FOTD -- August 17, 2007 (Rating 6) Fractal visionaries and enthusiasts: With today's image we once again enter hyperspace, (not that we ever left it). And just what is hyperspace? It is space of a dimension greater than 3. The most familiar hyperspace is of dimension 4, which just happens to be the space in which the Julibrot figure of the formula Z^2+C exists. Where is this hyperspace? In theory it is all around us, touching every point of our familiar 3-D space. We might say it is sideways to our insides. But what does a four-dimensional object look like? To begin, the question is a wrong one, for hyperspace is not a visual experience, it is a mathematical abstraction. But if we were able to visualize a 4-D object, we would see some kind of unimaginable monstrosity with a surface of three dimensions. On the surface of a four-dimensional hyperplanet, we could walk upright in three mutually perpendicu- lar directions, with the sky always remaining straight above us. Many other impossible things become possible in 4-D space. In my opinion, one of the most unusual is something called a double rotation. In 4-D space an object may be subject to two completely independent rotations at the same time. In 3-D space there is nothing corresponding to this curious motion. In 3-D space, a sphere, when illuminated by a point source at infinity, casts a shadow in the shape of a cylinder with a circular cross section. If this cylinder is sliced straight across, the section is a circle. If it is sliced at an angle, the section is an ellipse. If it is sliced along its length, the section is an infinitely long band of constant width with straight and parallel edges. Without going into hyper-detail, one of these straight edges is what appears on the southeast side of the Julia-like feature in today's image. Even though the image resembles a Julia image, it is actually sliced very close to the Oblate orientation. Today's image is a scene in the vicinity of the East Valley of the large midget on the main stem of the Mandelbrot set. I have rated it at a 6. I might have rated it a point higher, but I feel I have done a bit too many of this type of image over the past few years. I named it "Hyperspace Figures", because it is a scene in hyperspace. Unfortunately, with an maxiter of 1,500,000, the image is very slow -- 42 minutes on the P4-2000mhz and over 2 hours on the P200. To avoid fractal frustration, I recommend viewing the image on the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Perfection once again proved elusive here at Fractal Central on Thursday. There was simply too much heat (90F 32C) and humidity to call the day perfect. The fractal cats seemed unconcerned. They are still waiting for another bat to come out for them to chase. My day was slow but strangely unsettled. The next FOTD will appear in 24 hours, and that's pretty well settled. Until then, take care, and keep checking back for more fractals. Jim Muth jamth@mindspring.com jimmuth@aol.com START PARAMETER FILE======================================= Hyperspace_Figures { ; time=0:41:58.83-SF5 on P4-2000 reset=2004 type=formula formulafile=allinone.frm formulaname=SliceJulibrot2a passes=1 center-mag=+0.00328020489643859/+0.664356005504111\ 00/2545329/0.2808/-152.728346262165076/-72.8642931\ 627442465 params=2.75/90.11199999999999/0/90/\ -1.749/0/0/0/0/0 float=y maxiter=1500000 inside=0 logmap=46 periodicity=10 colors=000I0GF1HB2I73J54K25L06M07N08O09P0AQ0BR0CS0\ DT0EU0FV0GY0H_0Id0Le0Oi0Rl0Un8XqF_tKbxOezRhzUkzWnz\ Yqz_tzawzczzezzgzzizzmzzqzztzzwzzzzzmzzczzUtzFmzKc\ zQUmXUmdOziOzoOzvOztOzqOzoOvnOqnOllOgiOdgO_gOVeOQd\ gMagI_gF_g9Yg5Xg2Vg0Vg0Tg0Qe0Od0Oa0K_0HY0DX09T05g2\ 3g50g90lF0qI0xO0zT0zX0zQ0zK0zF0z90z30z00z00z02z07z\ 3Bz7HzDMzHQzMXzQazXizaez_az_YzYXzYTzYOvXMqXInXFiVB\ eV9aV5YT2VT0QT0MQ0IQ0FQ0HT0HT0HV0HV0HX0HX0HY0HY0H_\ 0H_0Ha0Ha0Hd0Hd0He0He0Ia2IY3KX5KT7MO9MMBOIDOFFQDHQ\ 9IT5KT3MV0OV0QX0TX0VY0XY0Y_0__0aa0da0_a2da5ga9iaDn\ aHoaKtaOxaTzaXza_zgdzagzXlzVozVtzVxzTzzQzzTzzVzzXz\ zYzz_zzazzdzzezzgzzizzlzznzznzznzznzznzznzznqzngzn\ YznOznFzn5zn0zn0zn0zn0zn0zq0zt0ox0dz0Tz5HzB5zI0zO0\ zX0za0zd0ze0te0lg0dg0Yi0Qi0Il0Dl05n00n00o00o00q00q\ 00a05M0M90Q70V70Y70_50d50g50i50n30q30v30x20z20z20z\ 20z00z00z00z0DqV7laQ7xM9v } frm:SliceJulibrot2a {; draws most slices of Julibrot pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1*0.0055555555555556), b=pi*imag(p1*0.0055555555555556), g=pi*real(p2*0.0055555555555556), d=pi*imag(p2*0.0055555555555556), ca=cos(a), cb=cos(b), sb=sin(b), cg=cos(g), sg=sin(g), cd=cos(d), sd=sin(d), p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd), q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd), r=u*sg+v*ca*sb*cg, s=v*sin(a), c=p+flip(q)+p3, z=r+flip(s)+p4: z=sqr(z)+c |z|<=real(p5+9) } END PARAMETER FILE=========================================