Chapter 6 comments

Leo Mariani

It is sure nice to see Professor Banchoff relaxing with Salvador Dali and a little wine. It is definitely in the St. Patrick's day spirit. I have been noticing that as we read each successive chapter I seem to understand more and more about four dimensional objects. Perspective obviously plays a very important role in understanding four dimensional objects because we can only really see three dimensions at once. I particularly liked the views of the cube with one, two, and three vanishing points. Even though I have seen these views many times it never occurred to me that this was true. I have never heard of a stereographic projection before, and it is definitely the most interesting part of the chapter. I like how you introduce it from a cartographer's point of view. I have probably seen some of these, but I just did not realize what they were called. What is the relationship between a stereographic projection and a mercator projection? Is a mercator projection simply two sterographic projections from the equator with the point source from the eastern hemisphere illuminating the western hemisphere and vice versa? Do schlegel diagrams also assume point light sources from a pole? What is the relationship between schlegel diagrams and sterographic projections? I was utterly confused by the relationships between the hypercube, hypersphere, and the torus. Is the Clifford torus just a polyhedral torus where the number of polyhedra limits to infinity? Is there another way of constructing the clifford torus?

Prof. Banchoff's response.