FOTD -- May 12, 2011 (Rating 8) Fractal visionaries and enthusiasts: Today's image is a view of the curious Julia rectangle as it appears in the Oblate plane, which is determined by the real(Z) and imag(C) axes. When viewed from this angle and given an appropriate amount of stretching and skewing, the rectangle pretty much retains its rectangular shape. The name "It Has No Symmetry" accurately describes the image. At first glance, the image does indeed appear to have origin symmetry, but a close inspection will reveal small discrepancies. The rating of an 8 could have been a half point or so higher if I had not re-used a previous color palette. The calculation time of 3 minutes is well within reason, but all the fuss and bother that goes with setting up and calculating a parameter file may be forgotten by viewing the finished image on the FOTD web site at: <http://www.crosscanpuzzles.com/Archives.html> The image may be seen in hi-def at: <http://www.emarketingiseasy.com/TESTS/FOTD/jim_muths_fotd.html> Paul's original FOTD web site may be accessed at: <http://www.Nahee.com/FOTD/> I find the inability of the human mind to visualize the fourth dimension most curious. Computers have no problem converting the math of the fourth dimension to spatial coordinates and manipulating the resulting spatial figures exactly as they do with everyday 3-D figures. It would even be possible to construct a monitor with a 3-D viewing surface, ready to be enjoyed by a 4-D viewer. Unfortunately, a normal 3-D man with his 2-D retinas would be able to view such a screen only from the 'edge-on' direction and would be unable to enjoy the 4-D show any more than he could enjoy viewing a normal flat screen edge on. Why are we limited to visualizing three dimensions, when such fascinating things as double rotation and undistorted Klein bottles exist only one dimension higher. It is impossible for us to imagine even a 2-D flatland or a 1-D line without automati- cally mentally embedding these lesser spaces in a surrounding 3-D space. The limitation is not due to the math, which merely grows more tedious as the number of dimensions increases, with nothing of a totally new nature appearing. There is a theory, not entirely scientific, that our inability to visualize 4-D figures is a limitation of our level of con- sciousness, which is stuck at the 3-D level. This would imply that a being with a higher level of consciousness might be able to visualize the fourth dimension, or might even interpret the sensory impressions of the world he lives in as possessing four spatial dimensions. If so, he would not be around too long to enjoy his advantage. Stable planetary orbits are impossible in a space of four dimensions with its inverse-cube law, and if our math is correct, a planet in 4-D space would either spiral into its star or spin off into interstellar space. But then again, maybe a 4-D man would have different laws. The days keep getting better here at Fractal Central. Wednesday featured cloudless skies, gentle breezes and a cat-pleasing temperature of 77F 25C. The fractal cats, totally pleased, spent the day sleeping and dreaming of what they might do if they ever get to go outside. The humans had more mundane things to attend to. The next FOTD will be posted in 24 hours. It will likely be another view of the rectangle. There is still much yet to be discovered in the vicinity. Until then, take care, and Nostradamus says the world will end in 589 days. It's got ME worried. I thought we had 5-1/2 billion years. Jim Muth jamth@mindspring.com START PARAMETER FILE======================================= It_Has_No_Symmetry { ; time=0:03:04.77-SF5 on P4-2000 reset=2004 type=formula formulafile=basicer.frm formulaname=SliceJulibrot4 passes=1 center-mag=0/0\ /1130/0.02072/0/77 params=0/90/90/90/-1.7435/0/\ 0.00019/0.07388/2.003/0 float=y maxiter=15000 inside=0 logmap=62 symmetry=none periodicity=6 colors=000FDHFDJFELFCNFCPFCRGCSIETKGUMIXOK_QLbSNeU\ OhWQkYSm_ToaUqcTseSugRuiRukQvmPvoOvqOvpStrWst_rvcq\ xhpzmozrpzvszzvzzzzzzzzzzzzzzvzzrzzmzzgzzdzyazxazw\ azw`zv`zu`zt_zt_zs_zr_zqZzqZzpZzoYznYznYzmYzlXzkXz\ kXzjWziWzhWzhWzcSzZOzUKzPGvKCrF8mB5hC8cDBZEEUFHPGK\ OGMRHPUISXJV_KYaK_dLbgMejNhmOkoOmmRkkTjiVhgXgfZed`\ dbbb`daZf_YhZgh_ph_dcTTZMHUF5P99SEDUJHXOLZTO`XN_YN\ ZYNYYNYYNXYNWZNWZNVZNUZNUZNTZMS_MS_MR_MQ_MQ_Q9mQ9m\ P9cP9cO9cOAcNAcNAcMAcMAcLAcLAmKAmKAcJAcJAcIBcIBcHB\ cHBcGBcGCcFDcFEcEFcEGcDHcDIcCJcCKcBKcAKmAKmAKm9Kz9\ Kz8Kz8Kz7Kz7Kz6Kz6D65D65D54D54D53D53D42D42D42D47U8\ CiCHyGGzFGzFGzFGzFGzFGzFFzFFzFFzEFzEFzEFzEFzEEzEEz\ EEzEEzDczDczDczDczDczDczDczDczCzzCzzCzzCzzCzzCzzCz\ zCzzBzzBzzBzzBzzBzzBzzBzzBzzBzzmzzmzzmzzmzzmzzmzzm\ zzmzzmzzmzzmzzmzzmzzmzzmzzmzzzzzzzzzzzzzzzzzzzzzzz\ zzzzzzzzzzzzzzzzzzzzzzzzz } frm:SliceJulibrot4 {; draws all 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), esc=imag(p5)+9 c=p+flip(q)+p3, z=r+flip(s)+p4: z=z^(real(p5))+c |z|< esc } END PARAMETER FILE=========================================