; Date: Fri, 15 May 2009 22:52:05 -0400 ; ; To: fractint@mailman.xmission.com ; ; From: Jim Muth ; Reply-To: Fractint and General Fractals Discussion ; ; ; Subject: [Fractint] FOTD 16-05-09 (Unity and a Half [6.5]) ; ; Id: <1.5.4.16.20090515225304.2a2728c6@pop.mindspring.com> ; --------- ; ; FOTD -- May 16, 2009 (Rating 6.5) ; ; Fractal visionaries and enthusiasts: ; ; The complex logarithm is multi-valued, so how then do we know ; which value is the correct one? The answer is that they are all ; 'correct'. The square roots of 4 are plus-2 and minus-2, and ; they are both 'correct'. We use the root that works best in the ; situation at hand, which is almost always the plus-2 root. The ; same is true with the complex logarithms, but the other values ; are always there, and they also make fractals. ; ; The MandelbrotBC3 formula makes use of these other values of the ; complex logarithm. Each value draws a different fractal, even ; when the exponent of Z remains unchanged. In today's image we ; have gone 30 levels up the logarithmic ladder to a level where ; the Z^(1.5)+C parent Mandeloid is totally changed. Today's ; image is located in the 'Seahorse Valley' of its parent. ; ; Since most of the effect is a result of the coloring, which I ; put at least 10 minutes work into, I have added the usual 1/2 ; point bonus for the coloring, bringing the overall rating to a ; less-than-inspiring 6-1/2. The name "Unity and a Half" refers ; to the exponent of Z. ; ; The calculation time of a shockingly fast 44 seconds will bring ; no discomfort to even the most fastidious. The trip out to the ; FOTD web site at: ; ; ; ; to see the finished image there will be equally enjoyable. ; ; The showers moved out of the Fractal Central area early Thursday ; night, leading to a foggy start for Friday. By 10am the fog ; cleared out however, and the rest of the day was near perfect, ; with blue skies, puffy clouds, and a temperature of 77F 25C. ; The fractal cats enjoyed the weather conditions, but they were ; not happy when a lovesick tomcat wandered by and sprayed the ; wall right beneath their favorite window. A quick squirt from ; the hose removed the offensive odor and restored the cats' good ; moods. My day was busier than expected, in fact it was barely ; under control. ; ; Unless FL gets an antiquing urge, the next FOTD will be posted ; in 24 hours. Until then, take care, and be conservatively ; liberal. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE======================================= Unity_and_a_Half { ; time=0:00:44.80-SF5 on P4-2000 reset=2004 type=formula formulafile=basic.frm formulaname=MandelbrotBC3 function=ident float=y center-mag=+0.1641879378341295/+0.775442975243285/\ 200823.9/1/-5/0 params=1.5/0/30/0 maxiter=1800 inside=0 logmap=105 periodicity=10 colors=0000HU000K0U0000KU000N0U0000HU000K0Ul0fi1cf\ 2`c3Y`4VY5SV6QS7NP8KM9HJAEWU2UR4SP5QN6OL7NJ8LH9JFA\ HDBzm8wi9te9qb9n_AkXAgUAcRB_NBWLBSICOGCKDCoMMfJJYG\ HPDEZOFYNEXMEWMEVLEVLEUKETKESJERJDRIDQRzPHzOGtNGmN\ KcMFcLEZKEUJDPJDKICFHCCGBCcuAaqB_mBYiBWeBUaBSYBQUC\ OQCMMCKICIECz5Az6Bz6Bz6Bv6Br7Bm7Bh7Bc7BZ8BU8BP8BK8\ BJ9CI9CI9CI9CIACHACHACHACGBCGBCGBCf5Ga6FY7ET8EP9DK\ ACH09G2AG3AG5BG6BG7BG9CGACPu`OrZOpYNnXNlWMiVMgTMeS\ LcRL`QKZPKXNJVMJSLJQKIOJIMHHJGHHFGFEGDDF5mG6lH7lJ9\ lKAkMBkNDkOEkQFjRHjTIjUJiVLiXMiYNh_Ph`QhaRhm`kxjnv\ hotgosfoqeopdoncombokaoj`oh_ogZoeYocXpbWp`Vp_UpYTp\ XSpVRpUQpSPpROpPNpOMpQOnRQlSSjTUhUWfVYdW_bXa`Zc__e\ Y`gWaiUbkScmQdoOeqMfrLeqKeqKepJdpJdpJdoIdoIcoIcnHc\ nHcmGbmGbmGblFblFalFakEakEakEdzMezOezPezQfzRfzSfzT\ gzUgzVgzXhzYhzZhz_iz`izaizbizc`zYSzSJzM0z7LzuJzeHz\ RZzEXzDWzDVzDTzDSzDRzDQzD } 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|