; Date: Wed, 05 Oct 2011 20:46:31 -0400 ; From: Jim Muth ; Subject: [Fractint] FOTD 05-10-11 (The flying Titanic [Rating 4]) ; Id: <1.5.4.16.20111005204639.137f34f8@pop.earthlink.net> ; --------- ; ; FOTD -- October 05, 2011 (Rating 4) ; ; Fractal visionaries and enthusiasts: ; ; Today's image rates only a 4, hardly up to FOTD standards, but ; the scene is so funny that I felt obliged to make it FOTD for ; today. To me the scene shows the ghost of the long-lost ship ; HMS 'Titanic', barely visible through the mists, sailing calmly ; over some far-away heavenly sea. Unlike that other ghost ship, ; the 'Dutchman', a sighting of this ship does not portend ; disaster. It is an omen of good fortune. The name "The Flying ; Titanic" gives recognition to both ghost ships. ; ; (Well, maybe the ship in today's image does resemble something ; more like a small destroyer than a mighty ocean liner, but at ; least it's something better than a tugboat.) ; ; The scene is located in the East Valley area of the large ; minibrot on the main stem of the Mandelbrot set. It is oriented ; in 4-D space at an angle that would be impossible to describe, ; and is stretched and skewed beyond imagining, yet here it is, a ; very recognizable image of a ghostly ship passing silently in ; the night fog. ; ; The image is an example of serendipity. When I stumbled upon ; the hole that resembles a ship, I was searching for something ; entirely different. I still am searching in fact, and if I find ; it, I will make it tomorrow's FOTD. ; ; Today's image finishes in 6-2/3 minutes, a little on the slow ; side for a scene that rates an unusually humble 4. But this is ; where the FOTD web site comes to the rescue. ; ; The web site may be accessed at: ; ; ; ; The high-definition version of the image is at: ; ; ; ; The original FOTD web site is at: ; ; ; ; The space of four dimensions, in which this month's featured ; fractal object, the Julibrot, lies, is curious in many ways, ; especially when compared to space of three dimensions. In 3-D ; space only five regular polyhedrons, the five Platonic solids, ; are possible: the tetrahedron, with four triangles as faces; the ; cube, with six squares as faces; the octahedron, with eight ; triangles as faces; the dodecahedron, with twelve pentagons as ; faces; and the icosahedron, with twenty triangles as faces. ; ; And in space of five dimensions as well as spaces of all higher ; dimensions, the increasing numbers of lower dimensional elements ; usually do not match up correctly, and only three regular ; figures now called polytopes are possible. These are the ; analogs of the tetrahedron, cube and octahedron. So what about ; space of four dimensions? ; ; Intuition suggests that also in 4-D space, only three regular ; polytopes are possible, just as in spaces of all higher ; dimensions. But in this case intuition is wrong. In space of ; four dimensions, where the polytopes are bounded by three- ; dimensional boundaries called cells, six regular polytopes are ; possible: the hyper-tetrahedron, bounded by five tetrahedrons; ; the hyper-cube, (sometimes called the tesseract), bounded by ; eight cubes; the hyper-octahedron, bounded by sixteen ; tetrahedrons; the hyper-dodecahedron, bounded by 120 ; dodecahedrons; and the hyper-icosahedron, bounded by 600 ; tetrahedrons. There is also a figure bounded by 24 octahedrons, ; which has no regular analog in 3-D space, but is related to the ; semi-regular rhombic dodecahedron. ; ; So what would life be like in 4-D space, living on a ; hyperspherical planet most likely subject to a dizzying double ; rotation? Would the stars in the 3-D sky trace out helixes? ; Unfortunately, no such planet would be possible. In 4-D space ; the force of gravity would obey an inverse-cube law, making ; stable planetary orbits impossible. A planet would either ; quickly spiral into its star or sail off into interstellar ; space. So much for fantasy. ; ; The weather here at Fractal Central today was certainly no ; fantasy. We enjoyed cloudless skies and a temperature of 70F ; 21C. The fractal cats, now settling into winter mode, found ; comfort on their shelf in the sunny southwest window. ; ; The humans, FL and I, found comfort in the bright sunlight ; streaming through the windows, which has been pretty scarce the ; past couple months. Even more comfort will be found when the ; next FOTD is found and delivered, hopefully in 24 hours. Until ; then, take care, and the formula for the hypervolome of a ; five-dimensional hypersphere is 8/15*(pi^2)(r^5). Let me know ; when this useless bit of information might become important. ; ; ; Jim Muth ; jimmuth@earthlink.net ; ; ; START PARAMETER FILE======================================= The_Flying_Titanic { ; time=0:06:39.42-SF5 on P4-2000 reset=2004 type=formula formulafile=basicer.frm formulaname=SliceJulibrot2 passes=1 center-mag=-0.00048410686492484/+0.089235490629652\ 03/262770.2/0.00651/-90/89.1258940753795486 params=30/90/0/90/-1.7489/0/0/0 float=y maxiter=4200 inside=atan logmap=206 periodicity=6 colors=000GJV_5W_AZ_Ca_Ca_D`_D`_D_ZEZZEZYFYXFYWGXW\ GWVGWUHVTHVTIUSITRITRJSQJSPKROKQOLQNLPMLPLMOLMNKNN\ JNMDONFOMHOMJNMLOMNOMPPMRPMTQLVRLXRLZSL_SLXTMUPORK\ QMJSKKUNKTQKSTOQWTOZWMaXKdYJgYIjZHk_Hm`Gp`FsaEtbDt\ cDtcCtdBueAue9uf9vg8vh7vh6wi5wj5wk4xk3xl2xm1xm1wl2\ wl2vk2vk2uk2uj2tj2tj3si3si3rh3rh3qh3qg3pg3pg4of4of\ 4ne4ne4me4md4md4ld5lc5kc5kc5jb5jb5ia5ia6ha6h`6g`6g\ `6f_6f_6eZ6eZ7dZ7dY7cY7cY7bX7bX7fU0dW4bX7`ZAZ_EY`H\ WbKUcOSeRQfUPgYNi`LjcJlgHmjBsnDqmEpmGnmHmmIkmKjmLi\ mMgmOfmPdmRcmSbmT`mV_mWYmXXmZVm_UmaTmbRmcQmeOmfNmi\ KphLohLngMmgMlgMkfNjfNifNheOgeOgeOfdPedPddPccQbcQa\ bR`bR_bR_aSZaSYaSX`TW`TV`TU_UT_US_USZUXYUaYUfXU`XU\ WWUQWULVUFUUAUU5UUAUUFUUKUUPUUUUUVUUWUUXUUYUUZUU_U\ U`UUaUUbUUcUUdUUdUUcUUbUUaUU`FU`EU_DUZCUYBUYAUXAUW\ AUVAUVAUTAUSAURAUPAUOAUNAUMAUPAUSAUVAUYAU`AUcAUeAU\ cAUbAUaAU`AU_AUZAUXAUdAUd } frm:SliceJulibrot2 {; 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|<=9 } ; END PARAMETER FILE========================================= ; ; ; ; _______________________________________________ ; Fractint mailing list ; Fractint@mailman.xmission.com ; http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint ;