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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: The formula Z^2+C draws Mandelbrot sets or Julia sets, depending on the way it is iterated. If both parts of Z are fixed, the formula draws Mandelbrot sets; if both parts of C are fixed, it draws Julia sets. But what if different parts of both Z and C are fixed? This can be done in four different ways. And in these cases the formula draws four entirely new types of fractals, which I have named Oblate sets, Rectangular sets, Parabolic sets and Elliptic sets. These four new groups of sets, along with the familiar Mandelbrot and Julia sets, mark the six perpendicular orientations through the four-dimensional Julibrot figure. Just as Mandelbrot and Julia sets have their distinguishing characteristics, so do these four new families of sets. In the new sets we find few Mandelbrot features, but many of the familiar features of the Julia sets are there, and also many new features, such as stretching, and the narrow, straight-edged bands I call bridges. Two of these bridges are visible near the top and bottom edges of the midget in today's picture. Just as it is possible to draw perpendicular slices through the Julibrot figure, so is it possible to draw slices oriented obliquely. Today's picture is one of these oblique slices. The hole at the center is actually a midget in the Scepter Valley area of the M-set. It also appears as a fancy hole in many Julia sets. But this midget, which in its fullness is actually a four-dimensional hole in a 4-D object, has been sliced in an unfamiliar direction. At this angle we do not see a Mandelbrot midget, nor do we see a fancy hole in a Julia set. We see an entirely new type of hole, a roughly rectangular thing with those filamentary bridges at the top and bottom. The orientation of this slice is within 6 degrees of the oblate direction. Many of the familiar features of Scepter Valley are visible in the picture, but in places they are stretched beyond recognition. I named the picture "This is not a Midget" after Magritte's famous painting of a pipe, which he named "This is not a Pipe", implying that it is only a picture of a pipe. The hole in today's picture is not a midget because only holes sliced in the Mandelbrot direction can be called midgets. The artistic value of the picture is below average, permitting a rating of only a 3. Don't let the render time of 1 hour panic you. This time is on an ancient 486-66mhz machine, which was the only machine available today, my Pentium being tied up with a very slow fractal. On a Pentium 200mhz the parameter file will finish in around 8 minutes. As always, the GIF image has been posted in its full glory to: The fractal weather was once again perfect, with crystal blue skies and a temperature of 84F (29C). The fractal cats approved of the perfection by catching a cicada. The fractal philosophy is still slumbering, but before long it will awaken with a mighty roar, enlightening all who read it. However today is not the day on which that will happen, so until next time, take care, and happy fractaling. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
ThisIsNot_A_Midget {;time=1:01:39.95--SF5 on a 486-66mhz
; Version 2000 Patchlevel 9
reset=2000 type=formula formulafile=multirot.frm
formulaname=multirot-XZ-YW passes=1
center-mag=+0.00000014017740139/+0.00000000818683823\
/91390.37/0.9263/-169.71/82.281
params=84.124/-1.342/-1.2510561243/0.0133126434/-1.2\
510561243/0.0133126434 float=y maxiter=15000
inside=0 logmap=478 periodicity=10
colors=000goEfpDgn6<10>etbeueeuh<3>ewt<21>LLbKJaJH`<\
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}
frm:multirot-XZ-YW {; Jim Muth
; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect
e=exp(flip(real(p1*.01745329251994))),
f=exp(flip(imag(p1*.01745329251994))),
z=f*real(pixel)+p2, c=e*imag(pixel)+p3:
z=sqr(z)+c,
|z| <= 36 }
END 20.0 PAR-FORMULA FILE==================================
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