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.
Thinking further about the kaleidocycle --- one could of course freeze it in (say) the position with all hinge angles equal, then replace the neighbourhoods of the hinge edges by (congruent) trapezohedra having 2 opposite faces rectangular, and 2 opposite pairs of congruent trapezia. By adjusting the angles of these last, with a little spherical trig. it should be possible to persuade the polyhedral vertices to have the required sum; but just now I don't think I can summon the required enthusiasm ... WFL