; Date: Wed, 10 Sep 2008 17:09:20 -0400 ; ; To: fractint@mailman.xmission.com ; ; From: Jim Muth ; Reply-To: Fractint and General Fractals Discussion ; ; ; Subject: [Fractint] FOTD 11-09-08 (Age of Iteration [6.5]) ; ; Id: <1.5.4.16.20080910171028.231f11e0@pop.mindspring.com> ; --------- ; ; FOTD -- September 11, 2008 (Rating 6.5) ; ; Fractal visionaries and enthusiasts: ; ; The complex logarithm is multi-valued. To take advantage of ; this, the MandelbrotBC (Branch-Cut) formulas are supposed to ; make visible all the fractals that result from the infinity of ; values that may be given to the 'k' part of the logarithm. This ; situation exists because the formula for the complex logarithm ; is: ; ; (1/2)ln(x^2+y^2)+i(atan(y/x)+2kpi) ; ; where the value of 'k' may be given any value from plus-to-minus ; infinity. ; ; I am not sure if any of the MandelbrotBC formulas calculate the ; different values of 'k' correctly, or even if a correct way ; actually exists. In fact, I'm not even sure of where to look to ; find a correct way. The correct way, if one exists, would ; probably fall in the field of 'complex variables' or something ; like that, but information on 'complex variables' is not easy to ; find. ; ; One thing I am sure of however is that the MandelbrotBC ; formulas, with the various functions applied, make some ; incredible fractals, so who cares if the images are ; mathematically correct. ; ; Today's image lies in the Z^(1-2/3)+C Mandeloid as it appears 83 ; levels up the logarithmic ladder, with the 'floor' function in ; effect. At this level, the Mandeloid somewhat resembles a ; distorted Mandelbrot set standing on its nose. The scene is ; located deep in the chaos on the western shoreline of the large ; period-2 bud, which is actually shaped rather like a nose. ; ; The minibrot at the center of today's image suffers from an ; insufficient maxiter, but its surrounding pattern is unusually ; well-defined for a minibrot in this range. ; ; All things considered, I rated the image at a 6-1/2, not quite ; as high as I would have liked, but high enough to make the ; calculation worth the 4-1/2 minutes required. ; ; The name "Age of Iteration" has no significant meaning. ; I simply pulled it out of the air. ; ; Finally, if all else fails, the completed GIF image is available ; on the FOTD web site at: ; ; ; ; In comparison to Tuesday, Wednesday's weather began peacefully ; here at Fractal Central, with singing birds and wisps of fog ; hanging over the river. When the fog cleared away, we were left ; with a day of mostly clear skies and a temperature of 72F 22C. ; The fractal cats purred approval, then meowed for breakfast. ; ; After such a hectic week, Wednesday, which actually was about ; average, seemed unusually slow. But psychological time ignored, ; the next FOTD will be posted in 24 hours. Until then, take ; care, and when it comes, recognize it for what it is. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE======================================= Age_of_Iteration { ; time=0:04:30.82-SF5 on P4-2000 reset=2004 type=formula formulafile=allinone.frm formulaname=MandelbrotBC3 function=floor passes=1 center-mag=-0.2811119334670882/-0.991563627943696/\ 5.35274e+011/1/10/0 params=1.6666666666667/0/83/0 float=y maxiter=2000 inside=0 logmap=428 periodicity=10 mathtolerance=0.05/1 colors=000LTdKUbIV_HWYFXVEYTDZQBZOAZL8ZJ7ZG5ZE4_B3\ `96a88`8B_8D_8F_8I_8KZ7NZ7PY7RY7UX7WW7ZV6`V6bV6eV6\ gV6iV6`PFTKOKFXCAe12p45n68l8AkADiCFhEIfGKeINcKQaMS\ `OVZQXYS_WUaVWdTXeTYfSHdR1bQ3_P5YP7WP9UOBSODQOFONG\ LNIJNKHMMFMODMQBLS9LS4JT7LUANVDPWFUWIZXLcYOhZQmZTr\ _Wv`Yza`zbczbfzchzdkzemzbozeqzgsziszlsznrzpqzsozum\ zwkzrizmgvifrdem`ehWecRdZNdUIdPEdKFcFFcAGc5Gc6GU7H\ c8HU9HcAIUBIUCJUDJUEJUFKUGKbHKbILbJLbKMbLMbMMbNNbO\ NbPNbPPcOQdNSeNTfMVgLWhLYiKZjK`kJalIcmIdnHeoGgpGhq\ FjrFksEmtDnuDpvCqwBsxBtyAuzAswCrtEqrGpoHomJnjLqkJm\ hMjePfbSc`V`YYXV`UTcQQfNNiMLmKLkJLiILhHLfGGdFGcEBa\ DB`CBZBBXABW90U8UzUUzzUzzUzzUzdKzaKzZJzWIzTIvPHrMG\ mJFhGFeDEdAEb70_90XA0UB0UD0UEQYFSXHUVIWUJYSL_QMaPN\ cNPeLQgKRiISkHTmHUnHVoHXpHYqHZrH_sH`tHauHbvHcwHdxH\ byIczJmzKqzKvzLzzMzzMzzNzzOzzzzzzzzzzzzzzzzzzzzzzz\ zzzzzzzzzzzzzzzzzzzzzzzzz } frm:MandelbrotBC3 { ; by several Fractint users e=p1, a=imag(p2)+100 p=real(p2)+PI q=2*PI*fn1(p/(2*PI)) r=real(p2)+PI-q Z=C=Pixel: Z=log(Z) IF(imag(Z)>r) Z=Z+flip(2*PI) ENDIF Z=exp(e*(Z+flip(q)))+C |Z|