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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: As has been obvious for some time, I am fascinated by things that cannot be comprehended by the human mind. When these things are irrational, some on the Philofractal list feel that I am indulging in fantasy, but none can dispute that the Mandelbrot set and its associated Julia sets comprise a single four-dimensional object that in a sense most certainly does exist, yet cannot be comprehended by the human mind. This 4-dimensional object is sometimes called the Julibrot figure. Depending on the direction and location in which this 4-D figure is sliced, either Julia sets or perturbed Mandelbrot sets will appear. But in addition to producing Julia and Mandelbrot sets, the Julibrot can be sliced in other directions which produce hybrid sets that are neither Julia nor Mandelbrot sets, but combine the features of both. Today's image is one of these hybrid sets. I could have named the image "Julibrot", but that word has become hackneyed, so I named the image "Mandelia", a name which combines the names of both families of sets and also has a nice sound to it. I rated the image a 6. To produce today's image, I began with the Julia set of Seahorse Valley, which normally appears as a chain of empty bays, but I did not slice the Julibrot in exactly the Julia direction. Instead, I double rotated the slice 1/100 degree and 1 degree from the Julia direction. This tiny rotation produces a tremendous difference in the resulting image. (Curiously, the values of the rotation are quite critical this close to the Julia direction.) The overall shape of the fractal is that of the Julia set of Seahorse Valley. But the chain of bays is no longer empty. Instead of a dull, blank inside, we see an inside filled with detail quite unlike any normally seen in Julia or Mandelbrot sets. The valley itself appears at the exact center of the image, and the great spirals that lie in the area are also in evidence, appearing where the many unadorned valleys of the pure Julia set would normally lie. I have tilted the image to give it a more dynamic feeling, and colored it with the basic equal-iteration-band method. I used the basic coloring method to assure that the unusual effects in the image are actually a part of the fractal and not a result of a fancy coloring system that distorts and sometimes totally hides the original fractal. Due to the unusually high maxiter and the periodicity setting of 15, both of which are necessary, the parameter file is a rather slow one, taking 25 minutes on my machine. But Paul and Scott are here to save such frustration, and will soon have the completed GIF image file posted to their web sites at: The fractal weather Tuesday was chilly (64F 18C) and very windy, with heavy threatening clouds that finally released a damaging rain squall at 7pm. When the squall passed, a glorious double rainbow appeared. (All rainbows are double, but the second bow, which lies outside the main bow, is usually too faint to be seen.) Then the sun set peacefully. The dynamic duo, who were still shell-shocked from the hailstorms of Sunday and Monday, dashed for cover when the wind started roaring and things started flying around again, and didn't reappear for several hours. Due to a power failure Sunday evening, I am a bit behind on the FOTD CD. But a half-hour extra work today will make up the shortfall. And it's now time to get busy. Until next time, take care, and it all depends on which direction you view your fractals. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Mandelia { ; time=0:25:37.91--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=multirot-XY-ZW-new function=flip/ident
passes=t center-mag=3.19189e-016/6.66134e-016/0.99\
3388/1/55/3.88578058618804789e-016 params=89/89.99\
/2/0/0/0/-0.75/0 float=y maxiter=75000 logmap=yes
inside=0 periodicity=15 colors=000N7PL8SL9SKGSJ9TI\
CUHDVGEWEFXCGYAH_8L_6P_4Wa2_a0ca0ha0nc3tc9ycDzcFzf\
HzfJzfLzfNzfPzfRzfUzfWzfUz_UzUUzNRzHRzBRz5_z0fv0mn\
0tf0zY0zR0zJ0zD0zH0tL0fN0RR1FWB3YJ7cFBjBDn9Hv5Jz3N\
z0Pz0Uz0Wz0Yz0Yz5YzB_zJ_yP_tWrnBzj0zh0zz0zz0zz0zz3\
zv7zrBzmFzhJzcNz_RzWWzR_zNczJfzHhzHhzHhzHhzHjzHjzH\
jyHjvHmtHmrHmnHmmDjjBhj7fh5ch3af0_f0_c0Ya0Wa0U_0R_\
0PY0NY0Nj5YyUhzttzzzzzyzzrzzjzzcvzYrzRmzchzlcyjlwj\
tvjyumzsnzqrznrzktzitzfvycvtaymayh_za_zY_z__aa_Hc_\
0f_0hY0jW0mU0nT0rS9tRJvQUyPPzOLzRHzUDzXBz_7vb3re0n\
h0jk0fn0cq0_t0Ww0Uz0Nz0Hy0Dv07t01r00n00m00j00h00f0\
0c00a00_00YLYWzWUyURrRPmWPhYPc_P_aPWfRUhRPjRLmRHrU\
FtUBvU7yU3zU1nW3fY3YY3P_3Ha39a3J_FUYPcWanUmyRyfYvR\
cvBjt0rt0yr0zr0nh0ea1gUBhNNhHYc9jU3vK0zA9z0Jz0Uz3c\
zDnz9mz5mtAmjHmaLmWOjNRjFUj9Xj1_j0bh0eh0hh0kh0nh0q\
j0tm0wn0zr0zt0zv0zy0zz0zz0zz0zz0zz0zt0zm0zf3zf3zf }
frm:multirot-XY-ZW-new {; draws 6 planes and 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))+p3,
c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p4:
z=z^(p2)+c,
|z| <= 36 }
END 20.0 PAR-FORMULA FILE==================================
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times.
