; Date: Mon, 19 Dec 2005 21:36:28 -0500 ; From: Jim Muth ; Subject: [Fractint] FOTD 20-12-05 (Mysterious Goings-On [NONE]) ; Id: <1.5.4.16.20051219213833.37df7520@pop.mindspring.com> ; --------- ; ; FOTD -- December 20, 2005 (Rating NONE) ; ; Fractal visionaries and enthusiasts: ; ; Today's fractal is not much to look at, but it holds deep math- ; ematical 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 Mandel- ; brot 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 signifi- ; cant 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 unexpec- ; ted 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-quali- ; fied, 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: ; ; ; ; from where it may swiftly be downloaded. ; ; 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========================================= ; ;