; Date: Sun, 30 Sep 2012 23:10:12 -0400 ; From: Jim Muth ; Subject: [Fractint] FOTD 30-09-12 (Fractals-101 [7.5]) ; Id: <1.5.4.16.20120930231416.2a8754de@earthlink.net> ; --------- ; ; FOTD -- September 30, 2012 (Rating ?) ; ; Fractal visionaries and enthusiasts: ; ; The day we spent yesterday back at Old Fractal Central went ; quite well. But the fractal cats disapproved of our all day ; absence and our 10pm return. A quick treat of tuna and cheddar ; restored their good spirits. ; ; Today's image shows both the Mandelbrot and Julia aspects of the ; same Julibrot scene together at the same time. Since I will ; attempt to explain how this is possible, I have named the image ; "Fractals-101". The reddish outer parts outline the Julia ; aspect of the scene, while the bluish inner parts show the ; Mandelbrot bud aspect. ; ; The bluish Mandelbrot bud near the center is about 60 times its ; apparent size when it is viewed as a straight Mandelbrot shape. ; To see the pure Mandelbrot view, reset the real(p2) and real(p3) ; parameters to zero and reset the magnitude to 60. (The value of ; 60 is necessary because it is close to the value of the tangent ; of 89.) But how can a Mandelbrot image be enlarged by merely ; slicing it at a very sharp angle? ; ; The answer lies in the 4-dimensional shape of the high-iteration ; Mandelbrot features of the Julibrot, which are extended to ; infinity in two dimensions, and display the typical compact ; Mandelbrot shapes in the other two dimensions. ; ; Imagine an infinitely long cylinder in three dimensions with a ; circular cross-section. If it is sliced straight across, the ; slice is a circle. But if it is sliced at an ever increasing ; angle, the slice is an increasingly stretched oval. And slicing ; the cylinder along its length produces an infinitely long ; straight rod. The angle of the slice causes the expansion of ; one dimension of the slice. ; ; In four dimensions we may have an object that is circular in its ; two compact dimensions and extends to infinity in its two ; extended dimensions. Such an object may be sliced at two ; independent sharp angles. Doing this will give a 2-D slice that ; is stretched in two directions, and if the angles are equal, the ; cross section will be enlarged proportionally without being ; stretched. We have a Mandelbrot microscope. The Mandelbrot ; part of today's image was enlarged just this way. (Unfortunate- ; ly, we can get no extra magnification of Mandelbrot objects by ; doing this.) ; ; The image itself is pretty routine. The most interesting part ; is the curiously shaped bluish Mandelbrot bud at the center. ; ; The rating of a question mark means I don't really know how to ; rate the image. ; ; The calculation time of 2 minutes is reasonably fast, but the ; web sites are always there to make things even easier. ; ; Be happy! View the finished on the official FOTD web site at: ; ; ; ; without the need to calculate it. ; ; View the image in high-definition, with lots of added bells and ; whistles such as anti-aliasing at: ; ; ; ; All the thousands of FOTD back images are online at: ; ; ; ; A mix of sun and clouds, with a temperature of 64F 18C made ; today quite typical of the end of September here at Fractal ; Central in Central Pennsylvania. The fractal cats enjoyed the ; sunny periods on their shelf, and endured the cloudy times in ; sulk mode. The humans spent most of the day recovering from ; yesterday's trip to Old Fractal Central, where my sister is ; putting out food for the neighborhood cats that are starting to ; gather in increasing numbers for a free meal. The next FOTD ; will be posted in a reasonable amount of time. Until whenever, ; take care, and if we suddenly lost all our advanced technology, ; how many, I wonder, would survive and how many would curl up ; and expire. ; ; ; Jim Muth ; jimmuth@earthlink.net ; ; ; START PARAMETER FILE======================================= Fractals-101 { ; time=0:02:00.00 SF5 at 2000MHZ reset=2004 type=formula formulafile=basicer.frm formulaname=JulibrotMulti2 function=recip passes=1 center-mag=0/0/1.572327 params=6.5/-6.5/89/0/89/0/\ -1.626/0/0/0 float=y maxiter=24000 inside=0 logmap=12 symmetry=xaxis periodicity=6 colors=000S05U17W28Y39_4Aa4Bc5Ce6Dg7Ei8Fk8Gm9HoAIq\ BJtCKwCLuEMsGMqINoKNmMOkOOiQPgSPeUQcVQbUPbUPbUOaUO\ aUNaTN`TM`TM`TM_TL_SL_SKZSKZSJZSJZSJVTKSULPVMLWNIX\ OFYPBZQ8_R5`S2`T9cQGeNNhKUjH`lEgoBnq8us5so9rlDpiGo\ eKmbNl_RkWViTYhQafMdeJhdGkdGkcHkcHlcHlcHlbIlbImbIm\ bImaJmaJmaJnaJn`Kn`Kn`Ko`Kz_Lz_Lz_Lz_LzZMzZMzZMzZM\ zYNzYNzYNzYNzXOzXOzXOzXOzWNzWNzWNzWNzWNzWNzWNzWNzW\ MzWMzWMzWMzWMzWMzWMzWMzWLzWLzWLzWLzWLzWLzWLzWLzVKz\ VKzVKzVKzVKzVKzVKzVKzVJzVJzVJzVJzVJzVEzVJzVJzVJzVJ\ zVIzVIzVIzVIzVIzVIzVIzVIzUHzUHzUHzUHzUHzUHzUHzUHzU\ GzUGzUGzUGzUGzUGzUGzUGzUFzUFzUFzUFzUFzUFzUFzUFzYKz\ XKzXKzXKzXKzXKzXKzWKzWKzWKzWKzWKzWKzVKzVKzVKzVKzVK\ zVKzUKzUKzUKzUKzUKzUKzaKz`Kz`Kz`Kz`Kz`Kz_Kz_Kz_Jz_\ Jz_Jz_JzZJzZJzZJzZIzZIzYIzYIzYIzYIzYIzYIzXHzXHzXHz\ XHzXHzWHzWHzWGzWGzWGzWGzVGzVGzVGzVFzVFzUFzUFzUFzUF\ zUKzKKzKKzMKzLKzLKzKKzKKz } frm:JulibrotMulti2 {; draws all slices of Julibrot pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p2*0.0055555555555556), b=pi*imag(p2*0.0055555555555556), g=pi*real(p3*0.0055555555555556), d=pi*imag(p3*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), aa=-(real(p1)-2), bb=imag(p1), c=p+flip(q)+p4, z=r+flip(s)+p5: z=(bb)*(z*z*fn1(z^(aa)+bb))+c |z|< 6 } ; END PARAMETER FILE========================================= ; ; ; ; _______________________________________________ ; Fractint mailing list ; Fractint@mailman.xmission.com ; http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint ;