The chapter begins by saying that until this point the approach has been synthetic--using coordinates sparingly. . . in some respects, I think the Configuration Space chapter should come after this one. . . I understand that chapters 1 through 6 progress forward, and that the last three chapters seem to be organized backwards--the three topics which should be dealt with last, successively, but. . . Well, you asked Alison in class what coordinates she used to create her set of strings--she said she didn't--you said she did. . . There are numbers and algebra underlying most of this book, but most configuration spaces seem not to exist on a level independent of coordinates. Their value seems to be representing data, or coordinates, in a space which makes sense either graphically or algebraically. Consider the case of the stage lighting which can be used to map the lighting on a stage using coordinates for center and radius. Not only does this provide a graphic representation of what it looks like, but in addition, with a few simple equations, the director can tell if lights overlap, go off stage, etc. Spaces of segments and lines are best understood form the approach made in class--how many coordinates are needed to represent this config. space.
At the same time I can understand how one would like to make the argument that the book is primarily synthetic; moreover, that coordinates come in only at the end--this argument is made when you have a publisher. . . moreover, when your publisher is Scientific American. I'm amazed that they had a problem with using coordinates throughout the book since this chapter pretty much lingers on unitary objects. If I could have thought of the simplexes or even the hypercube in this way before, I wouldn't have been confused, only helped along in visualizing.
On the subject of visualizing, I think the picture on page 171 illustrating slices of the 24-cell is actually one of the most difficult images to see what is going on--I don't know why...
Prof. Banchoff's response