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: <http://www.crosscanpuzzles.com/Archives.html> The high-definition version of the image is posted at: <http://www.emarketingiseasy.com/TESTS/FOTD/jim_muths_fotd.html> The classic FOTD web site may be accessed at: <http://www.Nahee.com/FOTD/> 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=========================================