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