; Date: Sat, 14 Jul 2007 00:00:12 -0400 ; ; To: fractint@mailman.xmission.com ; cc: philofractal@lists.fractalus.com ; ; From: Jim Muth ; Reply-To: Fractint and General Fractals Discussion ; ; ; Subject: [Fractint] FOTD 14-07-07 (Straight-Edged Julia [8]) ; ; Id: <1.5.4.16.20070714000531.2b87cfa4@pop.mindspring.com> ; --------- ; ; FOTD -- July 14, 2007 (Rating 8) ; ; Fractal visionaries and enthusiasts: ; ; Today's image is almost a Julia set of East Valley of the ; Mandelbrot set. But in this case, the word 'almost' means a ; huge difference. True, the scene is sliced within 1/10th of 1 ; degree of the true Julia orientation, and the outer shape of the ; fractal is what would be expected of an 'East Valley' Julia set. ; But the inside is unlike any Julia set that I have seen. ; ; The corresponding Mandelbrot point is located inland very near ; the edge of a tiny bud on the northern shore of East Valley. ; The diagonal straight edge is the remains of that shoreline. ; This edge inspired the name "Straight-Edged Julia". But how can ; a near circular shoreline become straight? ; ; The answer involves a four-dimensional abstraction. But a ; pretty good analog can be given in three dimensions. Imagine a ; long cylinder. When this cylinder is intersected by a plane at ; a right angle, the section is a circle. As the intersecting ; plane rotates, the section becomes an increasingly stretched ; oval, until finally, when the plane is parallel to the cylinder, ; the section consists of parallel straight lines. When a ; dimension is added, this is what causes the straight edge inside ; today's fractal, as well as the 'bridges' that appear so often ; in the four odd orientations of the Julibrot. The irregular ; disks in the upper left portion of today's image are created by ; rendering the image with the inside set to 'fmod'. ; ; Since I think the image is pretty good and I also enjoy the math ; part of today's image, I rated it at an 8. The parameter file ; takes just over 6-1/2 minutes to calculate on my fast fractal ; computer. On the slow machine it would take around 1/2 hour. ; ; But why not avoid calculation and view the image on the FOTD web ; site at: ; ; ; ; A near perfect day prevailed here at Fractal Central on Friday, ; with lots of sun, blue skies, puffy white clouds and a tempera- ; ture of 81F 27C. A few drops of rain and a couple grumbles of ; thunder in the evening did not spoil the day. The fractal cats ; were rather restless most of the day, though I noticed nothing ; out of the ordinary in the vicinity. ; ; My day was about average. With a slow day expected tomorrow, ; fractal lady and I will likely take a short trip. It probably ; will not affect the FOTD, but if the FOTD is late, you will know ; we were late returning to F.C. Until next time, take care, and ; see the world with 4-D eyes. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE======================================= StraightEdgedJulia { ; time=0:06:40.74-SF5 on P4-2000 reset=2004 type=formula formulafile=allinone.frm formulaname=SliceJulibrot2a passes=1 center-mag=0/0/1.103753/1/-90/0 params=89.9425/\ 90.0832/90.1493/90.0832/0.266/0.0042/0/0/0/0 float=y maxiter=100000 inside=fmod periodicity=10 colors=000cIKeLKgOKiRKkUKmXKo_KqbKseKuhKwkKynIyqHv\ oGsmGpkFmiFjgEgeEdcDaaDZ_CWYCTWBQUBNSAKQAIPAMOCQNE\ TMFXLH_KIcJKgIMjHNnGPqFQoEOnFMlMKkSJhULeWMbYN__PYa\ QVcRSeSPfUMhVKjWHlXEnZBp_8r`6saFl`Oe`XZ`eT`dVacXbb\ Yba_c``c_bdZdeZeeYgfXhfWjgVlhUmhToiTpiXlg`hfceegad\ kYcnVbrRavN`yK_wL`vM`uNatOasOaoXgkdlglqctvbusaupau\ m`uj_vg_veZvbZv_YwXXwUXwSWwPVxMVxJUxGUxEWuFYsGZqH`\ oIamJckKdiLfgMheNicOkaPl_QnYRoWSqUTrSUpSVnSWlTXKdI\ QbMWaPa`TgZXmY`sXcpY`nYZlYWjYUhZSfZPdZNbYL`YIZXGXX\ DVWBTW9RV6PV4NU2QT4TS6WR8ZQA`PCcOEfNFiMzlLznKzqKzt\ KzwKzyKzvKzsKzqKznKzkKziKzfKzcKzaKzZAzW8zU6zPKzLKz\ HKzDKzBKzAKz9Kz8Kz7Kz5Kz4Kz3Kz2Kz1Dz5Ez8FzCGzFHzIH\ zMIzPJzTKzWLzZLzbMzeNziOzlPzoPzcAzTAzPAzMAzJAzGAzD\ KzASzERzIRzLRzPQzSQzWQz_PzbPzf0zi0zm0zp0zs0zw0zu0z\ t0zs0zr0zq0zp9zo9zn8zm7zl7zk0zj0zi0zh0zg0zf0zd0zb0\ z`0zZ0zX0zW0zU0zS0zQ0zO0z } frm:SliceJulibrot2a {; 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|<=real(p5+9) } ; END PARAMETER FILE========================================= ; ; ; _______________________________________________ ; Fractint mailing list ; Fractint@mailman.xmission.com ; http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint