; Date: Mon, 30 May 2011 09:00:07 -0400 ; From: Jim Muth ; Subject: [Fractint] FOTD 30-05-11 (Uncharted Territory [7]) ; Id: <1.5.4.16.20110530090010.151fe1fa@pop.earthlink.net> ; --------- ; ; FOTD -- May 30, 2011 (Rating 7) ; ; Fractal visionaries and enthusiasts: ; ; Today we enter "Uncharted Territory", which is what I named the ; image. Never before have I checked a fractal with an exponent ; as close to unity as 1.009, making today's image uncharted for ; sure. ; ; The graph of Z^1.009 is virtually a straight line, so how is it ; possible that a fractal as rich and varied as today's could come ; about? The answer is the multi-valued nature of the complex ; logarithm. ; ; The complex log is multi-valued. This means that it has many, ; in fact an infinity of values, and all of them are 'correct'. ; When applied to fractals with fractional exponents, these ; different values create different fractals, all of which are ; correct. But curiously, when applied to exponents very close to ; unity, as is today's, these different values create some ; unexpectedly rich images, in effect turning a 'sow's ear' into a ; 'silk purse'. ; ; Today's image lies in one of these 'silk-purse' fractals. The ; parent fractal is an apparently featureless ellipse, but a close ; examination reveals a tiny bit of chaos on the southeast shore ; line. Today's image lies in this chaos. ; ; I used the built-in black-and-white color palette to create the ; 3-D effect. Other palettes might have worked better, but an ; attack of terminal laziness prevented me from trying to find ; one. This laziness held the rating to an everyday 7. ; ; The finished image may be seen on the FOTD web site at: ; ; ; ; and in high definition at: ; ; ; ; The original FOTD web site may be accessed at: ; ; ; ; Typical midsummer weather made Sunday a typical midsummer day ; here at Fractal Central. The fractal cats spent the day doing ; what cats do when the temperature reaches 88F 31C -- they ; stretched out and went to sleep. ; ; The humans took it just as easy, though we did stay awake at ; least most of the time. The next FOTD is due to be posted in ; 24 hours. Until then, take care, and don't ask me what I think ; is wrong with the new world being brought about by the recent ; advances in technology. I just might start to answer. ; ; ; Jim Muth ; jimmuth@earthlink.net ; ; ; START PARAMETER FILE======================================= UnchartedTerritory { ; time=0:12:09.74-SF5 on P4-2000 reset=2004 type=formula formulafile=basicer.frm formulaname=MandelbrotBC3 function=recip passes=1 center-mag=-0.132209132/-1.699446787/4304/1/97.5/0 params=1.009/0/4.75/1500 float=y maxiter=4200 colors=000fffgggggghhhhhhiiiiiijjjjjjkkkkkkllllllm\ mmmmmnnnnnnooooooppppppqqqqqqrrrrrrssssssttttttuuu\ uuuvvvvvvwwwwwwxxxxxxyyyyyyzzzyyyxxxxxxwwwwwwvvvvv\ vuuuuuuttttttssssssrrrrrrqqqqqqppppppoooooonnnnnnm\ mmmmmllllllkkkkkkjjjjjjiiiiiihhhhhhggggggffffffeee\ eeeddddddccccccbbbbbbaaaaaa``````______ZZZZZZYYYYY\ YXXXXXXWWWWWWVVVVVVVVVUUUUUUTTTTTTSSSSSSRRRRRRQQQQ\ QQPPPPPPOOOOOONNNNNNMMMMMMLLLLLLKKKKKKJJJJJJIIIIII\ HHHHHHGGGGGGFFFFFFEEEEEEDDDDDDCCCCCCBBBBBBAAAAAA99\ 99998888887777776666665555554444443333332222221111\ 11000000000000000111111222222333333444444555555666\ 666777777888888999999AAAAAABBBBBBCCCCCCDDDDDDEEEEE\ EFFFFFFGGGGGGHHHHHHIIIIIIJJJJJJKKKKKKLLLLLLMMMMMMN\ NNNNNOOOOOOPPPPPPQQQQQQRRRRRRSSSSSSTTTTTTUUUUUUVVV\ VVVVVVWWWWWWXXXXXXYYYYYYZZZZZZ______``````aaaaaabb\ bbbbccccccddddddeeeeeefff } 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|