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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Every fractalist is familiar with the classic Mandelbrot set. The most popular area of the M-set is the valley at -0.75 on the X-axis, otherwise known as Seahorse Valley. This valley is filled with fractal scenery of the most interesting kind, making it the most rewarding area to explore, especially for the fractal newcomer. Seahorse Valley also has a Julia aspect, which appears as a string of roughly circular bays, the shorelines infinitely divided into identical smaller bays. because of its self-similarity, this Seahorse Valley Julia set is less interesting than the Mandelbrot aspect of the valley. The Seahorse Valley Julia set is merely the completely perpendicular plane of the four-dimensional Julibrot that intersects the M-set at the point -0.75 on the X-axis. But since Seahorse Valley is actually a four-dimensional object, it can be sliced in infinitely many other directions. Today's FOTD is a slice through Seahorse Valley in one of these new directions. The orientation of today's image is but 0.1 degree removed from the Julia direction, but look at the difference that 1/10 degree has made. The familiar Seahorse Valley Julia figure is there, rotated to a 35-degree angle, but it is filled with a most incredible background consisting of parallel bands of color. This background is actually a grossly enlarged side view of the wall of Seahorse Valley. The diagonal pink strip marks the line where the north and south branches of the valley almost meet. The maxiter of the image is a whopping 2,000,000, and every iteration is used, since the part of the valley that we're viewing from the side lies extremely close to the X-axis of the M-set, where the iterations are well over 1,000,000. I named the picture "Seahorse Valley" because even though it's not immediately apparent, it's the Seahorse Valley part of the Julibrot that we're viewing. Since the interest of this image is more mathematical than artistic, I can rate it only an average 5. With such a high maxiter, the parameter file takes well over an hour to render even on a fast Pentium, but being merciful, I have posted the GIF image to: The formula that drew the image and will draw many more FOTD's this month of August was posted to the Fractint List last July by John Goering. It is the only formula that draws all possible oblique angles through the 4-D Julibrot. Unfortunately the formula is limited by its inability to enter variable starting points of Z, but this is a limit of the Fractint program, a limit which could be easily overcome with 2 more variable parameter entries for type=formula. The fractal weather today was mostly cloudy and very muggy, with only a little sun. Later in the afternoon a light thundershower passed over, but by that time the fractal cats had already taken their daily romp in the grass. In the sultry conditions the philosophy languished, but tomorrow is but 24 hours off. Check then for a 25 percent chance of finding philosophy. Until then, take care, and don't work yourself into a fractal sweat. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
Seahorse_Valley { ; time=1:03:21.27 -- SF5 0n a P200
; Version 2000 Patchlevel 9
reset=2000 type=formula formulafile=julibrot.frm
formulaname=SliceJB passes=1
center-mag=-0.00309701/-0.00597368/0.9861933/1/-35
params=0.4995/0.5/0.499/0.5/-0.75/0 float=y
maxiter=2000000 inside=0 logmap=yes periodicity=0
colors=000KSh<3>KbmKdnKgpKjqKlrKnsKovKptKorKopKnn<12\
>bdRccPebNfbLhaJi`HkZFmXEoVErTKuRQ<4>pHVoFWmDX<18>SB\
oRBpQBq<3>MBt<8>IV_IXYIZW<3>GfN<2>G`J6ZU8XSAWR<10>Ti\
DUjCWkB<2>`o7bp6bp6cq5<3>OY4KT4TP4VT3WT3VU3XV3ZV3YW3\
ZW3_W3`W1_X3<2>a_7aa9adAafC<3>bkIblJcmLcnMcpO<2>dtSd\
vUcxWdwV<7>k_NlXMmULnRK<2>oII<4>eNScOU<4>UTrSSwTQrTN\
m<2>UGZUDUUBRU9R<5>_CK`DJaDI<3>eFEfKJ<3>jK_kKclKd<3>\
pKhqKirKi<3>vMmwNnwNo<5>wQuwRvwRwwSxwSy<33>whz
}
frm:SliceJB {; by John R. H. Goering, July 1999
pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1),
b=pi*imag(p1), g=pi*real(p2), d=pi*imag(p2), ca=cos(a),
cb=cos(b), sb=sin(b), cg=cos(g), sg=sin(g), cd=cos(d),
sd=sin(d), p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd),
q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd), r=u*sg+v*ca*sb*cg,
s=v*sin(a), c=p+flip(q)+p3, z=r+flip(s):
z=z*z+c
|z|<=9
}
END 20.0 PAR-FORMULA FILE==================================
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