Chapter 8: Coordinate Geometry
Wow. I think it was worth the wait to get to Chapter 9 at long last. Finally, all those concepts we wondered about are coming together. Like Euclid's axioms. I wondered about what they were, especially the 5th one, at the beginning of the semester. And here they are! And alternate geometries--not just alternate dimensions, but alternate types of spaces--well, that's just a cool idea.
Have any of you ever been to the Boston Museum of Science? I'm sure some of you have. I was always intrigued by the MATHEMATICA room (predating the software, of course). I have always been a fan of pretty theoretical math, such as can be illustrated with models and cool exhibits and pictures and such...particularly if no proofs are involved on my end :-). Well, I have a tendency to stop and read all the little amusing blurbs in that room, and I remember looking at the model of a pseudosphere and thinking it was really neat. And wondering how it got that way. The picture on page 187 reminded me. For those who haven't been there, there's a great wealth of other math models (giant mobius strip, statistical probability curve illustrated, etc.). Good way to visualize complex projects somewhat simply.
All the business about nonorientability has applications in some very unusual places. For example, part of my work as a research assistant required me to make some stimuli for visual psychophysics experiments using a drawing program. The stimuli often involved simple shapes or combinations thereof. For one of them, we wanted to sort of invert the figure we were drawing, and we had to rotate the corners defining the figure in different directions. We ran into all sorts of problems using the commands "Flip Horizontal" or "Flip Vertical" and then trying to rotate the object clockwise or counterclockwise around its center point in the two-dimensional plane. It was quite confusing, keeping the objects' original geometry straight while trying to perform some sort of rotation/orientation.
I've always found the hand-flipping problem to be interesting. What happens to handedness in higher dimensions? Would there be more than two (or three, if you count ambidextrousness) handedness types? What are the implications of being right or left handed in higher dimensions?
--Michelle
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