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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: With today's Rectangular Plane image we enter new territory. In the previous images we saw that the shape of the Mandelbrot set, with all its buds, exists as an interrupted cylinder in the 3-D section of the Julibrot defined by real(C),imag(C),real(Z). In fact the cylinder shape exists in all 3-D slices of the Julibrot that contain the whole C-plane of the classic Mandelbrot set. In yesterday's 4D-04 image, we saw that by slicing this cylinder at an increasingly acute angle, the cross section becomes an increasingly eccentric ellipse. The limiting case comes in today's image, where the slice has been rotated a full 90 degrees from the Mandelbrot orientation, and cuts the cylinder along its length. In this case, we see the walls of the cylinder as straight edges. The lower wall appears as what I call a bridge. This bridge is actually a side view of the lower valley of the original period-4 bud, which has now stretched to its limit. Such bridges and straight-edged areas exist in all the odd planes, (Elliptic, Oblate, Parabolic, Rectangular), of the Julibrot, in all the rotations between the odd planes, and in all the simple rotations between the odd planes and the Julia plane. It is immediately apparent that Mandelbrot midgets are impossible in the images that contain these straight-edges and bridges, for in such images the Mandelbrot shape has been stretched to infinity, and therefore any midgets will also be stretched to infinity. I have attached the parameters for two versions of the same image. The second image is identical to the first except that the center around which the image rotates has been dropped so that it falls on the straight upper edge of the lower bridge. It is the center of this second image that we will use as the center of the rotation to the Julia orientation. Notice also in today's image that the banded Julia stuff does not stretch to infinity, but it does twist and distort in a most curious manner in the Rectangular plane. For tomorrow's image, the 6th of the series, we shall leave the comforting familiarity of 3-D objects and launch out into the fourth dimension. The surprises we will find will make the trip worth the effort. The parameter files of both of today's images render in well under one minute. To save bandwidth, I have posted only the first image to: The fractal weather today here at Fractal Central was sunny and not too cold. The afternoon temperature of 45F (7C) and warm sun lured the cats outdoors for a brief romp before that began worrying for their meal. That's it for another day, but I'll be back in 24 hours, when we'll take off on a trip to Julia land. Until then, take care, and keep those extra dimensions coming. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
4D-05 { ; time=0:00:40.42 -- 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=0/90/0.36775/0/0.281/0.531 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
}
4D-05a { ; time=0:00:44.46 -- 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=0/90/0.36775/0/0.281/0.4871 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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times.
