## Response from Prof. B.

Tracking motions of factory workers, in so-called time and effort studies, became quite popular in some circles, to the extent that efficiency experts even tried to apply the same techniques to professors. It didn't quite work. There aren't goniometers for everything.

It was partially in response to your question that I talked a bit about the Dimensions dance. There is a videotape, but not a very good one unfortunately. We hope to do better sometime in the future.

I do like the interplay between the videos and the material in the book. What we really need to supplement both is some interactive demonstration capability, and we might get that soon, even as part of the final projects. The wave front example is a good one to include.

Almost all ways of describing lines miss some of them, or treat them in special ways. Even if we want to describe the oriented lines through the origin, we can identify them by indicating the points where they pierce through the unit sphere, but then we still have the problem of putting coordinates on the sphere and that always leads to special points like the north and south poles ("singularities of the parametrization"). The attempts to find good parametrizations of spaces of lines have led to some very interesting mathematical research.

The waves a sphere produces in three-space can be stacked up in four-space
the same way the circular waves in the plane can be stacked into a right circular cone in three-space. So the analogue is a hypercone, and we can then look at hyperconical sections. The analogy actually takes us pretty far in this situation, and to that extent we might say that we are visualizing the waves in four-space, however imperfectly.