FOTD -- September 03, 2006 (Rating 6) Fractal visionaries and enthusiasts: Today's colorful image is another unfamiliar view of Seahorse Valley, which appears as the straight-edged bluish horizontal band extending nearly the full width of the frame. If the Mandelbrot set were expanded to include the part of the Julibrot extending behind and in front of the screen along the real-Z axis, today's view would be looking down from above. (Yester- day's view was looking from the left.) Being a four-dimensional object, Seahorse Valley has more different views than can be imagined. And even when all the views have been studied, the viewer will have no better idea of the actual overall shape of the valley than he did at the start. Four-dimensional objects are not mere everyday three-dimensional objects with extra complex shapes. They are totally and absolutely different. The most basic feature of 4-D things is that their surfaces are three-dimensional, with 3-D volume. When looking at the surface of an opaque 4-D object, a 4-D man would see a 3-D volume projected onto his 3-D retinas. When observing a 4-D hypersphere, he would see what we would consider every point on the inside of a normal 3-D sphere. Rotation in four dimensions is about a plane as axis, and a 4-D object may be subject to two totally independent rotations simultaneously, with the gyroscopic effect of one rotation having no effect on the other rotation. All this is entirely unimaginable, but may easily be confirmed with mathematics. I named today's image "Seahorse Valley-02". I rated it at a 6. The render time of 1-1/2 minutes is brief enough to cause no one impatience. But the image has been posted anyway to the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Heavy rain fell all day here at New Fractal Central on Saturday. The fractal cats watched with some interest for a few minutes, then got about their normal cat business. My day was accept- able. The next FOTD will appear in 24 hours. Until then, take care, and step likely. Jim Muth jamth@mindspring.com jimmuth@aol.com START PARAMETER FILE======================================= Seahorse_Valley-02 { ; time=0:01:36.83--SF5 on a P200 reset=2004 type=formula formulafile=allinone.frm formulaname=multirot-XY-ZW-new function=flip/ident center-mag=-0.0375587/0/13.33333/0.06756/90/6.3997\ 41848e-014 params=90/0/2/0/0/0/-0.75/0.1 float=y maxiter=16000 inside=0 logmap=yes periodicity=10 colors=000qcUoXVmQWkJXiCYh6YlAeoDlrGsuJzpLtlNogPic\ QdZSZVUUQWOMXJHZDD`89a3Cc6Fe8HgAKiCNkEPmGSoIUqKXsM\ _uOawQdySfzUhuVjqWkmXmiYneYpaZqX_sT`tP`vLawHbyDcz9\ cyHcyOcyWcybcyjcyqcyxcsu`msYgqVanSWlPQjMKgJEeG8cD2\ aAhANkPancolbmkakj`jh_hgZgfYeeXdcWbbVaaU_`UZZVYYVX\ XVWWWWUWVTWUSXTRXTPXSOYRNYQMYQRVWWT``QfeOkjMpfPobR\ o_ToWWoSYnP_nLbnIdnEfmAim7km3mm0om8hkGbjOWiWQgcJmk\ DrbHvVKzNNz`NzOAzkdziezhezffzefzdfzbgzagz`gzZhzYhz\ XhzXgzYgzYfzYfzZezZezZdz_dz_cz_cz`bz`bz`az`azWYzSU\ zOQzKNzGJzCFz8Bz48z7AzABzDCzGDzJEzLFzOGzRHzUIzXJzZ\ KzaSzd_zggzklzoqzsuzrvzUwz6xzPyzgzzizzjzzkzzlzznzz\ ozzpzzqzzrzzdzzfzzgzzizzjzzlzzmzzizzfzzbzz_zzZzzZz\ zZzzYzzYzzYzzYzzXzzXzzXjzXkzWlzWzzWzzWzzXzzYzzZzz_\ zz`zzazzbzzczzdzzezzfzzgzzhzzizzizzjzzkzzlzzmzzzzz\ lzzVzzDzzHzzKzzNzzQzzTzzVzzXzzZzz`zzbzzgzzlzzqzzvz\ zszzqzzozzmzzkzzizzgzzdzz } frm:multirot-XY-ZW-new {; draws 6 planes and rotations ;when fn1-2=i,f, then p1 0,0=M, 0,90=O, 90,0=E, 90,90=J ;when fn1-2=f,i, then p1 0,0=M, 0,90=R, 90,0=P, 90,90=J a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994, z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p3, c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p4: z=z^(p2)+c, |z| <= 36 } END PARAMETER FILE=========================================