; Date: Wed, 04 Dec 2002 10:48:10 -0500 ; From: Jim Muth ; Subject: [Fractint] FOTD 04-12-02 (New View of the M-Set [3]) ; Id: <1.5.4.16.20021204105023.39276b36@pop.mindspring.com> ; --------- ; 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: ; ; ; ; and at: ; ; ; ; 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. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE================================ New_View_of_M-Set { ; time=0:00:11.70--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=multirot-XZ-YW-new passes=t center-mag=-2.10942e-015/0.61004/0.721356/1.2195/-\ 90/3.88578058618804789e-016 params=4.5/22/2/0/0/0/\ -1.25/0.03 float=y maxiter=5000 inside=0 logmap=yes symmetry=none periodicity=10 colors=000GABJAENAHQAJTAMWAPZARaAUdAXhAZkAanBdqCft\ DiwEksHjoKjkNjhQjdTj`WjYZjUajQdjMgjJjjFmjBpj8rj9ok\ 9lk9ik9gk9dl9al9cl9elAgmAimAkmAmmAonAqnAsnAun9wk9y\ i8zg8ze8zc7z`7zZ6zX6zV6zT5zR5zO4zM4zK4zI3zG3zE2zB2\ z92z71z51z31z15zB9zLDzVHzd`zVszMqzNpzOozPmzQlzRkzR\ izShzTgzUezVdzWczWbzX`zY_zZZz_Xz`Wz`VzaTzbSzcRzdQz\ dLzcHzbDzaDzaDzaDza2zP2zO2zOqz2jzKdzafz_gzYhzWizVk\ zTlzRmzPnzOYzYHzfAzbszbszbszaszarz`rz_rz_rzZrzYrzY\ rzXrzWrzWrzVszXrzVrzTrzRqzPqzNqzLpzJpzIpzGozEozCoz\ Anz8nz6oz4nz5nz6nz7nz7nz8nz9mzAmzAmzBmzCmzCmzDlzEl\ zFlzFlzGlzHlzIkzIkzJkzKkzKkzLkzMkzNjzNjzOjzPjzQjzQ\ jzRizSizSizTizUizVizVhzWhzXhzYhzYhzZhz_mzdjzahz_ez\ XczV`zSZzQXzNUzLSzIPzGNzDNzDMzCLzBKzAKzAJz9Iz8Hz7H\ z7Gz6Fz5Fz5UzLjzbgz`ez_czZazYZzXXzWVzVTzUSzURzTRzT\ QzTQzTQzTPzSPzSPzSOzSOzSOzSNzRNzRNzRMzRMzRLzQLzQLz\ QKzQKzQKzQJzPJzPJzPIzPIzP } frm:multirot-XZ-YW-new {; Jim Muth ; 0,0=para, 90,0=obl, 0,90=elip, 90,90=rect e=exp(flip(real(p1*.01745329251994))), f=exp(flip(imag(p1*.01745329251994))), z=f*real(pixel)+p3, c=e*imag(pixel)+p4: z=z^(p2)+c, |z| <= 36 } ; END PARAMETER FILE================================== ; ;