Response from Prof. B.

Thank you for your (almost) on-time reflections, which reached me at 12:22 a.m. It really is a pain when the web server breaks down on the day that final submissions are due, and the deadline will actually be extended to Monday midnight so you were in fact early! I suppose that the technological difficulties at the end are an omen--there still has to be a lot worked out before the paperless course becomes a smooth operation.

I agree that there is no overwhelming justification for the paperless aspect from the viewpoint of ecology, even though I myself have a good deal less paper to store as a result of this class. In the basement there are several boxes of old projects from previous Math 8 classes, many falling apart or disintegrating in other ways. For this semester, nearly everything is on line. That definitely is an advantage, although electronic archives are subject to decay as well. At least we have a good record of the experiement.

The suggestion of a formal training class on html and related technical questions is the most frequently proposed improvement on the structure of the course, and we will definitely include such a session in the future. It took us a good month this time before we decided on what we wanted to use, and that certainly inhibited some class members. The main advantage of the web approach, I agree, is that it has the potential for facilitating interaction among the class in a much different way than in the past. With more of that happening earlier in the semester, I expect that the final projects will become even more satisfying an experience.

Good work, especially on your successful final project segment. Course Grade: Satisfactory.

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.

Final Grade for the Course: Satisfactory. I will be happy to amplify that in a CPR or on a later letter of recommendation whenever you like. Good work, and thanks.