On 8/7/09, James Propp <jpropp@cs.uml.edu> wrote:
... Find a toroidal polyhedron of genus 1 whose corners are all "flat" in the sense that the angles of the faces meeting at each corner add up to 360 degrees. ... I'm curious whether equally nice but less degenerate examples are known.
On 8/8/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
... I'd guess that you found an example with 12 vertices, 24 edges, 12 faces which are all self-intersecting quadrilaterals --- though I must admit to not having actually computed any coordinates!
A construction similar to that suggested earlier for the kaleidocycle should successfully de-degenerate this --- always assuming an example with the correct --- i.e. equal --- angle-sums exists initially. The 4 false vertices are excised, and replaced by triangular antiprisms, with sufficient freedom to juggle the new angle-sums into equality. The result would have 48 faces, 96 edges, 48 vertices --- the same statistics as has a surgically enhanced 6-cell kaleidocycle. And I still don't fancy actually doing it! WFL