FOTD -- December 06, 2002 (Rating 5) Fractal visionaries and enthusiasts: Today's image catches one of those Mandelbrot midgets at an angle where it is most distorted. The actual midget can be seen by checking the M-set at the coordinates of (p4) of today's image. At the angle of today's slice, little is left of the hole, which once was the midget and now lies invisibly at the center of the image, or even of the basin itself, for that matter. At this angle, the basin appears to have split into two branches, though actually it is the angle of the slice that creates the split, cutting the single curved basin in two places. The surrounding areas of chaos, which somewhat resemble comets, are far better defined in the M-set. I named the picture "Passion Flower". At one time I thought I saw a flower blossom somewhere in the scene, though the bloom is now far less apparent. I rated the image at an average 5, which is all I could do for it. (The corresponding midget, which appears in the M-set, would rate a 6 or 7.) With a 3-1/2 minute render time, the parameter file is one way of viewing the scene. A more efficient way is to download the completed GIF image from Paul's web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from Scott's site at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> Today's image is a slice of the four-dimensional Julibrot. The Julibrot cannot be accurately considered an object, because objects are things that exist objectively in the objective world, and to our specialized senses at least, the objective world is three dimensional. I have spent many years trying to visualize a four-dimensional hyper-object, knowing all the while that my effort was futile. The greatest problem is that the surface boundary of a 4-D figure must be three-dimensional, and my imagination invariably produces flat two-dimensional bounding surfaces for all my mental figures. A 4-D hyper-object must have on its bounding outer surface as many dimensions as there are altogether in our space. On the 3-D surface of a 4-D planet, there would be six cardinal directions. Three parameters would be needed to fix a location, and two parameters would be necessary to define a direction. The horizon would not be a circle, but a sphere that totally surrounds and encloses a visitor from the third dimension, as the heavens surround the planet Earth. On such a world, the sunrise would first appear as a brightening concentrated in a particular area. Then a brilliant star-like dot would suddenly pop into existence at the brightest point, grow to the familiar blinding disk, then shrink again to a dot and vanish. At sunset the same thing would occur at another point of the field of view. At night, assuming perfect clarity of the atmosphere, the stars would appear as twinkling fireflies as they rose above and sank beneath the horizon, passing through the 3-D visitor's field of view. The view on a 4-D world would be interesting indeed -- provided, that is, that the visitor could find a way to shield his retinas from light from the side, and also a way to avoid crumpling into a wrinkled heap as a 2-D sheet of plastic wrap does in our 3-D world. It snowed at a moderate rate all day Thursday here at F.C., piling up 8in 20cm by the time it ended at 11pm. Combined with a temperature of 27F -3C, the conditions proved far too harsh for the intrepid duo, who passed the day alternating between the window and the radiator. All things considered, their moods were surprisingly cheery. Unable to get out yesterday, I passed the day getting ahead in the work. As a result, after clearing the sidewalk, I will have extra time this afternoon to devote to fractal hunting. Hope- fully, I'll find something worthwhile. Check back tomorrow to see whether I actually find it. Until then, take care, and fractals can and do make life interesting. Jim Muth jamth@mindspring.com jimmuth@aol.com START 20.0 PAR-FORMULA FILE================================ Passion_Flower { ; time=0:03:35.04--SF5 on a P200 reset=2002 type=formula formulafile=julibrot.frm formulaname=SliceJB-new passes=1 center-mag=0/0/2.\ 632523e+010/0.006156/-8.21933390591836854/89.48120\ 55080059565 params=0.22/0.51/0.505/0.326/2/0/-0.15\ 40762385267798/1.030994820300086/-0.15407623852677\ 98/1.030994820300086 float=y maxiter=2500 inside=0 colors=000WIFXJGYKHZLHZMI_NJ`OJ`PKaQKbRLbSMcTMdUNd\ VOeWOfXPd`XeZTfXPgVLhUHiSDjQ9mR3jP6hN9eLBcKE`IGZGJ\ XEMUDOSBRP9TN7WJ2_L6YNAWPDURHSTKQVOOXRMZVL`ZJbaHde\ FfhDhlBjo9ls7ms6kqDimKgiRkkVmmWfeXbYW_YXYYYWYYUYZR\ Y_PY_NY`Laa9ceEcgJYaNRWSLQXEKd4Gb6F`8FZAFXCEVEETGE\ SIDQKDOMDMOCKQCISCHUBFWBDYBB_A9aA7cA3dD6gA9j7Cl4Fm\ 2Mk8SiEZfKecMeaOd`PcZQbYRaXS`VT_UUZTVYRWXQXWPYWN_V\ M`ULaTJbSIcRHdQFePEfODgNBhMAiM9jPIlSQmVToYWp_Zrbas\ ecuhevjguiiuhkshlrglpfmofmmenlenjdnicogcoebpdbpbaq\ a`q_`rZ_rX_rWaoXbmYckYdiZfgZgd_hb_i``kZ`lXamUbnSbp\ QcqOcrMdtJdvHewFixDly8lxBiwEfvHcuJ_sJWqJQoJPmJOkJN\ iJMgJLeJKcJJaJJ_JIYJHWJGUJFSJEQJDOJCMJBMBDLFCKJBKN\ AJR9IV8IZ7Ha6He5Gi4Fm3Fq2Eu1Cz0Ex1Fw2Gu3Ht4Ir5Jq6K\ o7Ln8Ml9NkAOiBPhCQgDReESdFTbGUaHV_IWZJXXKYWLZUM_TN\ _RL`SNaTPbUQcVSdWTeXVfYWgYYhZZi_`j`akaclbemcfndhod\ ipekqflrgnshotiqujrvktzmw } frm:SliceJB-new {; by John R. H. Goering, July 1999 pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1), b=pi*imag(p1), g=pi*real(p2), d=pi*imag(p2), 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)+(p4), z=r+flip(s)+(p5): z=z^(p3)+c |z|<=9 } END 20.0 PAR-FORMULA FILE==================================