Response from Prof. B.

Thank you for your responses and for your contribution to the final presentation. I hope that you get the information to Keith and that he will arrange to have your links put into the page on the web. I have looked at them myself and I have followed some of the links rather far. There are really some nice resources there.

I agree that the fixed desk situation in Foxboro is execrable for this kind of course. I hope that soon we will have all sorts of rooms that are outfitted for the graphics we needed so that seminar-style sessions can also be accommodated.

I like your suggestions for helping people navigate through the weekly submissions without having to go through all of them without guidance. Perhaps if you end up at EBT you can find ways to enhance this kind of opportunity through developing new software? (I was at their building this morning and I noticed your name on the sign-in book.) By the way, before I forget--earlier this semester I loaned to someone my copy of "The Dot and the Line" by Norton Juster. I seem to recall that it was one of the trio? Can you help me locate that? Thanks.

Once again, I appreciated your comments about the course, and if you run into some new ideas, please let me know. 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 survey of geometry for children was very effective in providing colorful renditions of the axioms as well as an excellent collection of links to other sources. This will be valuable for future versions of the course, especially for those planning careers in teaching.

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.