; Date: Sun, 29 May 2005 08:54:19 -0400 ; From: Jim Muth ; Subject: [Fractint] FOTD 29-05-05 (Cocoons [7]) ; Id: <1.5.4.16.20050529085551.2aa7f5da@pop.mindspring.com> ; --------- ; ; FOTD -- May 29, 2005 (Rating 7) ; ; Fractal visionaries and enthusiasts: ; ; In a recent letter to the Fractint list, Vortex S. asked what is ; the hyperspiral I keep talking about. The hyperspiral is simply ; the set of all the fractals that can be created with the same ; fractional exponent of Z. ; ; Many fractals are possible from the same fractional exponent ; because the complex logarithm function is multi-valued. The ; formula for the complex natural logarithm is: ; ; (1/2)ln(x^2+y^2)+i(atan(y/x)+2kPI) ; ; The hyperspiral arises because k can have any value from zero to ; +- infinity, the different values producing different fractals. ; I like to picture this infinity of fractals as a single object ; -- a 5-dimensional spiral, each level of which may be explored ; by entering the number of that level as the real(p2) parameter ; of the MandelbrotBC2 formula. ; ; But none of this has anything to do with today's fractal, which ; was calculated by the M-Mix4 formula, when it combined various ; portions of Z^(-1.5) and Z^(-4.5), then added (1/C). I use ; (1/C) instead of plain C because fractals with negative ; exponents of Z are inside out and (1/C) turns these fractals ; inside out again, which results in a rightside-out fractal. ; ; In addition to being turned inside-out twice, today's fractal ; has been partially evaporated by raising the escape radius to ; 2*10^14. In fractals created with negative exponents of Z, the ; points never reach infinity, but they can travel quite far in ; their limited range. As the bailout radius is increased, more ; points are trapped, until eventually every point is trapped and ; the screen is filled with nothing but a flat color consisting ; totally of trapped 'inside' points. An active inside fill such ; as 'bof61' brings this flatness to life. ; ; In today's image a few points still manage to wander beyond the ; escape radius, and appear as 'outside' stuff. These points ; comprise the dark, banded cocoon shapes that give the image its ; name. Unfortunately, all the cocoons have hatched and the ; butterflies have flown away. ; ; The rating of a 7 seems appropriate for today's image. The ; render time of under 3 minutes is not bad either. And those who ; do not render may find the finished image posted to the FOTD web ; site at: ; ; ; ; Saturday started out fair enough here at Fractal Central, with ; lots of sun and a temperature of 79F 26C. But soon after the ; cats started their daily afternoon outing, a rain squall moved ; in and sent them scurrying for cover under the porch. The sun ; returned 30 minutes later, but by then the temperature was down ; to 59F 15C, the wind was blowing and the grass was soaked -- ; unfit conditions for the sensitive fractal cats, who came inside ; to sulk. ; ; Today is starting like Saturday, but no rain is in the forecast. ; Luckily, the duo has a short memory. For me, it looks like I ; will be unable to talk my way out of a trip with fractal lady to ; one of the local antique emporiums to look at the stuff someone ; else got rid of. The next FOTD will appear almost by magic in ; 24 hours. Until then, take care, and the trouble with fractals ; is that one can never be sure of what a fractal will do next. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE======================================= Cocoons { ; time=0:02:46.42--SF5 on a P200 reset=2004 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=-28.44386721751713000/-11.7012260054644\ 8000/9.165932e+009/1/-70/0.166537863130951536 params=-5.46/-1.5/-2.67/-4.5/0/2e+014 maxiter=2500 float=y inside=bof61 outside=tdis periodicity=10 colors=222xS2xL2xG2xA2x72x72x72x72x72x72x72x72x72v\ 72k72`72S72J72B723722Q22O32L72KB2KF2IL2GP2EU2EZ2g2\ 5m7DsGLxPUz`bznlzzXzvLzvBvs2ps2kq2gq2sxLzxhzxxzxxz\ xxzxxsxxgxxaxxSxvLxvExs8xs7xq7xn7xn7xk7xk7xf7x`7xX\ 7nU7bP7UL7LJ7DF85BC29I25L22S22X22Z22Z22Z22a22a22a2\ 2Q52GJ77ZN7qL7hJAbGGZFKUDQPBXL9aG7gD5p93v52z22z22z\ 22z22m22a22Q22E227227227227227527927F27J27N27U27Z2\ 8b2Ak2Cq2Gv2Ix2Kx2Lx2Ix2Gh2EU2AG287272272272272272\ 272272zz2zz2zz2z02z72z7227227527927D27G27L27N27J27\ F27D2792752732722722722722722722A22z22z22s22g92aG2\ XJ2SJ2QL2LL2KN2GN2EP2AP27S77SD7UJ7UP7XX7X27k27b27Z\ 27U27P27L27G27D2772732722722722722722822S22pJLzfBz\ P2zB2z22z22z22z22z22z22x22m22e227327727B27F27G37LB\ 7PJ7SSCX`I`kObvUhxanxeqxcsxcsxavxavxZxxZxxXhxXUxUG\ xU5xZ2vc2qe2kk2fp2`s2Na2DL528B27J27P27N27N27L27L57\ JB7JF7GJ7GP7GU7F`7Fxa2xa2 } frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } ; END PARAMETER FILE========================================= ; ;