Response from Prof. B.

Your discussion of the edge-truncation of the hypercube is really nice, and you have done it in a different way than I did in _B3D_, where the approach was more in terms of coordinates, in Chapter 8. By analyzing things synthetically, beginning with the cuboctahedron in three-space, you have come up with an excellent way of visualizing the twenty-four octahedra, eight of them easy to see and sixteen others that are more obscure. The way the corner octahedra are presented as four pieces fitting together, in analogy with the three pieces of triangle fitting together to give a triangle in the cuboctahedron, is especially instructive.

One thing that might be worth pursuing further at this point is the possibility of dividing the sixteen octahedra into two sets of eight, each of which would have the same setup as the eight you started with, namely that any two of them would either be disjoint (if they were in opposite cubes of a hypercube) or touch at a single vertex (if they were in adjacent cubes). This fits in with an idea I heard at a conference in Canada two summers ago where it was observed that the 96 edges of the 24-cell could be split up into three sets of 32, each being the set of edges of a hypercube. I'll bet you come up with that if you continue your investigation in this way.