Response from Prof. B.

Your point is well taken. By the time we get to the eighth chapter, the usefulness of a consistent coordinate representation is more than clear. In fact it is quite frustrating to try to describe all of these things in paragraph form, and to be using the notion of "strings of numbers" implicitly before we can introduce ordered n-tuples. Perhaps the ideal way to write this kind of book is to use hypertext from the beginning so that anyone can switch over to coordinate representation at any time, whenever the urge comes.

It was unfair to insist that Lisa E. (not Alison, by the way) was using coordinates when in fact she was merely sewing her successive threads in an equally-spaced pattern along two non-intersecting edges of a cardboard figure. More accurately I could have pointed out that there is an abstract geometric model of the actual object she had created, and the equal spacing idea translates into a uniform scale on the two lines. Of course that correspondence becomes much more easy to state and to see if we have coordinates in three-space to use in the description, and in a sense, that formed the bridge between chapters 7 and 8. I suppose I think of configuration spaces as independent enough from coordinatization that I might resist switching the order of the chapters. The more I think about it, I realize that the two notions of dimension are interlocked but separate--for example you have a circle which is a one-dimensional configuration space since each point is determined by a single real angle, but the circle itself exists in a two-dimensional space. The coordinate space notion in Chapter 8 is a rather rigid one, with one fixed space for each dimension, while the configuration spaces have a greater variety of topological forms. Still, most of these topological forms can be visualized as sitting in one of the fixed coordinate spaces, and it might be very useful to have that method around. It isn't quite a toss-up, but I'm not sure right now how I would rewrite it, if I were considering rewriting these ideas in a traditional book form. Is hypertext the answer, or is it just a way of avoiding the question?

The 24-cell is by far my favorite polytope, precisely because it is challenging, at least one step beyond the hypercube. But the hypercube seemed mysterious at first and now it has become familiar. With enough practice it would almost certainly be possible to gain some similar appreciation of the 24-cell.