Chapter 8: Coordinate Geometry
First of all, I'd like to say that my long-overdue chapter 7 response is now in its proper place. My thesis is slowly consuming my entire academic/personal/mental life! Help! Aughhhh!!!!!!! Ahem, sorry. But if anyone is interested in reading my Ch. 7 ramblings, I invite you to do so.
I like the beginning of Chapter 8 because it doesn't start with a cute anecdote. Well, that's not quite true; as I declared in my Chapter 7 response, I enjoy reading the little interdisciplinary examples that tie math together with everything else in the known universe. But what I like about Chapter 8's beginning is that it's sort of like a "breather" for the reader: it reminds us where we are and where we are going. It places a lot of the math stuff we''ve been looking at in context again, and I think that's useful.
A lot of coordinate geometry is very pretty. It is quite elegant that a formula or theorem can be translated from dimension to dimension--generalized, to use Prof. Banchoff's term--without losing that sense of beauty. Of course, added layers of complexity arise with each dimension. But everything seems so nice, regular, and predictable. I have to say that such predictability always appealed to me. Don't laugh, but in times of greatest emotional turmoil throughout high school, etc., in a family crisis, for example, I would frequently get the urge to do my math homework before the rest just for the comfort of watching the predicted numbers arise from a nice, logical thought process. It was indeed somehow comforting to see the rules of math behaving exactly as expected, to see that after pages of work and complex trains of thought, I had arrived (hopefully) at the same solution as the authors of the book. I get surprisingly warm memories of that as I look at the even translations of the Pythagorean theorem from two-space to four-space.
On an equally warm note, who was it that noticed that the golden ratio is what appears in a sunflower head? Very poetic. Whence the name?
I'm not sure I'm understanding the shading in the picture on the bottom of page 174 correctly, but the one on the left (the graph of the real part of the complex parabola projected from four-space into three-space) reminds me a lot of a figure in a cosmology textbook from Prof. Brandenberger's Physics 15, _A Brief History of Time_. The figure I am thinking of illustrated a possible configuration for a negatively curved space-time. Is there actually any relation between these two figures? Is this a saddle shape, or does the yellow bit at the front represent some kind of curvy loop? Is there any reason why there SHOULD be a mathematical relationship between these two concepts?
Why does the graph at the bottom of page 175 look like a corkscrew (the imaginary part of the cube root function), and how does that fit in with the real part? Does it fit in somehow?
Much as I think it might be a little "painful," I think the reader would benefit from actually doing the derivations of some of these coordinates (i.e., for the vertices of shapes in three-space or for the Pythagorean theorem in four-space). Maybe the next time there's a family crisis....
--Michelle
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