; Date: Mon, 31 Dec 2007 23:34:58 -0500 ; ; To: fractint@mailman.xmission.com ; cc: philofractal@lists.fractalus.com ; ; From: Jim Muth ; Reply-To: Fractint and General Fractals Discussion ; ; ; Subject: [Fractint] FOTD 01-01-08 (No Name [No Rating]) ; ; Id: <1.5.4.16.20071231233713.29e77b20@pop.mindspring.com> ; --------- ; ; FOTD -- January 01, 2008 (No Rating) ; ; Fractal visionaries and enthusiasts: ; ; I have been lagging in the FOTD discussions recently, and giving ; too much attention to the images, so what better day than the ; first day of a new year to begin a discussion of the fourth ; dimension as it relates to the section of Seahorse Valley that ; appears as today's FOTD. ; ; What do I mean when I claim that today's image is a new view of ; perhaps the best known of all fractal 'objects', Seahorse Valley ; of the Mandelbrot set? It bears no resemblance at all to the ; valley or its Julia sets. Despite appearances, I mean exactly ; what I say. Today's image shows part of Seahorse Valley. ; ; What then is Seahorse Valley? The first impression is that it ; consists of two tapering wedges that approach but never actually ; reach the point at -0.75 on the real axis of the M-set. This ; much is true, but these two wedges are only a small part of ; Seahorse Valley. What about the Julia set of the point -0.75? ; This Julia aspect is as much a part of Seahorse Valley as the ; Mandelbrot aspect. And when we consider Julia sets, what of the ; countless other Julia sets associated with the other points of ; Seahorse Valley? They are also a part of the valley. ; ; We now find ourselves with a unique two-dimensional Julia image ; associated with every point of Seahorse Valley, and all these ; Julia images stack together in four-dimensional space to form a ; single four-dimensional assemblage, which is the hyper-Seahorse ; Valley area of the four-dimensional Z^2+C Julibrot figure. ; ; Since it is a 4-D object, the full Seahorse Valley cannot be ; visualized in its entirety at a single moment of time, but it ; can be analyzed and statements can be made about it that can be ; demonstrated to be true. In its full 4-D aspect, Seahorse ; Valley consists of two tapering hyperwedges that approach but ; never quite reach the plane of the Julia set of the point -0.75 ; of the M-set. ; ; Instead of terminating in two sharp points, as do the two ; branches of the familiar Seahorse Valley of the M-set, these two ; hyperwedges terminate in two sharp planes, which are cuttingly ; sharp over their entire 2-D surfaces. In our three-dimensional ; space, sharp planes are an absurdity. Planes are flat and there ; is nothing sharp about a flat surface. In 4-D space however, an ; unlimited plane does not form an impassable barrier, and one may ; simply step around the plane and continue on. If the two ; additional dimensions of the plane are extremely small, the ; plane will act in 4-D space as a razor edge does in 3-D space, ; slicing through any reasonably soft 4-D object it comes in ; contact with. ; ; So the full Seahorse Valley consists of two hyperwedges ; terminating in two sharp planes. What shape then is the eastern ; surface of the valley, which faces the main bay of the M-set? ; Like the surface of any other 4-D object it is a surface with ; three dimensions, and the buds that appear to be flat circles ; are actually 4-D hyper-cylindrical shapes with two extended ; Julia dimensions and two small Mandelbrot dimensions. Like ; regular 3-D cylinders, these hypercylinders may be sliced in two ; dimensions to give circles, ellipses and parallel-edged stripes. ; In today's image the eastern surface of Seahorse Valley has been ; sliced in the Oblate direction, which shows the buds there as ; black stripes. ; ; To make the stripes more clear I have stretched the image quite ; a bit in the vertical direction. And to add interest I moved ; the center of the slice 0.35 in the imag(z) direction. For some ; reason the whole thing ended up rotated 180 degrees. Oh well, ; nobody's perfect. ; ; Since today's image is more a study than a finished piece of ; art, I gave it no name or rating. And its calculation time of ; 20 minutes is admittedly slow for a mere 4-D study. I seriously ; recommend visiting the FOTD web site at: ; ; ; ; and viewing the finished image there. But it is New Year's ; time, so be patient if Paul has not yet posted it. ; ; Three wet slushy inches or 7cm of snow fell overnight Sunday ; here at Fractal Central. Monday was sunny with a temperature of ; 41F 5C, which melted a good part of the snow. The fractal cats ; noticed the snow with curiosity, then returned to their normal ; activities. The next hyper FOTD will be posted in 24 hours. ; Until then, take care, and to find the fourth spatial dimension, ; look sideways to your inside. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE======================================= FOTD_for_Jan_01_08 { ; time=0:20:13.47-SF5 on P4-2000 reset=2004 type=formula formulafile=slices.frm formulaname=Oblate passes=1 center-mag=+0.02067323\ 061209259/+0.01838987041970301/1425.157/0.002231/\ -180/-32.619243071192308 params=0/-0.35/-0.7498/0 float=y maxiter=32767 inside=0 logmap=325 symmetry=none periodicity=10 colors=000cpaco`cn_cmZclXcjWciVchUcgTcfScdRccQcbPb\ aNb`MbZLbYKbXJbWIbVHbTGbSEbRDbQCbPBbNAbM9bL8bK7aH1\ bJ6bLAbMFbOJbPOcRScSXcU`cVecXicYmdZldZkdZjeZieZheZ\ gfZgfZffZefZdgZcgZbgZahZahZ`hZ_hZZiZYiZXiZWjZWjZVj\ ZUjZTkZSkZRkZQlZQlZPlZOmZNmZMmZLmZKnZKnZJnZIoZHoZG\ oZFoZEpZEpZDpZCqZBqZAqZ9qZ9pYAoYAnYAmYAlYAkYBjYBiY\ BhYBgYBfYBeYCdYCcYCbYCaYC`YC_XDZXDYXDXXDWXDVXEUXET\ XESXERXEQXEPXFOXFNXFMXFLXFLXFMWEMWENWENWEOWDOWDPWD\ PWDQWCQWCRVCRVCSVBSVBTVBTVBUVAUVAVVAVVAWU9WU9XU9XU\ 9YU8YU8ZU8ZU8_U7_U7`T7`T7aT6aT6bT6bT6cT5cT5dT5dT5c\ S4dT5dT6dU7dU8dV8dV9dWAeWBeXBeXCeXDeYEeYEeZFfZGf_H\ f_Hf`If`Jf`KfaKgaLgbMgbNgcNgcOgdPgdPgdUfdZfccfcheb\ mebrdavdazc`zc`zc_zb_zbZzaZzaYz`Yz`Xz`Xz_Wz_WzZVzZ\ VzYUzYUzYTzXTzXSzWSzWRzVOzWRzXTzYWzYYzZ_z_bz_dz`gz\ aizakzbnzcpzcrzbqzaqzaqz`qz`qz_pzZpzZpzYpzYpzXozXo\ zWozVozVozUnzUnzTnzTnzSnz } frm:Oblate {; Jim Muth real(z),imag(c) z=real(pixel)+p1, c=flip(imag(pixel))+p2: z=sqr(z)+c, |z| <= 16 } ; END PARAMETER FILE========================================= ; ; ; _______________________________________________ ; Fractint mailing list ; Fractint@mailman.xmission.com ; http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint