Response from Prof. B.

Thanks for the comments, throughout the course. I'm glad that you have had the opportunity to see Gauss showing up in so many different ways. He really is a mathematical hero.

I agree that the notion of a "geometry"is subtle, more of a game. There are some people who don't believe that there is such a thing as an elliptic geometry because it doesn't meet some criterion or other. But I like elliptic geometry better than the hyperbolic sort, at least from the point of view of interaction on a computer screen.

Skew planes are hard to visualize, that is definitely true. The best analogy I can think of for the difficulty is trying to explain to a Flatlander how a line and a plane can intersect in one point. That leads in two directions--to see how a hyperplane and a line can intersect in four-space at only one point, and then to split off one line from the hyperplane and attach it to the line to get two planes intersecting just at one point. I'm not sure that helps.

I'd be interested in your answer to discussion topic 9.2, especially as you prepare to enter the teaching profession. There are some problems, and not just logistical ones, but I for one am convinced that the weight will dominate on the plus side. We'll see in a few years how the educational trends will go.

I'm looking forward to seeing your final project, and all the others.

Prof. B.