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: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> 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=========================================