It's good that a formal language for mathematics evolved. I was told or read someplace that proofs and algorithms used to be written out in paragraph form. What a mess! This is part of the problem I had in reading chapter 8. It is impossible to grasp all the concepts through a cursory reading. In order to appreciate what is going on, one must really get a pencil and paper out (or a java applet) and work through all the intricacies of the objects being described in the coordinate system.
I found the way of describing a simplex using simple coordinates from the next higher dimension cool, yet another way higher dimensions are useful. I also found the adding of hypercube coordinates to whole numbers a great way to relate dimensionality to Pascal's triangle.
A question - The golden ratio intrigued me, where else in nature does it appear?
Prof. Banchoff's response.