Response from Prof. B.

Thank you for the reflections, just under the wire too, at Monday midnight! It is interesting to see how much variation there has been in the class, with several persons wishing for more exercises and others just as happy to have the freedom to decide how much effort, mathematical or otherwise to put into the class. One of the virtues of the hypertext medium is that we should be able to come up with series of exercises of varying levels, suited to the backgrounds and tastes of just about anyone who might be interested in the course. As it was, the weekly assignments were places where individuals could follow ideas as far as they wanted, guaranteed in most cases with a response at the same level. Thus it was possible for several students to pursue ideas, mathematical or humanistic, to a depth that would almost certainly not have been appropriate for general class discussion. This has always been true in my classes, and the difference this year is that these dialogues are open to be read by anyone, and anyone who wants to enter into the discussion can do so. That didn't happen all that often, however, and in the future it might be possible to foster that more explicitly.

It is always going to be a challenge to hit a good balance between the solid mathematics and the less solid-looking applications and related topics. My originally stated aim was to get each person to stretch his or her mathematical abilities, and it is hard to do that all together in the class periods. Breaking down into subgroups more often might have worked with a different collection of people, but that didn't seem to catch on this time.

I might add that the approach in Math 9 will be more structured, but you still will have to depend on your own initiative to stretch yourself with respect to the standards of rigor. I hope that some students will go after the rigorous parts of the subject from the beginning, although I suspect they might form a small subgroup. I hope we will be able to use the hypertext technology in that course too, especially for the purposes of getting people to work together on the non-standard and non-routine parts of the subject.

By, the way, Course Grade: A. Good work.

Now for the general comment page on your group:

Commentary on the Extended Geometry Group

Your setup and your presentations were good, and you all showed effort in trying to arrange material for a broad audience. The subject itself is limitless of course, so you had to make choices. By and large they worked well, and it would be even better to see how electronic technology could enable you to create further linkages among your various presentations, and to those of other members of the class.

The history component will be important for any future development of this course, since it does seem to be a good subject for those planning to teach mathematics, at any level. It would be useful to have some linkages to particular parts of B3D or other works that treat dimensionality. I am curious about what parts of Manning's introduction were left out (other than the sections on non-Euclidean geometry). Abbreviating the biographical references from Smith's History of Mathematics is a good idea, since the longer sources contain much material not relevant to the subject at hand. This is true as well for the general mathematical archives at St. Andrew's in Scotland, one of the most popular of the biographical sources. In a sense you want to include the background most pertinent to the study of dimensions, shortening or omitting other aspects. It would be good to link to some primary sources as well, for example excerpts from Sylvester's inaugural lecture where he raises questions about the conceivability of higher dimensions (cf. the book on Victorian mathematics by Joan Richards).

The more impressionistic view of history and literature and art suffered by staying too close to material already contained in B3D although the links to some of the celebrity pages, like Madeleine L'Engle and Salvator Dali, were welcome additions.

As mentioned in class, in the lists of geometric properties, it would be helpful to highlight the sections where the postulate of the existence of a space of four dimensions changes the statements about lower dimensions, or recasts them. It might even be good to come up with some n-dimensional statements, for example to state that a flat space of dimension k through the origin will intersect a flat space of dimension m in a space of dimension at least m+k-n, so for example two planes through the origin in three-space meet at least in a line, but in four-space, they might meet just in a point.

In the section on parallelism, it is important to illustrate the various concepts pictorially as well as analytically. It should be possible to give equations for two 2-planes that are semi-parallel for example, as well as to use diagrams such as the ones presented for which the darkness of the shading indicated the position in the space perpendicular to the 3-space containing the viewer. The drawings were quite effective, although it would be better if the viewer had the option of making individual panels larger. It is not so easy to read the words on the console screen, and it was difficult to make anything out on the image projected to the screen in the auditorium.

The short story about the substitute teacher was very nicely presented, reminiscent of the movement of objects that we are now familiar with from watching motion of four-dimensional objects on the screen. "Substitute" has an alternate meaning of "ersatz" or "artificial", which underlines the reduction of the human form to a polyhedron. It might be nice to see some geometric forms on the side of the page undergoing the same deformations as the figure of the substitute teacher, if such illustrations are not there already. That could help tie the concepts together and reinforce what had been presented in the class.