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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: For today's fractal we return to that most basic of all fractal formulae, Z^2+C. Normally, this formula would draw some nice Julia sets, or a nice scene in the Mandelbrot set, but today it has drawn nothing like the familiar fractals. Something has gone wrong. What we see does not resemble a Julia set, nor does it look like a Mandelbrot image. It resembles a Mexican pattern, but that pattern has been distorted in a strange way. The secret is that I have drawn a slice in a new direction through the four-dimensional Julibrot figure. What appears on the screen is actually part of the valley at -1.368 of the M-set. It appears so strange because it is sliced in an unfamiliar direction, which is oriented 52-1/2 degrees from the rectangular direction toward the Julia direction. This direction is impossible to visualize with our 3-D minds, but it cuts the Julibrot in some quite unusual slices, a few more of which I may use as FOTD's in the near future. The narrow vertical open area along the picture's Y-axis is actually the narrow space between the two branches of the valley located at -1.368 on the X-axis of the M-set. I named the picture "Juliana" because its orientation is closest to the Julia direction. The draw time of 10-1/2 minutes is my best estimate, since the image was rendered on three different machines running at three different speeds. But regardless of the draw time, there is no uncertainty about the minute or so it will take to download the GIF image from: The fractal weather today was variably cloudy and very humid, with a temperature of 85F (29.5C) that drew the cats into the yard, where they chattered at the house sparrows. The day ended with a torrential thunder-storm, which drove the cats into hiding and re-flooded the fractal basement. Due to the very busy day, the philosophy went nowhere. Tomorrow I'll once again have to dry out the basement, but while doing so, I could think of some incredible philosophy. If I do, it will appear in tomorrow's FOTD. Check back in 24 hours to see how the day will evolve. Until next time, take care, and fractals forever. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Juliana { ; time=0:10:31.91 -- SF5 on a P200
; Version 2000 Patchlevel 9
reset=2000 type=formula formulafile=multirot.frm
formulaname=multirot-XY-ZW function=flip/ident
passes=1 center-mag=-0.17524/2.35922e-016/58.12843/0\
.3623/90 params=52.5/90/0/0/-1.3681038846/0 float=y
maxiter=120000 inside=0 logmap=25 periodicity=10
colors=000bVzaUzaUzaSz<3>aRzaQzaQz_Qz_Oz<5>_Mz_Mz_Kz\
<3>ZJzZIzZIz<2>ZHzZHzYIz<4>YJzYKzYKzYKzYMzYMyXMyXMw<\
2>XNuXOuXOsXOsXQrXQrXQqXKnXQqXVsXZvXcyXizXmzXrzXwzXz\
zUvzSrzRmzOizNczM_zIOzKXzMbzNjzOqyQywSzyRzvQzsOzrNzo\
MznKzkJzjIygHwfFwcEvbDu_BuZAsX8rU7qS6qQ4oO3nM2nK0mI0\
kH0kE0jD0iA0s00i80ZI<2>B4fA8b8BZ7EX6IS6KO4NM3RI2UE0a\
D2XB8RAFN8MI7RD6Y84c33j02o00u44yE7zOAzYDzVFzSIzRK<2>\
zKSzIVzHYzE_yBbyAew7gz7mw6ju6is6gq4fo4em4cj3bi3af3_e\
3Yb2X_2VZ2UX0S<2>Q0OO0NM0KK0JI0IF0HE0FB0EA0D70B60AA3\
7EA6HF4KM3NR2RY0Ub0Yi0_n0cu0cz0fz0iv0kr0mm2oi4re7uaA\
vXDySFzOIzKKyNOvORsRVoSYmU_jXcfYfc_jaamYboVesSfvOizM\
jzMqzJkzIgzHczE_wDXuBSsAOq7Kn6Hk4Di38g46m44qB2rI0s<2\
>a0wg0yo0zm0yj0vi0sf0qe2nb3ka6iZ7fYAcVBaUE_RFYOIVNJS\
KMQJNNHQKFRIDUFBVD8YA8a4<2>4UE0700400306DNMMj
}
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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