Subject: Re: flat triangulation of torus
Date: Thu, 8 Feb 2001 14:57:40 -0600 (CST)
From: Dave Rusin <rusin@math.niu.edu>
To: dmd1@lehigh.edu
CC: rusin@math.niu.edu

>We would be particularly
>interested in a proof that there exist *any* polyhedron at all
>homeomorphic to an orientable surface, regardless of the genus, that can
>not be realized in 3-space by flat faces.

For genus 6 I believe it is combinatorially possible to glue 44 triangles
along their 66 edges leaving just 12 distinct vertices and a surface of
genus 6. (Here 12 is the minimal number of vertices for which this
could be possible.) As far as I know, no one has found 12 points in R^3
which realize this simplicial complex.

Flabbergasted that there could exist a polyhedral torus with only 7
vertices, I took the trouble a few years ago to construct such a thing
out of cardboard as well as making a Mathematica model. Links to these
and pointers to research on this topic are at
    http://www.math-atlas.org/index/52BXX.html

dave