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: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> 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=========================================