Coordinate geometry definitely fills up one of the gaping holes in our discussion of dimensions. The introduction is poignant and it expresses one particular idea that I had never thought of trying before. You say, "In analytic geometry, the most powerful results occur when we express some geometric relationship in coordinate form, then manipulate the number pairs or triples algebraically, and finally reinterpret the effects of these transformations on the original points in the plane or in space." This is a very powerful statement, and one that leads to much experimentation. Though I studied analytic geometry in high school, I do not recall doing this at all. The progression from coordinates on the number line to coordinates representing the vertices of the hypercube is clear and user-friendly. Why do you vacillate between the expression for four coordinates by using both four-tuples and quadruples? What` is a parallelotope? In the section on the coordinates for the n-simplex, how does the intersection of the two tetrahedra forming the stella octangula take the form of an octahedron? I am having difficulty in visualizing this representation, and I can not percieve this. The introduction of the golden ratio is a fascinating one. It is something that I was never aware of, but I have probably used the number in calculations many times. It is an example of yet another beautiful mathematical symmetry that arises in different situations. I am glad that there is a section on complex numbers because it is a topic that everyone should be familiar with. I liked the rule of multiplication for the quaternion. I like how you related this back to the two pendulums from the previous chapter.
Prof. Banchoff's response.