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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: With today's image we begin the journey into the fourth dimension. The journey will not be an easy one to visualize, but by taking it one step at a time, we may come to at least a vague understanding of that next higher space. One of the best ways of understanding four-dimensional geometry is by breaking it down into three-dimensional sections and comparing it to the solid geometry of three dimensions. Keeping this in mind, today's image and discussion remains within the realm of three-dimensional space. I have given today's image a new color palette, which emphasizes the contrast between the low-iteration features that I call Julia stuff and the high-iteration features that I call Mandelbrot stuff. In today's image, as well as in the images to follow, the Julia stuff, which lies mostly in the foreground, is colored with a banded dark blue and red palette. By comparison, the Mandelbrot stuff, which lies mostly in the background, is colored a brilliant bluish-white. Today's image is a perturbed version of the period-4 northeast bud of the Mandelbrot set, which appeared in all its perfection in yesterday's FOTD. The scene has been perturbed by setting initial Z to 0.36775+0i. This perturbation moves the position of the illustrated slice 0.36775 along the real(Z) axis, while keeping the orientation of the slice unchanged. The most striking feature of the new image is the ghost of the original bud, which is still there in the same position, glowing a brilliant blue-white, partially obscured behind the broad but dull Julia-stuff arms in the foreground. In fact, unless the Julia stuff totally obscures the background, giving a blank screen, the remnants of the original bud are always visible in the same position regardless of the value of initial real(Z). Since changing the value of initial Z moves the illustrated slice without rotating it, and the ghost of the period-4 bud always retains the same circular shape, size and position, the three-dimensional shape of the Mandelbrot bud must be a cylinder. This cylinder is not continuous, but broken in many places by the low-iteration Julia stuff. Wherever it reappears however, the cylinder is always in the same location and of the same size and shape. The entire border of the classic Mandelbrot set (but not the filaments) behaves in the same manner, thus the Mandelbrot set can be taken as a three-dimensional collection of parallel tangent cylinders lying along the real(Z) axis. To keep things three-dimensional I have ignored the imag(Z) axis, but I will discuss this aspect when we shift to higher space in a few days. Today's image renders in 30 seconds from the parameter file, and downloads in a minute or so from the Usenet binary group: The fractal weather today was cloudy and cold, with a few flurries of snow, which amounted to little. The temperature of 37F (3C) was too chilly for the cats, who passed the day by the radiators of their choice. It's now time for me to shut down the fractal shoppe and call it a night. But I'll be back tomorrow with a most curious continuation of the 4-D saga. Until then, take care, and don't let hyperspace make you hyper. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
4D-02 { ; time=0:00:30.34 -- 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=0/0/0.36775/0/0.281/0.531 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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