Very interesting historical information here...
I enjoyed the analogy between mathematicians accepting noncommutative algebra and their acceptance of non-Euclidean geometries. I can see why it was a little harder to understand, since geometry has always had certain obvious links to the physical world and non-Euclidean thinking requires the use of imagination--experimentation is impractical.
Gauss was the greatest. I enjoyed this basic explanation of intrinsic vs. extrinsic--a helpful reminder. It is interesting to think about how Ptolemy saw his knowledge of the relationships among the points of a sphere. We have a tendency to think that it would be obvious that what he was considering was a geometry, but then again hindsight is 20-20. All of these great minds thought of these things which are presented to us as true. It is good to know basically how hyperbolic and elliptic geometries were set up and how they work, since those are definitely terms that one hears often, and yet I, myself, seem to forget how each of Euclid's axioms work (or do not work) within each system. The equator in spherical geometry seems to be formed in much the same way as a Mobius band, but it does not have any width so is actually quite different, right? But if one includes a small strip in the equatorial region as the book suggests, a Mobius strip would result--does this make a difference? (I was reacting to this as I read the chapter--having come to the section where it is explained that this space is a disc attached to a Mobius band, I understand more clearly, but I am still interested to know if it matters if the strip has no width at all.)
The skew planes are indeed hard to visualize. I may be looking for the diagram in the wrong place, for I cannot seem to find it.
An excellent finale for the book! It is amazing that the Klein bottle can be modelled so beautifully with a computer. Leaving the reader on this note certainly allows for endless contemplation--especially coupled with the three-dimensional view of the same object that heads the chapter. The book is very well punctuated with such an exclamation point.
Topics for discussion:
9.1 The Mobius Band and its relation to the Klein bottle. Build models of the Mobius Band and look at them in the mirror. Draw figures on the surface and study those in the mirror. Cut them down the middle--what happens? why? (I cannot tell you why I think what happens happens because I have never come to any solid conclusion, so I would love to hear explanations!) Cut them again--what happens then?
9.2 The modern graphics computer and its effect on mathematics. Are there down sides to their use? How do the benefits compare to these negative impacts (if there are any)?
Prof. Banchoff's responseAlexis Nogelo