FOTD -- December 04, 2002 (Rating 3) Fractal visionaries and enthusiasts: When is Seahorse Valley not a valley?...When it's a bridge, of course. Today's image shows a new view of the Mandelbrot set. But when seen from this angle, it is not the M-set. It's an odd slice of the Julibrot, a slice in some unimaginable direction oriented roughly halfway between the Mandelbrot and Julia directions. To picture the orientation of today's view, imagine the M-set on the computer screen, with normally unseen attached parts of the Julibrot extending invisibly in front of and behind the screen. Then imagine the top half of this invisible 3-D part of the Julibrot cut off and the newly-exposed 2-D surface rotated 90 degrees around the X-axis until it fills the screen. Today's image illustrates that slice. The image vaguely resembles the M-set, but it is not the M-set. The main bay is represented, as are the primary bud on the negative stem and the next bud beyond. Farther west, there is a slice of the prominent midget at -1.75, which barely skims that midget's main bay. Only hints of the stem itself are visible. The large period-2 buds that lie at the north and south edges of the main bay of the M-set are nowhere to be found. They are now lying invisibly in the space before and behind the screen, in the position where the slice now on the screen was originally located. Along the eastern edge of the large bay on the screen, which is actually East Valley, a few of the 'elephants' that roam there can barely be discerned, grossly distorted at this strange angle. The most surprising thing about the new slice is what has happened to Seahorse Valley at -0.75 and Scepter Valley at -1.25. They are in the image, but they have turned into the narrow straight rods which I have named bridges. I have colored these bridges a brilliant green for emphasis. These straight rods are not artifacts. The Julibrot is filled with them. They are caused by the fact that, regardless of where iterated points start, if they do not escape first, they will eventually settle into the same orbits as points that start at zero. The immediate implication is that, when its total shape in the Julibrot is considered, a Mandelbrot valley does not come to a sharp point, but rather a sharp knife-edge. But even this idea is incomplete. The Julibrot is four-dimen- sional, and there are many other orientations in which the Mandelbrot valleys appear to come to sharp knife-edges. In the four-dimensional world, these knife-edges stack together edge-point to edge-point, not side-by-side, forming sharp two- dimensional cutting surfaces, such as a 4-D man would use to shave his hyperface. Like so much that can exist only in the world of four dimensions, these sharp 2-D surfaces are impos- sible to visualize, and the complete shape of the Mandelbrot valleys will therefore never be imagined. I named the lightning-fast image "New View of the M-set". The rating of a below-average 3 might be a bit pessimistic, but hopefully the render time of only 11 seconds will make viewing it worth the effort. As always, the pre-rendered GIF image is available on the FOTD web sites at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> and at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The fractal weather Tuesday was cold and sunny, with blustery winds and a temperature of only 23F -5C. Needless to say, the fractal duo turned around and came back inside as soon as their noses felt the outside air. They passed the day sleeping by the radiators, thankful that they have such a cozy place to take shelter in. This morning is even colder, but there is little wind. It's a fine day to tackle some work. Until next time, take care, and don't get taken for a ride by a Seahorse Valley in disguise. 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