Yes, nice! That's a particularly beautiful torus in the regular 6-simplex living in 6-dimensional space R^6. Call this torus T_7. Its vertices and edges form the complete graph K_7, and its triangles consist of exactly 2/5 of the possible 35 triangles made of 3 vertices. As a combinatorial torus, it is dual to the 7-hexagon (hexagonal) torus, whose hexagons form a maximally symmetrical 7-coloring of the torus. This shows that T_7 with its intrinsic metric (distances measured along its 14 triangles, not across 6-space) is also a hexagonal torus: One obtained by identifying opposite edges of a regular hexagon. Which suggests the question: Can the 7-hexagon hexagonal torus itself be embedded in R^6 so that each hexagon is a bona fide planar regular hexagon? —Dan Adam Goucher wrote: --------- For n >= 6, you can take a Császár polyhedron sharing the vertex-set (and indeed edge-set) with a regular 7-vertex simplex. The problems for n in {3, 4, 5} are certainly interesting! I wrote: ------- RWG wrote: ----- Subject: Re: [math-fun] Can an equilateral toroidal polyhedron have fewer than 32 faces? —rwg ----- If it's not embedded anywhere, it can have just two faces. If it's embedded in n-space for fixed n, this is a terrific question! (Likewise for other surfaces, like the Klein bottle.) ------- ---------