I notice that Banchoff, in this chapter, has chosen to examine only the Schlegel diagrams of those where "the image of the face closest to the north pole contains the images of all the other vertices." Only on the tetrahedron does this entail putting a vertex at the north pole itself; I propose that this is closely related to its self-duality.
Here's why: If we position each shape in the Schlegel diagram so that a vertex is at the north pole instead, we get a whole variety of interesting diagrams with neat symmetries. Let's call taking the position Banchoff describes the "surround" position, just for reference's sake, and the one I have described the "centered" position. In the case of the tetrahedron, though, we get exactly the same picture under the "surround" position as the "centered" position. (Which faces are "in front" is not addressed in Schlegel diagrams.) Hmm... I'm not clear on how to prove this yet, but I think we can show that when an object A's Schlegel "centered" image and Schlegel "surround" image are congruent, A is self-dual. Comments?
I think I have just disproven my own conjecture. I have looked a long time at the picture of the Schlegel polyhedron of the 24-cell shown on page 118. This, of course, is a good test-site for the conjecture above; we know from other sources that the 24-cell is self-dual, so if it demonstrates this property we have a piece of support, and if it fails to demonstrate this property then we have disproven the conjecture. We can see fairly quickly that the 5-cell actually does support this conjecture, but there is probably something in common here among the simplexes instead.
The picture, I believe, disproves the conjecture, because at the very center of the Schlegel projection of the 24-cell (projected from the region of the south pole, presumably) is not a point but another octahedron. Since in the simplex reversals we see above one of the reasons this switch can happen successfully is that the north and south poles are transparent to each other, we can tell that if we were to put a vertex in the north pole, we would have something that was _not_ an octahedron in the south.
I am a bit confused by this stuff. I can see the extension of the relationship between circles in the sphere and circles in the plane to the relationship between circles in the hypersphere and circles in space (the torus). Especially pretty are the parallels between the sphere projection (with rings drawn on) to the plane and the Clifford Torus 3-space projection changing 90 degrees too. I imagine the series continuing and watching the bulge appear on the left of the new torus and the red rings would expand out to infinity and return to surround the green bands, moving the red to the outside and the green to the inside, neatly reversing the north-south position (which when projected to our 3-space is reflected as red-within-green, when red is south, and green-within-red, when red is north, and as loops-going-through-the-bagel-hole when red and green are on the equator. Pretty.
Jeremy Kahn x6753
Prof. Banchoff's comments