FOTD -- February 08, 2003 (Rating 5) Fractal visionaries and enthusiasts: Today's image is yet another of my evaporated ones, created this time by subtracting a portion of Z^(-7.17) from an equal portion of Z^(-1.717) before adding 1/C. The scene was then 'evaporated' by raising the bailout radius to one-googol and rendering the resulting blank screen with the inside option set to < bof61 >. The real(p3) parameter is so awkward because its value was determined using the evolver feature. I use this feature in my fractal travels more often than is apparent, but in most cases I round off the value to something more aesthetic. In today's case, I was searching for the point where the entire fractal expands to infinity and returns from the opposite direction, and once I had found this point with the evolver, I did not bother with the rounding. Infinity is a curious concept, in some ways quite counter- intuitive. The arithmetic of infinity is very simple -- regardless of what is done to it, infinity remains unchanged. When geometry is expanded to infinity, some even more surprising things happen. An infinitely long line has a single point at infinity. It does not have the two points that intuition would demand. This can be better understood by thinking of two infinitely long parallel lines lying in a flat Euclidian plane, which can be considered either as never intersecting or as intersecting at infinity. (This is two different ways of saying the same thing.) Then imagine a finite point where the two lines intersect. It is only a single point, but as the lines are rotated, this point of intersection moves off in one direction, reaches infinity, then returns from the opposite direction. The point at infinity of the line lies in both directions. At no time during the rotation does a second point of intersection magically appear. The same situation prevails with an infinite plane, which has a straight line at infinity. Points that go off in any one direction and pass beyond infinity return from the opposite direction. (Actually, we're getting into projective geometry, and such a plane is actually a projective plane, but that's a different topic.) I named today's image "Lava-Ring". If enough imagination is used, the boiling pit of molten lava can actually be seen. But since molten lava is not my favorite substance, I could rate the resulting image no higher than a 5. The 24-minute render time also holds down the rating, but this wait can be avoided by downloading the completed GIF image from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The snow ended by noon on Friday, but the damage had already been done. With 8 inches 20cm of ice-crystals covering the yard, and a temperature around freezing, the fractal cats were confined to the near side of the porch, under the cover, where only a dusting of snow had settled. With such cramped quarters and cold paws, they stayed outside only a few minutes. The snow is still there this morning, so I suppose it will be another bad day for the aging duo. For me, it will be a passable day, but only if the small amount of work goes well. But regardless of how the work goes, I shall return in 24 hours with the next in the near-eternal series of fractals. Until then, take care, and be kind to web-footed fractals. Jim Muth jamth@mindspring.com jimmuth@aol.com START 20.0 PAR-FORMULA FILE================================ Lava-Ring { ; time=0:24:05.69--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+0.79396508590478570/+0.640345353073420\ 90/5.261436e+008/1/-15/1.94124406955459294e-006 params=-1/-1.717/1/-7.17/-5.453445844904936/1e+100 float=y maxiter=1200 inside=bof61 logmap=122 periodicity=10 colors=00000X34A3JD3Fc3AQ36f9A`DDXIGSMJOSMJXPF`S9d\ V4jY0o`0sc0xf0za0s_0vX0oS0lP0vM0YI0lF0PC0V70_40P10\ V00T00Q01P01Q09Q3FQALSIQSPXSYcTfiTooTxuTzzSzxSvuSr\ rSmoSlmXgi_cfc_afV_jQVmOSpJOuFLxAGz6Dz19z06z7CzGGz\ PMzXQydVxm`vvduzixufylczd`zXYzOVzGTxJPoLMgMJ_OGQPD\ JQAAS73T44V76YA6_C7`F7cG9dJ9fLGdTOd`Vdgadogdvfcsdf\ pcimall`ri_vfYmcQzaJraJzaJzaJzaJmcJzcJzcJzcJzcJzcJ\ mcJmcJzcJmcJzcJzcJzcJzcJzcJzcJmcImdIzdImdIzdIzdIzd\ ImdIzfIzfIzfIzfImfIzfIzfIzfIygIygIygIxgIxgIxgIvgIv\ gIviIuiIuiIuiIsiIsiIsiIriGrjGrjGpjGpjGpjGojGojGolG\ mlGmlGmlGllGllGclGjlGjmGjmGimGimGimGgmGgmGgmGfoGfo\ GfoGdoGdoGdoGcoGcoFcpFapFapFapF`pF`pF`pF_rF_rF_rFY\ rFYrFYrFXrFXrFXsFVsFVsFVsFTsFTsFTsFSsFSuFSuFQuFQuF\ QuFPuFQuFTsDXoC_gAa`9dT7gM0jF0iC0fG0dL0cP1`V3__4Yc\ 7VgFTlMTpTTr`TsgTuoTusTuxTszVjzXazYTzYMz_Dz`4z`0z`\ 0y`0CmYAlX7jT4iS3gQ3aT3YV } 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 20.0 PAR-FORMULA FILE==================================