Response from Prof. B.

Actually the Dali encounter took place closer to Easter, a bit after St. Patrick's Day. And it was a martini, in front of me anyway. It wasn't exactly relaxing, as I recall. We were sizing each other up at that time, although we did relax after a while. He was very wary of people trying to take advantage of him, and I didn't feel like being exploited either. But then it became clear that we could avoid that problem and both of us could end up with some good stories. He is certainly one of the most interesting people I ever expect to meet.

For a hypercube, you can even get four vanishing points. Think about it.

The Mercator projection is a pretty subtle way of mapping, and it can't be described as simply as the stereographic one. Stereographic projection is a central projection, so that we choose a point source of light and a plane and send each point in space to its shadow on the plane, or more precisely, to the intersection with the plane of the line from the light source through the point. If the point happens to be at the North Pole and the plane goes through the equator, then we get stereographic projection. If we place a polyhedron so that its vertices are on the sphere, then the images of these vertices will be the vertices of the Schlegel diagram of the polyhedron.

But the Mercator projection starts by wrapping a cylinder around the equator, then spreading the parallels of latitude upwards and downwards so that the -angles- between curves are preserved, although lengths and areas are quite distorted as we get farther and farther away from the Equator. It might seem at first that all we are doing is taking a central projection from the center of the sphere outward onto the surface of the cylinder, but as a matter of fact it is a much more complicated function that determines the spacing of the images of parallels of latitude.

After I wrote this, I read Lisa Eckstein's week 9 piece and found a link to some interesting cartographic material: To see it, click here.

Clifford constructed his torus using equations, I believe. Polyhedral approximation wasn't all that important in the nineteenth century, more a computer graphics phenomenon. I hope we will be able to make such things clearer when we introduce coordinates in Chapter 8, although there is a danger that the formalism will threaten to take over. We will stand firm, however, and maintain geometric descriptions while we use the algebra to describe things more explicitly.