Beyond the 3rd Dimension, Chapter the Last

OK, let's skip straight to my favorite part of the chapter: the Klein Bottle. It was very interesting to see perspectives of it that I hadn't seen before. I had never really thought of a three dimensional representation of the Klein Bottle other than the one hanging at the Providence Art Club (an image, by the way, which I think should be included in future editions of the book). I have a question about the computer generated image in the margin of page 198. Obviously, the two ends of the figure are different colors because the final "gluing" has not yet been done. However, would it make more sense to color it similarly to the picture above it--with sides to be glued the same color? I think that would be neat. Would that still be valid? Could I draw a line, reddish in the middle and bluish on the ends, and say that it is really a circle in 4 space? That is a similar construction.... I have a little trouble "seeing" the image on the bottom of page 198. I guess it makes more sense after I review the pictures in the margin. Why this view for the book? Why that weird figure 8 thing?

I'm a little bit fuzzy on the projective planes made from simplexes. Is it that... oh, nevermind. I just don't get it. When we were actually working with hemispheres, I was with you. I'll have to re-read this section at a later time.

Beltrami's pseudosphere doesn't have to be thought of as having a sharp edge. I think it suggests infinity.

What would a spherical image mapping on a surface look like? What did Gauss do with the points he found? Did he, for example, take 2 connected points from a surface, find their corresponding points on the sphere, and then map them? That's what it seems like. I'd like to see an example of that, if someone can point me in the right direction.

Oh, back to the Klein Bottle thing. In Jeffrey Weeks' book, there is a great diagram illustrating the construction of the Klein Bottle. It's basically a bunch of slides from "the movie". First we have a square, then a tube, then one end of the tube pierces the side of the tube, then the other end on the tube flips outside in, then the two ends are connected. I like that. It would be neat to see a similar thing in B3D.

What would a mobius bad be like in real life? There seems to be so much assymetry in it. What happens when you are on the seam? are you half flipped, and half not? Your body would split in half--twisted to shreds? How does that work? It would make sense if there was no seam. Wait--I guess there is no seam. I could cut a mobius band at any point and call that a seam, so there is no ONE place where the flipping happens. It is a continuous process. That makes more sense. Is that right?

That's all for now.

Prof. Banchoff's response

Dan