## Response from Prof. B.

You are quite right that the Schlegel diagrams in the book were of a special sort, and it is also quite interesting to view the projections of the edges of a polyhedron from other from other points than those situated just about the center of a face. We get some very intriguing images by projecting from a vertex, as you said, and also from a point just above an edge. Sometimes these projections pick out a symmetry that would be otherwise missing. I hope that the final project on Polyobjects will consider this question and produce some images from other viewpoints, commenting on their relationship to symmetry and duality.

Comment--See the Keith Adams's Week 9 paper and the response.

There is also a good opportunity to illustrate the relationships among Schlegel diagrams by setting up some animations. We can consider stereographic projection from the North Pole on the sphere to the plane, and watch the images of the edges of a polyhedron inscribed in the sphere as the sphere rotates, keeping the point of projection and the horizontal plane fixed. That is essentially what we did in order to make the second movement in the hypercube film.

After examining these phenomena for the regular (and semi-regular?) polyhedra in three-space, you can go on to look at the changes in the Schlegel diagrams of the regular polytopes in four-space, at least in the simple cases. If you want to go for the whole prize, you can find coordinates for the other regular polytopes in the book Regular Polytopes by H. S. M. Coxeter.

The 24-cell is my favorite. It always holds surprises.

The Clifford torus question is somewhat subtle. The analogue of the circles in the sphere would properly be a collection of two-spheres in the three-sphere, so it is something of a leap to consider instead the decomposition into tori. It isn't all that hard to write down the parametrization, and the torus case seems a bit more symmetric in a way than the one with the family of two-spheres (with its two distinguished degenerate levels given by the poles). There are several ways of covering each of the tori with circles or circular bands, and each of these brings out something special about the symmetries involved. I hope to go further into this when we get to succeeding chapters.