FOTD -- July 24, 2003 (Rating ?) Fractal visionaries and enthusiasts: How does one answer the question, "Where is the fourth spatial dimension?" This, in my opinion, is one of the most puzzling questions that has ever been asked. We know where it would be if it existed, but it cannot exist. There is no room for it. Three-dimensional space is all there is. Yet it does exist. Using numbers, we can take apart and examine 4-D hyperspace as rigorously as we examine three-dimensional space. I suspect that the inability to perceive 4-D space is a limitation of our awareness, and that the fourth dimension is there all around us, if only we had eyes to see. How to open our eyes and adjust our numbers is anyone's guess. Today's image is a scene in the Z^2.003+C Julibrot. Julibrots are four-dimensional things that, sliced in one direction give Julia sets and, sliced in another direction, give perturbed Mandelbrot sets. Of course, being hypersolid, the 4-D Julibrot can also be sliced in oblique directions between the Julia and Mandelbrot directions. Today's scene is sliced in a direction only one degree from the Julia direction, but that single degree makes a world of differ- ence. (To see the pure Julia set, change imag(p1) from 89 to 90.) The Z^2.003+C Mandeloid is shaped almost exactly like the classic Mandelbrot set, whose formula is Z^2+C, the major change being along the negative stem, which has split into an infinity of curving, infinitely-divided filaments. The large midget at -1.75 has also started to split, and today's scene is located in the East Valley of this midget, which has broadened into a much- divided wedge-shaped thing. I have given the image the name "Four-Dimensional Scene", which is a fair description. But I am undecided on the worth of the picture, which comprises all there is of the fractal. Sometimes I think it is worth an 8, then I view it again and see an image worth only a 4. The image will likely go unrated. The modest render time of X minutes can be avoided by download- ing the finished GIF image from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Though it was quite humid, the temperature rose no higher than 84F 29C and no rain fell here at Fractal Central on Wednesday, giving the dynamic fractal cats a most pleasurable day. The good weather also helped conserve the stressed tuna population, since no treat was necessary. Today is promising to be a slightly less humid repeat. I expect a repeat from the fractal cats. More good news is that the work shelf is filling up. And the best news of all is that I shall return in 24 hours with another glorious fractal. Until then, take care, and when I started the FOTD on the Fractal-Art list in April of 1997, I expected to keep it going maybe a week or two. Jim Muth jamth@mindspring.com jimmuth@aol.com START 20.0 PAR-FORMULA FILE================================ 4-DimensionalScene { ; time-0:06:55.34--SF5 on a p200 reset=2002 type=formula formulafile=allinone.frm formulaname=multirot-XY-ZW-new function=ident/flip center-mag=-0.0156752/-0.0738926/7.021696/1/-90/3.\ 88578058618804789e-016 params=90/89/2.003/0/0.0001\ 9/0.07388/-1.7435/0 float=y maxiter=8000 inside=0 passes=1 logmap=yes periodicity=10 colors=0000000000000000200400600830A60C90EC0GF0II4\ KL6MOAOREQUGSXJU_LWbPYeT_hVakZcnbeqdgtgiwhjzikzjmz\ kmzlmzmmznmzomzpmzqozrqzssztuvuvnvogugTkbEbV0XR0VP\ 0VN0TN0TL0RJ0RJ0PH0PG0PG0RL2RP8RVERZJRbPRgVRk`RqdU\ uicxomzurzzzzzvzzrzzmzzhzzfzoosbxiPz`CzT0ob4Rk82uC\ 0zE0zG0xH0vJ0sL0qN0oP0mR4iR6gTAeVEdXG`ZJZ`NXbbGRPV\ bCim0xv0zz0zz0zz0zz0oz0dz0Ru0Eq00m00i40g`6ez0mzEeq\ R`ddVPqNCzH0z00zC8zLJzXTzebzoezigxeigbkRZk2NmAVoH`\ qPesXkudqvkvxszzzzzzzxuzvqzukzsezqbzoXxoRvmNukHuiC\ sg8qe2od0md0md0qd8udGxdNzdVzdbz``zXZzTXxPXuLVqJToG\ RkCRg8Pe4Nb0Na2N`H40RE0`N6iXCseJzoPzxXdbzdbzdbzdbz\ dbzdbzdbzdczcczcczcczcczcczcczcczcczcczcczcczcczcc\ zcczcczhc0mc0rc0vc0zV0zT2zP8zLEzJHzGNcczz8Zz6dz2iz\ 0oz0s20z40z60z60z86zAAzAEzCJzENvERuGZsGdqHioHomJuk\ JziLzgLzeNzdNzbPz`Pz`NzdNzeNzgNziNzkNzoNzqNzsNzuNz\ vGsqAmm4gi0be0X`0RX0LT0GP } 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 20.0 PAR-FORMULA FILE==================================