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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: There is a curious rectangle in the Julia set of the formula Z^(2.003)+C with initial C set to -1.7453,0. Its Z coordinates are 0.00019,0.07388. It is best seen at a magnitude around 100. I have featured this Julia rectangle, and various distortions of it, many times over the 10-year life of the FOTD. Not surprisingly, similar rectangles surround the Mandelbrot midgets in the -1.7453,0 area of the Z^(2.003)+C Mandelbrot set. But there are an infinity of other directions in which the 4-dimensional Julibrot can be sliced at this point. Does the rectangle exist in any of these odd slices? As today's image shows, it most certainly does exist in at least some of the odd slices. Today's scene is sliced very near the Oblate direction, and the rectangle, if anything, is even more spectacular. The appearance of origin symmetry in the image is deceiving. A close examination reveals subtle differences at the corners. But if the slice had been exactly in the Oblate direction, the symmetry would have been preserved. I rated the image at a 5, since there are only so many times the same fractal object can be shown and rated above average. I named the image "Once A Rectangle--", because no matter how the orientation is twisted, the rectangle is always hiding somewhere nearby. The calculation time of under 3 minutes on the P2000 would be around 12 minutes on the P200 unit. But why calculate when the finished image is posted for viewing on the FOTD web site at: Near perfect weather prevailed here at Fractal Central on Monday. Only a slight bit too much humidity kept it from being absolutely perfect. Of course, weather perfection is a subjective thing. With almost no rain for the past month, the local farmers certainly are not calling the weather perfect. The fractal cats spent the day being cats. I spent the day trying, but not quite succeeding, avoiding activity. The next FOTD will appear in 24 hours. Until then, endure. Jim Muth jamth@mindspring.com jimmuth@aol.com |
START PARAMETER FILE=======================================
Once_A_Rectangle-- { ; time=0:02:54.25-SF5 on P4-2000
reset=2004 type=formula formulafile=allinone.frm
formulaname=SliceJulibrot4 passes=1 periodicity=10
params=0/90/2/90/-1.7435/0/0.00019/0.07388/2.003/0
center-mag=-0.00000638398529728/-0.000133749671322\
46/3892.894/0.01489/92.009718764060608/76.75397332\
36568793 inside=0 float=y maxiter=10000
colors=0001In1In2Jo2Jo2Jo2Jo3Jo3Ko3Ko3Ko4Ko4Ko4Ko4\
Lo4Lo5Lo5Lo5Lo5Lo6Mo6Mo6Mo6Mo7Mo7Mo7No7No7No8No8No\
8Oo8Oo9Oo9Oo9Oo9OoAPoAPoAPoAPoAPoBPoBQoBQoBQoCQoCQ\
oCQoCRoDRoDRoDRoDRoDRoCSpCSpCSqCSqCSqCTrCTrCTsBTsB\
TsBUtBUtBUuBUuBUuBVvAVvAVwAVwAVwAWxAWxAWyAWyAWyCYx\
EZxF_wH`wIawKbvMdvNevPfuQguShuTitVjtXltYms_ns`osbp\
rcqresrgtqhuqjvqkwpmxpnyplvnjtlhrjgphengclebjc`haZ\
f`YdZWbXU`VSZURXSPVQOSONRNMQMKOLIMJHKHFIFDGEBECACA\
8A8687565343O21YASiDUiDTjESjERkERkEQlEPlFOmFNmGMnI\
MnKLoMKoOJpQIpSHqUHqVGrWFrXEsYDsZCt_Ct`BuaAub9vc8v\
c7uc4vc5vc6vc7vd7ve8vf9vg9vhAviBvjBvkCvlDvmDvmEvmF\
vmFvmGvmHvmHvmIvmJvmJvmKvmLvmMvmMvmNvmOvmOvmPvmQvm\
QvmRvmSvmSvmTwmUwmUwmVwmWwmWwmXwmYwmYwmZwm_wm_wm`w\
mawmbwmbwmcwmdwmdwmewmfwmfwmgwmhwmhwmiwnjwnjwokwpl\
wplwqmwqnwrnwrowspxtqwspwspvsovrovrournurnurnvrmvr\
mvrmwrmwrmwrmgDXhDWhDWiDV }
frm:SliceJulibrot4 {; draws most slices of Julibrot
pix=pixel, u=real(pix), v=imag(pix),
a=pi*real(p1*0.0055555555555556),
b=pi*imag(p1*0.0055555555555556),
g=pi*real(p2*0.0055555555555556),
d=pi*imag(p2*0.0055555555555556),
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)+p4:
z=z^(p5)+c
|z| <= 9 }
END PARAMETER FILE=========================================
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times.
