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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Today's fractal is not much to look at, but it holds deep mathematical mysteries. The iterated formula is Z^2+C, but the image is neither a Julia set nor a perturbed Mandelbrot set. It is a simple slice in the Oblate direction through the center of the large bud centered at -1 on the X-axis of the classic Mandelbrot set. The real(c) value of every point of the image is -1, while the imag(z) value is zero. It is one of the most familiar slices of the Z^2+C Julibrot. Since the image is so familiar and I put almost no effort into coloring it, rating it according to the usual standards would be unfair. I chose to leave the image unrated and concentrate instead on the mathematics, which at least to me is a mystery. The number 1.618... is known as the golden ratio, or just the greek letter 'phi'. This number has the property of showing up in the most unexpected places, not the least of which is the Z^2+C Julibrot. It appears very prominently in several forms in today's image. To start, the limit of today's fractal on the X-axis is exactly + and - 1.618... . Next, the two sets of most prominent arms meet at + and - 0.618... on the X-axis, a value which is the reciprocal of 'phi'. In addition, the pair of arms beyond these meet at + and - 1.272..., which happens to be the square root of 'phi'. The next pair of arms meet at -1.507..., which is the square root of 1+(sqrt(phi)). I assume the series of significant numbers continues through the entire set of ever smaller arms, though I have not attempted to track them. I named the image "Mysterious Goings-On" because of the unexpected numbers that appear in it. I assume that a math expert could easily supply a simple reason for this apparent mystery, but unfortunately I am no math expert. The render time of today's image is no mystery. It is one of the fastest FOTD images of all time. At a superluminal 2-1/4 seconds, it will try no one's patience. Those with over-qualified, handicapped computers may grow a bit impatient with their machines, but the completed GIF image is posted as always on the FOTD web site at: A bit of philosophy is brewing, but is not yet ready to be made public. Stay tuned. Limited sunshine and cold temperatures kept the fractal cats safely indoors here at Fractal Central on Monday. But they appear to have been spoiled by my sister, who gave them all the tuna they could eat as she looked after them over the weekend, while we were away. It's no problem. I am not fooled by their guilt-inducing sulky stares. My day was moderately busy; the fractal was moderately successful. The next fractal will appear in 24 hours. Until then, take much care, and remember that things are almost always better elsewhere. Jim Muth jamth@mindspring.com jimmuth@aol.com |
START PARAMETER FILE=======================================
MysteriousGoingsOn { ; time=0:00:02.15--SF5 on a P200
reset=2004 type=formula formulafile=allinone.frm
formulaname=SliceJulibrot2 symmetry=xyaxis passes=b
center-mag=0/0/0.7788162 params=0/0/90/0/-1/0/0/0
maxiter=1200 inside=255 float=y
outside=real periodicity=yes
colors=0007Qv7Mw7Ix7FxAGsCHnEIjGJeIK`KJXMHSODOSIUW\
NZ_UccZhgbmkfrojwnltmmqlonkpkjrhiseiubhv_gxXfyUezR\
dzOdzLezJfzIfzHgzGgzFhzDhzCizBizAjz9jz8bzEVzJNzOFz\
T7zYHzaRzd_zgczYgzOdzKgzNizQkzTmzVozYqz`szbpz`nz_k\
zZizYgzXdzWhzUmzTrzSvzRzzQzzPzzOzzQzzRzzSzzTzzVzzW\
zzXzzYzzZzz`zzazzbzzczzdzzbzzazz_zzZzzXzzWzzUzzTzz\
SzzQzzPzzNzzMzzKzzJzzIzzOzzUzz_zzezzkzzqzzpzzpzzoz\
zozznzznzzmzzmzzlzzlzzizzfzzdzzazzZzzXzzUzzRzzPzzM\
zzJzzHzzKzzNzzQzzTzzWzzZzz`zzZzzYzzXzzVzzUzzTzzRzz\
QzzPzzNzzMzzLzzJzzIzzHzzFzzEzzCzzBzz9zz8zz6zz5zz3z\
z2zz6zz9zzDzzGzzKzyNzxRzwUzvYzu`ztdzsgznjzimzdpz`s\
zakzbdzbYzcRzdKzdDzaCz_CzYCzVCzTCzRCzPCzKQzFczAqzB\
mzCizDfzEbzFZzGWzHSyHPvIQsIRqJSnJTlJTiKUgKVdLWbLX_\
LXYMYVMZTN_QN`ON`PM_QM_QLZRLZRKYSKYSJXTJXUJXUIWVIW\
VHVWHVWGUXGUXGUVFPTELREHPDCND8LC4KC0PH3TM5XR7aW9e`\
BidDDIIILMNNPSQTWSW`V_000 }
frm:SliceJulibrot2 {; 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=sqr(z)+c
|z| <= 9 }
END PARAMETER FILE=========================================
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times.
