Response from Prof. B.

Your investigation of the symmetries of the Schlegel diagrams does raise some questions about the way you want to count faces. Somehow you are losing aspects of duality that you might wish to maintain longer, and that can be done only if you allow some "infinite faces". One way to motivate this is to consider spherical polyhedra rather than ordinary rectilinear ones. There is an easy correspondence between the two types due to the fact that any regular polyhedron in three-space can be inscribed in a sphere. We can then think of the polyhedron as made of rubber, and then inflate it so that the flat faces become curved portions of the sphere. (If we think of a point on the polyhedron as a [non-zero] vector, then we can divide it by its length to get a unit vector with the same direction, thus pushing the whole surface onto the unit sphere.) Under this process, each edge of the original polyhedron is sent to a piece of great circle arc on the sphere.

Now think of what stereographic projection does to this collection of edges. The images of the vertices are as they were before, but now instead of being connected by straight lines in the plane, they are connected by pieces of circles (thanks to the felicitous property of stereographic projection that states that the image of any circle, great or small, is a circle or a straight line). Thus if the North Pole is situated at the center of a face, we get a bulgy version of the usual Schlegel diagram.

But maybe you see what the problem is going to be--if we look at the inflated version of the dual of the polyhedron, then one of the vertices is going to lie exactly at the North Pole, the one point of the figure that has no image down below! How can we handle this? Well, we note that a certian number of great circle edge come into this point, and the images of these edges, except for the North Pole point, will be straight rays shooting out to infinity. These rays form parts of the boundary of regions stretching out to infinity, the images under stereographic projection of the spherical polygons the meet at the North Pole.

Consider a specific case, the cube. If we "Schlegelize", we should get not only the four triangles you drew, but also four infinitely large triangles, each having one finite edge and two infinite edges, chosen from among four rays going out toward infinity from the four vertices in the boundary of the figure you drew.

I feel quite frustrated at this point that I am unable to include a gif file of the picture I have in mind, but I don't have the appropriate tool on this machine. Suffice it to say at this point that the process mentioned here will preserve duality in a rather strict sense. Compare this discussion with the comment I made at the end of Jeremy Kahn's Week 9 piece, and I think you will agree that this is a good topic to pursue further as we get into the final projects.

Good investigation--further comments?