Thanks, George! I look forward to watching the video, which unfortunately I don't have time to do this morning. But I'm wondering a couple of things from this post: 1. How do you define the product of two polygons as a polyhedron in 3-space? Abstractly, it makes perfect sense: The product of two vertices is a vertex; of a vertex and an edge (in either order) is an edge; and of two edges is a quadrilateral. So far this is just a combinatorial object, but it could even be an abstract geometrical object, as if each polygon were in a distinct plane and the product becomes a specific subset of 2+2 = 4-space. But that seems to allow a lot of possibilities for it to end up in 3-space. How do you see that any way of doing this results in a polyhedron that's self-intersecting? 2. What does "homogeneous" mean in this situation? It could refer just to the result being combinatorially homogeneous, in that the result has maps taking vertices to vertices, edges to edges, and faces to faces while preserving incidence, that will carry any vertex to any other (transitivity on vertices). Is that it? Thanks, Dan ----- The talk Jim had referred to is now online here in a more completed form. Polyhedra fans might enjoy it: https://www.youtube.com/watch?v=_PSdVX02Vbs The question which I thought might have been intended (during the CoM discussion, or initially by Jim) was whether one could make a *homogeneous* zonohedral surface in 3D which is topologically toroidal without self-intersection. By homogeneous, I mean following the algorithm described in the video from a given initial state. (One with all vertices of degree four would be homogeneous, but that isn't a necessary condition; homogeneous examples might have vertices of any order, e.g., polar zonohedra.) One can take a product of two polygons to create a toroidal surface composed of parallelograms and having all vertices of degree four, but it will be self-intersecting. And one can approximate any continuous surface by joining homogeneous patches, e.g., gluing together small cubes. (The video shows how to join larger homogeneous patches seamlessly.) But I would conjecture that a homogeneous zonohedral surface can only be toroidal if it is self-intersecting. -----