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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: In today's two images, we continue rotating the view ever closer to the Julia orientation. In fact, we are only 4 degrees from the Julia orientation. At this angle, the Mandelbrot stuff, which has already expanded to infinity horizontally, has stretched to over 14 times its normal measure vertically. Only the top part of the Mandel-bridge now appears in the close-up 4D-08 image, while the emerging spiral toward the upper right has become quite well defined. A narrow band at the top of the main band of brilliant Mandel stuff is the side view of a sub-valley in the main valley. A careful examination will show the Mandel bridge beginning to lose its definition as the increasingly narrow angle emphasizes the irregularities along its edges. However, a high enough maxiter would smooth these irregularities. The bottom of the image has been overwhelmed by an intruding mass of banded, low-iteration Julia stuff. This mass will clear away as we rotate ever closer to the Julia direction in future FOTD's. The image is filled with spirals, all of which converge clockwise. But look carefully in the lower left corner, where one small spiral converges counter-clockwise. The spirals are like people -- there are some that just have to be different. As I mentioned at the start, two images are attached to this discussion, the second image being a view of the entire fractal. This image, 4D-08a, which is already taking the shape of a Julia set, is perhaps more striking than the first. In it we see a proto-Julia set, with a grossly enlarged band of Mandel stuff cutting straight through it. This band is still a side view of the lower valley of the northeast period-4 bud, though it has long since lost any resemblance to a Mandelbrot valley. The supernumerary bands at the edges of the brilliant band are sub-valleys of the main valley. The first image renders in one minute; the second takes 1/2 minute. These times do not include the cutting, pasting and renaming necessary to change this letter into an executable parameter file. Those who would rather not cut, paste and rename may find the GIF image files posted to Usenet at: The fractal weather today was cloudy and a bit milder. But the temperature of 43F (6C) was still too cold for the pampered fractal cats, who spent the day by their radiators, wishing summer were here. Well, the day is done and I'm exhausted. It's time to shut down the fractal shoppe, call it a day, and watch the continuing election nonsense on TV. My guy has become as big a jerk as the other guy, so I've long since stopped caring who finally gets to be president. Until next time, take care, and have the patience of Job. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
4D-08 { ; time=0:01:09.25 -- SF5 on a P200
; Version 2000 Patchlevel 14
reset=2000 type=formula formulafile=allinone.frm
formulaname=multirot-XY-ZW function=flip/ident
passes=t center-mag=-1.11022e-016/8.32667e-017/10.41\
667 params=86/90/0.36775/0/0.281/0.4871 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
}
4D-08a { ; time=0:00:29.50 -- SF5 on a P200
; Version 2000 Patchlevel 14
reset=2000 type=formula formulafile=allinone.frm
formulaname=multirot-XY-ZW function=flip/ident
passes=t center-mag=-1.55431e-015/-0.344718/0.8623068
params=86/90/0.36775/0/0.281/0.4871 float=y
maxiter=3600 inside=0 logmap=yes
symmetry=none 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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