FOTD -- December 05, 2002 (Rating 4) Fractal visionaries and enthusiasts: Today's image resembles a disembodied grin -- a rictus someone who likes to use fancy words might call it. I simply call it "The Laugh", a name which properly describes the effect the picture creates. Turned upside-down, the image might be called "The Cry", but let's keep things as pleasant as possible. I'll try to be as pleasant as possible about the image, for which I make no excuses and which I can rate no higher than a somewhat inflated 4. But even a 4-rated image is worth the effort when the time required to render it is only 44 seconds. The hole at the center, shaped like a gaping grin, is actually an odd slice of a midget located near the tip of the northern- most filament of the M-set. At the hole's upper shoreline a bit of one of those ubiquitous straight bridges appears. (In the M-set a slice of this bridge appears as a valley that forms a lip of a bud.) The grin is actually part of a four-dimensional hyper-object. In a space of four dimensions, things that seem absurd in three dimensions become everyday facts. Consider rotation. In a space of zero or one dimension, rotation is impossible. In two- dimensional space, rotation takes place around an axis-point, while in three-dimensional space, rotation takes place around an axis-line. The general rule is that in n-dimensional space, a simple rotation takes place around an n-minus-2 axis. But a simple rotation requires only two dimensions. The remain- ing dimensions simply remain fixed while turning in place. In four-dimensional space, this leaves the two-dimensional axis of rotation motionless, its points rotating on themselves. But even while serving as an axis of rotation, this plane may itself rotate. The resulting motion, which is impossible in three dimensions, is known as a double rotation. When subject to a double rotation, every point of a four-dimensional hyper-object except the point where the two axis-planes intersect is in motion. If the two rotations are equal, all points move in a circle. If the rotations are not equal, the points trace out a surface of double revolution, which is shaped like a four-dimen- sional doughnut. This double-rotation motion can perhaps be more fully realized by considering the Mandelbrot set and Julibrot. With the M-set on the screen, which of the Z dimensions is perpendicular to it, real(z) or imag(z)? Of course, they both are. In fact, every line in the Z, or Julia, plane is perpendicular to the M-set. Keeping the M-set undisturbed on the screen, the Julibrot can be rotated around the M-set, so that every line of the Z plane in turn lies in our 3-space and at a right angle to the M-set. Looking at only the screen, one would never know that the Julibrot was rotating. Now, with the Julibrot still rotating, if the M-set on the screen were to begin rotating on the screen around its origin, carrying the rest of the Julibrot with it, the Julibrot would be in a state of double rotation. Do not try to mentally trace out the path the points would take, it is impossible. It is far easier to imagine the fun of rendering today's super- fast image from the attached parameter file. And those not particularly fond of rendering can download the completed image from one of the FOTD web sites at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> and at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> I'll have much more popularized 4-D curiosities in the weeks to come. Wednesday was hazy and cold here at Fractal Central, but the day was made interesting by a late morning apparition of two brilliantly colored sun dogs. The temperature of 26F -3C was far too chilly for the ears of the intrepid duo, who also dislike dogs of any variety. They wisely decided to pass the day stretched by the warmest radiator they could find. Gazing from the F.C. window this morning, I see snow falling at a moderate rate with 6in 15cm already on the ground and the snow expected to continue until evening. The fractal cats will need comforting. Regardless of the weather, the human's work still needs to be done however. And the best way to finish work is to start it. So until tomorrow, take care, and be of good cheer. Jim Muth jamth@mindspring.com jimmuth@aol.com START 20.0 PAR-FORMULA FILE================================ The_Laugh { ; time=0:00:44.70--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=multirot-XZ-YW-new passes=1 center-mag=-0.00000000000001003/-0.000000000000001\ 47/3.099006e+012/1.4001/0.404762088721556723/-20.7\ 50360169932005 params=-73.9/161.5/2/0/-0.207107799\ 5585138/1.122756154909389/-0.2071077995585138/1.12\ 2756154909389 float=y maxiter=12000 inside=0 logmap=79 periodicity=10 colors=000PKAQLARMASNATOAUPAVQAWRAXSAYTAZUA_VA`W6a\ X5bY5bZ5a_4``4`a4_b3Zc3Zd3Ye2Xf2Xg2Wh9ViFUiLTiRTiX\ SibRihQinPitPizSkyUmyWoyYqy`sybuydwyfxyetwdpvcmtbi\ safq`bp__nZWmYTjXTkXWmPZoFaq0ds1cq1bp2ao2`n3_m3_l4\ Zk4Yj5Xi5Wh5Wg4Yh4_i4ai4bj4dk4fk4hl4il9gjDehIcfMbd\ Q`bVZ`ZXZbWXZSRVOMRLHNHCKE7RCBXAEc8Ii7LQBeigqgclf_\ hdXdcT`aPX`MT_IOYEKXBGV7CU38T04Q4GO8RLBbJFmHIxaTpu\ bin_hhXgaVgWSfPPeJNeCKd6Id7Lg8Oj9QlAToBWrCYtD`wDby\ F_wHYuIVsKTrLQpNOnOLlQJkRGiTEgUCfch5qfDhdK_cSRaZI_\ fAZmKTnTNnbInkCnt7nkHkcQiWZfOhdGqa8z_7mZ6gz5Qz4Ez5\ Fz6Fz7Fz7Fz8Fz9FzAFzAFzBFzCFzDFzDFzDDzDCzGDzGDzFGz\ EJzCMzBPzASz8Vz7Yz6`z4cz3fz2iz5dz8_zAVzDQzFLzLRzQW\ zVaz_fzdkzcizbhzbfzaez`cz`bz_`z__zcbzfezihzlkzonzr\ qzutzxwzurzrnzojzleziazfYzcUz`PzYLzVHzSDz_KzgRznYz\ m_zl`zkazjbziczhdzgezfgzehzdizcjzbkzalzamz_lzZlzYl\ zWlzVlzUlzSlzRlzQlzPlzQfz } frm:multirot-XZ-YW-new {; Jim Muth ; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect e=exp(flip(real(p1*.01745329251994))), f=exp(flip(imag(p1*.01745329251994))), z=f*real(pixel)+p3, c=e*imag(pixel)+p4: z=z^(p2)+c, |z| <= 36 } END 20.0 PAR-FORMULA FILE==================================