; Date: Tue, 23 Jun 2009 17:48:31 -0400 ; From: Jim Muth ; Subject: [Fractint] FOTD 24-06-09 (Seahorse Valley-24 [Not Rated]) ; Id: <1.5.4.16.20090623175004.2ba7777e@pop.mindspring.com> ; --------- ; ; FOTD -- June 24, 2009 (No Rating) ; ; Fractal visionaries and enthusiasts: ; ; At first glance, today's image is a disappointing and not very ; inspiring debris-filled Julia set, but it is an image that is ; intended to be tweaked, so don't hesitate to start changing the ; real(p1) and real(p2) parameters. ; ; I have seen rumors circulating around the list lately that the ; fourth spatial dimension doesn't exist. Of course it doesn't ; exist. The orbits of the planets, among many other things, ; definitely obey the mathematically rigid inverse square rule of ; three-dimensional space. ; ; But does this prove anything. After all, we invented math ; ourselves and designed it specially to work with what we observe ; with our senses. ; ; Does the fourth dimension exist then? There is a big difference ; between not existing and being impossible. There is nothing at ; all impossible about four-dimentional space with time as the ; fifth dimension. There is no impenetrable barrier beyond the ; third dimension other than the inability of our minds to visual- ; ize spaces higher than the familiar everyday 3-D space. ; ; One- two- and three-dimensional objects can be mathematically ; modeled and manipulated by computers. With some additional ; complexity, four-dimensional objects can be modeled and manipula- ; ted by computers just as well. The problem here is that a three- ; dimensional screen surface would be needed to properly display ; the resulting images of 4-D objects, and this 3-D surface would ; need to be viewed from the fourth dimension. Since our minds ; have evolved to interpret our sensory input as a true image of a ; surrounding space having three spatial dimensions, we will never ; be able to visualize a fourth dimension, but this in no way pre- ; vents us from knowing what we would observe if we were able to ; do so. ; ; Two of the more curious possibilities in 4-D space are absolute ; perpendicularity and double rotation. Two planes are absolutely ; perpendicular when they intersect in a point, with every line in ; one plane perpendicular to every line in the other. Double ; rotation exists when a 4-D object is subject to two independent ; rotations at the same time. The points in the object move in a ; circular hyperhelix, somewhat like a slinky toy stretched out ; and curved into a circle with its ends connected. When the two ; rotations are equal, as with today's scene, all the points ; except the stationary point at the center move in circular ; arcs. Don't try to picture this motion. I've been trying for ; years with no success yet. ; ; In today's FOTD we have double rotated around the point at +003i ; in the north branch of Seahorse Valley, and stopped only 1/500th ; of one degree from the Julia direction, which is absolutely per- ; pendicular to the Mandelbrot. To see the 4-D double rotation in ; action, decrease real(p1) and real(p2) toward 0 and 0, keeping ; the two parameters equal. At (0,0) the Mandelbrot set will fill ; the screen, with Seahorse Valley at the center. For a quick but ; dizzying trip through four-dimensional space, check the slices ; at (90,90) (89.998,89.998) (89.99,89.99) (89.96,89.96) (88,88) ; (82,82) (75,75) (62,62) (50,50) (30,30) (10,10). Notice that ; the closer we come to the Mandelbrot orientation, the slower ; things go. The final (10,10) slice is barely distinguishable ; from the actual M-set. ; ; The scene may also be rotated from the Julia to the Mandelbrot ; by two simple 3-D type rotations. Gradually reduce real(p1) to ; zero, which reveals the Oblate aspect of Seahorse Valley. Then ; gradually reduce real(p2) to zero, which rotates the Oblate ; aspect back to the Mandelbrot. Doing it this way passes through ; some quite interesting slices, while introducing quite a bit of ; stretching distortion. Keeping the real(p1) and real(p2) para- ; meters equal will prevent this distortion. The passes=t option ; is needed to avoid major drop-outs. ; ; Since the image is actually one in a series of images, I gave it ; no rating. The calculation time of 2 minutes may be eliminated ; by visiting the FOTD web site at: ; ; ; ; where the finished image is posted. ; ; Picture perfect weather prevailed here at Fractal Central on ; Tuesday, with digitally-enhanced blue skies, puffy clouds, and a ; temperature of 81F 27C. Taking things for granted, the fractal ; cats slept through most of the day. My day was once again ; average and uneventful. The next FOTD trip to Seahorse Valley ; will be posted in 24 hours. Until then, take care, and today's ; series of images could make an interesting animation, but it ; would require choosing some carefully measured increments. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE======================================= Seahorse_Valley-24 { ; time=0:01:54.26-SF5 on P4-2000 reset=2004 type=formula formulafile=basic.frm formulaname=SliceJulibrot4 passes=t center-mag=0/0\ /0.7 params=89.998/0/89.998/0/-0.75/0.003/0/0/2/0 float=y maxiter=6400 inside=0 periodicity=10 colors=000QPGDC8MrzGdjBRV5DFTDsL9eE6S73E3gT2XL1ME0\ B7I_F9I7AG3581NPbRRlhJCM96TYzBWv8Og5GT28ExsjheZUSN\ FEBfHIWCDL89A44u0yRvByT8UU2FF13Nj1BNGwvpR4QD2HFteh\ 3SU2EF1wfLhWFULAFA5JsHC`B6I5W9_O6RG4I8294K02A09uV6\ aK3JAsYZ`MNIBBBvO7bG3J8a_UOyoJd9OHiC8NvmObXGJG8LfU\ np5PQ2XjKMVDBF6TACF5f72LW3_keid`CKU4AF2pMLQBAQy_Ji\ RDVI6F9UXwKMcABK49C268134rp5db3RQ2DD1J64C42621iu`f\ OiWIYLCNA6BjVkZN_NFOB7C5pIR2GI1A905RDxI8c94Kx81U40\ gEZDPT8GJ489zP9eG6L83p8`b6RQ4ID29hRrUI_F9IdYRtEqa9\ _J4IqUY_KMIABfaEWSALJ7A93e1WV0OL0GA08qSMREBxWnUGPZ\ k1HO0wEYc9MK4B7507pV1gR0MDEFDuMbTBJUGFKAAA55tlkikT\ NOEkYqOHRtZaeQSSHJE89VsnNeaFSP7ECWtoGSQqqY__MIIBHS\ S8EE1HK08As9pS4QVI7KC4A62sQK`HDI86wSGcIAK95gCOT8GE\ 48Xes`zzXfrTNjQ4c9XZZWLxV7qcOjldctukYnrChkP`e`UZlM\ TxFfFFWVVMjjeaTSPJFPMEC9V2uK1aA0J3Xk2MW1BGmfuPLTKe\ BFV8AL55A2DJw9Eh69U34FbaP } frm:SliceJulibrot4 {; 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=z^(p5)+c |z|<=9 } ; END PARAMETER FILE========================================= ; ; ; ; _______________________________________________ ; Fractint mailing list ; Fractint@mailman.xmission.com ; http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint