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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Today brings us to the fourth in the series of 12 images illustrating a double rotation from the Mandelbrot orientation to the Julia orientation. We started with an everyday image of the period-4 bud on the northeast shore of the main bay of the Mandelbrot set. Then we shifted our view 0.36775 along the real(Z) axis, and saw that the bud was gone, but its ghost remained, showing that the bud is actually part of a cylinder. Next we rotated the view 45 degrees around the imag(C) axis, which caused the ghost of the bud to elongate into an ellipse. In today's image, which is rotated 80 degrees toward the Rectangular direction, we continue that rotation. When the plane of our view is rotated 80 degrees, it cuts the Mandelbrot bud-cylinder at such a sharp angle that the once circular bud is elongated almost beyond recognition. The low-iteration Julia material in the foreground is also elongated, in this case to an incredible degree. And whereas the ghost-bud is elongated in an orderly, predictable manner, the foreground material is stretched haphazardly, with no sense of order. This random stretching is characteristic of all Julibrot slices except those oriented in the Julia or Mandelbrot directions, or in certain equal double-rotations between the two. The parameter file of today's image once again renders in less than a minute. Also, the GIF image has been posted to: The fractal weather today was mostly sunny but very cold. The temperature of 30F (-1C) kept the cats snug indoors. I see by the wall clock that it's 11pm -- time to shut down. But I'll return in less than a day with the surprising fifth in the series of images. Until then, take care, and you'll rarely get dizzy from rotating a fractal. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
4D-04 { ; time=0:00:55.04 -- 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=0/80/0.36775/0/0.281/0.531 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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