; Date: Mon, 23 May 2011 22:09:17 -0400 ; From: Jim Muth ; Subject: [Fractint] FOTD 24-05-11 (Bubbles in Numbers [6]) ; Id: <1.5.4.16.20110523220923.11b7c634@pop.earthlink.net> ; --------- ; ; FOTD -- May 24, 2011 (Rating 6) ; ; Fractal visionaries and enthusiasts: ; ; When certain people have trouble sleeping, they sometimes count ; sheep. When I can't sleep however, I begin mentally exploring ; an imaginary four-dimensional world. My imaginary world is ; truly four-dimensional; the problem is that I can mentally ex- ; plore it only one 3-D slice at a time. ; ; As our planet earth has a curved two-dimensional surface, my 4-D ; world has a curved three-dimensional hypersurface. On earth, ; two coordinates, latitude and longitude, are necessary to posi- ; tion a point; on my 4-D world, latitude, longitude and lithi- ; tude, (my own invention), are required. Moving in the lithi- ; tude direction does not take us into the sky or underground; it ; moves us in the third dimension, flat along the hypersurface. ; ; To reach a given location on this 4-D world, we start at the ; zero-zero-zero point on the equator, which is a great circle ; that circles the planet just as earth's equator does, and from ; there we travel until we reach the correct longitude. ; ; Now we find ourselves with a problem. Before we can set off ; toward the pole, we need to determine in which direction to go. ; On earth there are two directions, north and south, leading ; toward the two polar points; on my 4-D world there is a full 360- ; degree circle of directions extending from the equator, each ; direction leading to a distinct point on the polar circle. The ; direction in which we must travel away from the equator is the ; lithitude. Once we have chosen the lithitude, we need the lati- ; tude coordinate, which tells us how far to go toward the polar ; circle in the direction we are traveling before we reach our ; destination. ; ; But even stranger things than this happen on my planet. There ; is also the matter of rotation. On earth, rotation is a simple ; thing -- the planet rotates on its axis, a line that extends ; through the globe from pole to pole. But on my 4-D planet, the ; axis of rotation is a plane that cuts the hypersurface along the ; polar circle, a curiosity that causes complications. ; ; The problem arises because the polar plane exists in two dimen- ; sions that are stationary even as the planet rotates around it. ; The polar plane is therefore free to rotate on itself, with the ; equator as the axis of its rotation. And in a universe with 4-D ; gravity, it is unlikely in the extreme that the polar plane does ; not have at least some degree of rotation. The 4-D planet is ; subject to two independent rotations at the same time. Every ; point on the hypersurface as well as every point within the ; planet is tracing out what is known as a surface of double revol- ; ution, a kind of 4-D donut. ; ; Things on the surface of my planet move in paths resembling ; spirals, which curve back on themselves into closed loops. This ; means that the stars in the hypersky must move in helixes. What ; fun it must be to set up camp in some dark spot on the hypersur- ; face of my hyperplanet and point a hypertelescope at the hyper- ; stars and hyperplanets. The hypertelescopic clock-drive would ; be a complex device, but I'm sure that it could be made. ; ; If the two rotations are equal, things get even more complica- ; ted, but this is a story for another day. For now, we have a ; fractal to investigate. ; ; Today's image lies in the parent fractal that comes about when ; Z^(1.25) is subtracted from Z^(-1.75) before (1/C) is added on ; each iteration. This parent is more a field of debris with two ; large Mandeloids than a single object. Today's scene is located ; in some debris lying between the two largest Mandeloids. ; ; There is not much to be said about the fractal, so I'll let it ; speak for itself in visual terms. ; ; The name "Bubbles in Numbers" refers to the bubbly shapes in ; today's image and also to the fact that fractals are made of ; numbers. ; ; The rating of a 6 indicates that the image has room for ; improvement. ; ; To view the image, either run the included parameter file or go ; to the FOTD web site at: ; ; ; ; The high-definition version of the image is posted at: ; ; ; ; The classic FOTD web site may be accessed at: ; ; ; ; There were too many clouds at Fractal Central on Monday for the ; day to be considered a good one. The temperature of 75F 24C was ; fair enough however. The fractal cats spent the day plotting ; mischief, but fell asleep before they could put their plot into ; action. In the afternoon a tornado warning came over the radio, ; but the storm passed over with only a strong wind squall and a ; torrential rain. ; ; The day of the humans was one more in an unending series of ; similar days. But the older one grows, the more such uneventful ; days are appreciated. The next in the unending series of FOTD ; fractals will be posted in 24 hours. Until then, take care, and ; when do we get to see the all of fourth spatial dimension in one ; moment of time? ; ; ; Jim Muth ; jimmuth@earthlink.net ; ; ; START PARAMETER FILE======================================= Bubbles_in_Numbers { ; time=0:00:30.04-SF5 on P4-2000 reset=2004 type=formula formulafile=basicer.frm formulaname=MandAutoCritInZ function=recip float=y center-mag=+0.03748573643483764/-0.655971033500649\ /1.5667e+010/1/-35/0 params=-1/1.25/1/-1.75/0/0/0/0 maxiter=500 inside=0 logmap=57 periodicity=6 colors=0000440650962C74F86I9AL7DO5HV4Kd2Nl1cu2tz2h\ ZPK35Q46X47b48h59n5AuKBzPBsKGk8Lc9QXAVjBnkGelKYmOQ\ nSIoWAp_2sWLvSbyOtv`atlJqfMnaOkXQhRTeMVbHXaMa`Qe_V\ iZZmYcqXguXky`lndmdgnVkoLnoBLCIF_J9wKLaUXHceMjmRqu\ WxrSppOimKbkGWhCPf8Id5Bc6Eb7Ha8K`9M_APZBSYBUdjo_il\ WijRigNieIicEi`9iZ5iXPVfgGpz1yy2xx3wx4wvAutGtsLrqR\ qpWonanmfmkiqjlthowgqzajpXdgRZYMTPHNGVVNgbUti`neai\ aacZaZVaTRaOOaIKaDHaDGbDFbDEcDDcHCdLCdPBdTBdXBdYAc\ Z9cZ9cSCaLF_EIY7LW0NV4TS8YQCbNFgLJlINqGQvEMbCIJAF0\ 8E62TGJXLL`QMcVOg_PkdRniSrnUurVqrUmrTksYitagueevic\ wmbxq`toZqnXnlWkkUgjSdhQagPZfPYeQXcQWbQW`QVZQUYRTW\ RTVRSTRRSRQQRQPTOQVMQXKRZIR`GSbESdCSfATh8Tj6Ul4Um2\ Uj5Vh8WfBWcEXaHY_KYXNZVQZTT_QW`OZ`MaaKdaUH0`IDVH2W\ H3XH5YH6ZH8_H9`HBDSaHQYKPVNNSQMPTLLWJIZIFRx_StYTqW\ UnUVkTVhRWePXbOYZMZWKZTI_QH`NFaKD5Qd8PaBO_ENXGNVJM\ SMLQPKORKLUJJUTeOWXJZNDaD } frm:MandAutoCritInZ {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(((-a*b*g*h)^j)+(p4)), k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } ; END PARAMETER FILE========================================= ; ; ; ; _______________________________________________ ; Fractint mailing list ; Fractint@mailman.xmission.com ; http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint ;