Subject: question (fwd) Date: Thu, 8 Feb 2001 13:42:47 -0500 (EST) From: Joanna Ellis-Monaghan To: dmd1@lehigh.edu CC: Dan Archdeacon Hello, My name is Jo Ellis-Monaghan. A colleague, Dan Archdeacon, and I are working on a problem about flat triangulations of the torus. It seems to have a bit of a topology/geometry flavor, and we were hoping you might be able to help us please. I have described the problem to Jim Stasheff who told me that, in addition to your own expertise, you coordinate a topology newsletter that it might be worth posted this question to. Would you be willing to take a look at the question below and please let us know what you think, and/or post it to the news group if it seems it appropriate? This would be a great help. Thank you very much. Jo Joanna A. Ellis-Monaghan e-mail: joellis@emba.uvm.edu or jellis-monaghan@smcvt.edu website: http://academics.smcvt.edu/jellis-monaghan Department of Mathematics phone: 802 656 2940 University of Vermont fax: 802 656 2552 16 Colchester Avenue Burlington, VT 05405 or Department of Mathematics phone: 802 654 2660 Saint Michael's College fax: 802 654 2610 Winooski Park Colchester, VT 05439 _____________________________ Torus flat triangulation question We are looking at the following problem about polyhedra in Euclidean 3-space. Let us first describe an example. Consider the well-known triangulation of a torus by the complete graph K(7). This has 7 vertices, 21 edges, and 14 triangles. In the 50's Csaszar found what we call a _flat realization_ of this polyhedron in 3-space. That is, he located the vertices in 3-space and represented each face by the convex hull of its three boundary vertices, so that the intersection of any two faces in 3-space was exactly the convex hull of the intersection of their boundary vertices. Hence each face is contained in a plane, and two faces have no interior intersection. We would like to show that any simple triangulation of the torus has such a flat realization. This is in the flavor of Steinitz' Theorem, which says that a graph is the 1-skeleton of a (spherical) convex polyhedron in 3-space if and only if the graph is 3-connected and planar. Toroidal polyhedral, of course, would not be convex. We are focusing on triangulations as the most interesting (easiest?) special case. Does anyone know of work done in this area? 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. What happens if the faces are not constrained to be triangles? We would be interested in any results along these lines. If you know anything, please contact dan.archdeacon@uvm.edu or joellis@emba.uvm.edu. Thanks for your attention to our question. ------------------------------------------------------------- Dan Archdeacon Tel: x6-0850 (work) 204 Mansfield (office) 872-0023 (home) Dept. of Math. & Stat. (campus mail) x6-0696 (fax) dan.archdeacon@uvm.edu http://www.emba.uvm.edu/~archdeac/ -------------------------------------------------------------