On 8/19/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
... The situation is improving. There is now a still shot of a particular polytore (cubical crossover q = 0.674013, h = 1.285291, c = 1.0) at http://www.mapleprimes.com/files/8970_polytore2.pdf with an A4-sized planar net for the same at http://www.mapleprimes.com/files/8970_flattore2.pdf and the plots of h against q (both branches) for c = 1 at http://www.mapleprimes.com/files/8970_hq_plot2.pdf
Since I can't update these, I've changed all the filenames from ...2 to ...3 in the current versions.
There should also be a Maple worksheet with algebra and movies, just as soon as we can work out how to upload an updated file (!), and can persuade MapleNet to actually run the graphics over the internet as advertised ...
I've become brassed-off waiting for this to happen, so the current program is (hopefully) downloadable from http://www.mapleprimes.com/files/8970_polytore3.mw --- if anybody manages to make any kind of use of these, I'd be pleased to hear from them. On 8/19/09, Michael Kleber <michael.kleber@gmail.com> wrote:
... Fred's polytorus has two types of vertices -- (a) eight that are corners of the original cube, where meet two squares and three triangles, and (b) four more where six triangles come together. In the pdf above, the (a)s are all interior vertices of the net, which means you get a really visceral feeling that they're flat: you watch them go from planar to part of the polyhedron. The planarity of the four (b)s, on the other hand, is not at all obvious here -- the six incident triangles are divided up three and three.
Perhaps the model would be improved if the two mostly-interior type-(b) vertices were augmented with a little bit of webbing -- not a full extra copy of the three other triangles that meet there, but just a small stub with (two valley) fold lines, that the counterpart faces can snuggle into when the whole thing is assembled.
After a grotesque amount of finagling, I managed to put some rather crude dashed lines along the re-entrant edges. The "webbing" sounds a nice idea, but I'm not convinced it's worthwhile here: it is actually pretty obvious from the diagram that the whole net constitutes a tile which covers the plane in chessboard fashion. Or the triangular faces could all be transferred onto the same side of the squares, yielding bounding straight lines to left and right --- but spoiling the symmetry. WFL