On 8/18/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
The original problem was to construct a polyhedron, homeomorphic to a torus, with angular defect zero at every corner (and everywhere else): properly embedded in 3-space, so that the interiors are homeomorphic. ... My attempts to find somewhere public to put all this stuff having been so far cruelly stymied, I must again offer to send the files to anybody expressing an interest.
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 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 ...
The latter poses a nontrivial robotics question: is it possible to fold the polyhedron from a sheet on which opposite edges have already been joined [forcing it to be folded flat, but four layers thick]?
I dunno about that --- but it's pretty tricky trying to build the wretched thing from a conventional net anyhow! Dividing into (at least) two pieces is recommended, to anyone reckless enough to attempt the feat. Fred Lunnon