; Date: Thu, 16 Aug 2007 22:44:26 -0400 ; ; To: fractint@mailman.xmission.com ; cc: philofractal@lists.fractalus.com ; ; From: Jim Muth ; Reply-To: Fractint and General Fractals Discussion ; ; ; Subject: [Fractint] FOTD 17-08-07 (Hyperspace Figures [6]) ; ; Id: <1.5.4.16.20070816225013.2b271b0c@pop.mindspring.com> ; --------- ; ; 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: ; ; ; ; 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========================================= ; ; ; _______________________________________________ ; Fractint mailing list ; Fractint@mailman.xmission.com ; http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint