; Date: Fri, 30 Dec 2011 15:16:13 -0500 ; From: Jim Muth ; Subject: [Fractint] FOTD 30-12-11 (Diminishing Returns [7]) ; Id: <1.5.4.16.20111230151807.2bc781e0@earthlink.net> ; --------- ; ; FOTD -- December 30, 2011 (Rating 7) ; ; Fractal visionaries and enthusiasts: ; ; Start with Z^2, subtract Z^3, add Z^4, subtract Z^5 then add C ; on each iteration. The result is a fractal with a mis-shapen ; main bay and several large sub-bays. Large period-2 buds lie on ; both the east and west extremities of the parent, with spikes ; extending outward along the X-axis from both buds. Today's ; image is located on the spike extending west from the western ; large bud. ; ; Luckily, one of the critical points of today's iterated ; expression lies at zero, which would have saved some tedious ; calculation if I had not been curious about other critical ; points. (A second critical point lies at +0.729323..., which ; draws a very similar parent fractal. A point of inflection ; lies at +0.436117..., but this also draws a very similar image.) ; ; The name "Diminishing Returns" refers to the nature of fractals. ; The most interesting Mandeloid is created by raising Z to the ; power of 2. As the exponent of Z is increased, the Mandeloids ; start to look ever more similar, and by the time we reach Z^100, ; there is almost no difference in the overall appearance or in ; the smaller details. In short, the law of diminishing returns ; kicks in, and the fractals start to all look the same. ; ; The same law of diminishing returns applies to the magnitude. ; As we go deeper into a fractal such as the Mandelbrot set, the ; minibrots we find grow ever more interesting at first, but at a ; magnitude around 10^25 they begin to all look the same, and by ; the time we reach 10^100, with some few exceptions, little but ; nearly identical concentric circles surround the minibrots. ; ; The same diminishing returns appear as we pile on various powers ; of Z. Today's parent fractal combines four different powers of ; Z in each iteration, yet the image falls far short of some of ; the images created by the MandAutoCritInZ formula, which ; combines only 2 different powers. Out of curiosity, I have ; written formulas that pile on up to 15 different powers, only to ; discover that, once again, the fractals all start to look the ; same. ; ; The rating of a 7 is near FOTD average, but the calculation time ; of 12 minutes is too high a price to pay for a merely average ; image. Luckily, rescue from boredom may be found at the FOTD ; web sites. ; ; The official FOTD web site with the completed image is at: ; ; ; ; A more-detailed high-definition version of the image is posted ; at: ; ; ; ; The original FOTD web site may be accessed at: ; ; ; ; Another day of uneventful weather prevailed here at Fractal ; Central today, with mostly cloudy but dry skies, a little sun ; and an unremarkable temperature of 43F +6C. The fractal cats ; were satisfied that no draft fell on their shelf by the window. ; ; The humans, FL and I, had another in a long string of similar ; routine days, which are quiet enough, but leave little to ; report. The next FOTD, the last of the year, will be posted in ; 24 hours. I'm trying to think of a fractal theme for the month ; of January, so far without much luck. Until next FOTD, take ; care, and only 357 days remain before the world ends. ; ; ; Jim Muth ; jimmuth@earthlink.net ; ; ; START PARAMETER FILE======================================= DiminishingReturns { ; time=0:12:00.00 SF5 at 2000MHZ reset=2004 type=formula formulafile=slices.frm formulaname=MandelbrotMix5way center-mag=-0.764203\ 8506481917/+0.00004122174900695/1.586226e+012/1/\ -83.25/0 params=1/2/-1/3/1/4/-1/5/0/0 float=y passes=1 maxiter=3750 inside=0 logmap=460 periodicity=6 mathtolerance=0.05/1 colors=000mCcmCcmCcmCcmCcmCcmCcmCcmCcmCcmCcmCcmCcm\ CcmCcmCcmCcnCcnCcnCcnCcoCcoCcoCcpCcpBcpAcp9bq8`q7Y\ r6Vr2Tr3Vr3Wr4Xr4Yr5Zr5_r6`r6ar7br7cr7dr8er8fr9gr9\ irAjrAkrBlrBmqBnqCoqCpqDqqDrqEsqEtqFuqFvrDyqFwqHvq\ JuqLtqNspPqpRnpTkpUhpUeoUboU_lUXhUUcUUZUUUUUPUUNRU\ ONXOJ_PFbPBeQ7hQ3kO5nN7qL9qKBnIClHEjFGhEIfCKdBLeCK\ fCJfCIfCHfCGfCGfCFfCEfCDgCCgCCgCBgCAgC9gC8gC8gCAaB\ CXBERAGLAHGK4UH5QE5NC5K96G76D46A05A48AABAFEAKHAPKB\ UMCZODcQEcSFcRGcQHcPIcOJaNK_MLZLLSIOMGRGETACW4AY7C\ WAEVCGTFISHKQKMPNONPQMSSKUUJXWHZYGYVHXTHWRHbbAdb5g\ a6j`7l_8o_8qZ9tYAvYApV9jS9dP9ZM8TJ8NG8HD8VU9gj9tz9\ ryIpyRox_mxhlxpiahfG`WOgMWmCcs2kyGfuTareXodPecHXbA\ OaGK`MH_REZXBYa8Xc5Wc2Uc3Tc4Rc5Qc6Pc7Nc8Mc9Lc9FcBA\ cDGcELcFQcGVcH_cIdcJicKccSZc_UczZczcczhczmczqczmcz\ iczemzamzYmzUmzRmzZmzemzmnztzzmzzfzz`zzUzzNzzHzzAz\ z4zzLzzLzzVzzVzzUzzTzz_zz } frm:MandelbrotMix5way {; Jim Muth z=p5, c=pixel, a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=real(p3), h=imag(p3), j=real(p4), k=imag(p4): z=(a*(z^b))+(d*(z^f))+(g*(z^h))+(j*(z^k))+c, |z| <= 100 } ; END PARAMETER FILE========================================= ; ; ; ; _______________________________________________ ; Fractint mailing list ; Fractint@mailman.xmission.com ; http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint ;