; Date: Thu, 05 Dec 2002 09:24:18 -0500 ; From: Jim Muth ; Subject: [Fractint] FOTD 05-12-02 (The Laugh [4]) ; Id: <1.5.4.16.20021205092631.38f76d1a@pop.mindspring.com> ; --------- ; 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: ; ; ; ; and at: ; ; ; ; 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 PARAMETER 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 PARAMETER FILE================================== ; ;