the apparent width of the bridge has broadened to twice its previous value. The banded Julia stuff continues to shift and change in an almost random manner, though there is an apparent tendency for it to spread out along the bridge. In tomorrow's image, we'll continue the rotation, showing the changes that take place at an ever-increasing rate as we approach the Julia direction. The parameter file of today's image renders in one minute, about the same time it takes to get to and download the GIF image file from: The fractal weather today at Fractal Central was partly sunny but quite cold, with a temperature of 30F (-1C) that kept even the thoughts of the outdoors from the minds of the fractal cats. I'll return in 21 hours or so with further rotation of the image. Until then, take care, and practice fractal conservation. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
4D-06 { ; time=0:01:00.92 -- SF5 on a P200
reset=2000 type=formula formulafile=multirot.frm
formulaname=multirot-XY-ZW function=flip/ident
passes=t center-mag=-1.11022e-016/8.32667e-017/10.4\
1667 params=60/90/0.36775/0/0.281/0.4871 float=y
maxiter=3600 inside=0 logmap=yes periodicity=10
colors=000BAABWfCAACVfDAAEVfEAAFUfFAAGUfGAAHUfHAAITf\
IAAJTfJAAKSfKAALSfLAAMSfNAANRgOAAORgPAAPQgQAAQQgRAAR\
QgSAASPgTAATPgUAAUOgV9AWOgW9AXOgX8AYNgY8AZNgZ8A_Mg_7\
A`Mg`7AaMga6AbLgb6AcLgc6AdMhd8AdNid9AdOjeAAeQjeCAeRk\
eEFeSlfGKfTmfJPfUnfLUgVngOZgWogRcgYpgShhZqhUjh_rhWlh\
`siZniasi`pibtibricujdtjev<6>kiykiykjz<11>nqznqznrz<\
4>nsznsznsz<31>vzzwzzwzz<2>xzzxzzvyztwzruz<3>jmwhkwf\
iw<2>`cxZbxXbyVbyTby<9>NbzMbzMbz<2>KbzJbzKdz<4>PnzQp\
zRrz<2>UxzVzzWzz<3>Zzz_zz_zz<5>_zz_zz`zz<35>Nzz
}
frm:multirot-XY-ZW {; draws 6 planes and many rotations
;when fn1-2=i,f, then p1 0,0=M, 0,90=O, 90,0=E, 90,90=J
;when fn1-2=f,i, then p1 0,0=M, 0,90=R, 90,0=P, 90,90=J
a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994,
z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p2,
c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p3:
z=sqr(z)+c,
|z| <= 36 }
END 20.0 PAR-FORMULA FILE==================================
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