## Response from Prof. B.

Somehow in spite of my suggestions, there seems to be some strange characters appearing instead of quotation marks, but I suspect that thet will disappear in some later iteration."

With respect to the mathematical fashion that produces efforts on a specific subject at a particular time, it is often difficult to identify any special factors. It may be that an article appears in a journal that is read by a number of mathematicians, and several of them decide to work on this or that interesting aspect. For example, one of my German co-authors sent me not too long ago a reprint of an article first published in 1881 in which the author generalizes to four-dimensional space the relationship among the numbers of vertices, edges, and faces of a convex polyhedron in three-space. (Perhaps we can generate some evidence to conjecture what that relationship is, and why it is different in four-space?)

It might be interesting to do a brief literature search in England in the 1880's to see if there were any papers on the subject of regular polytopes at that time. There is a good resource to check: D. M. Y. Sommerville published a source book on non-Euclidean geometry that included nearly all the names of articles on that subject published before the early part of the twentieth century, and he had a section on four-dimensional geometry (which some people considered non-Euclidean simply because it wasn't treated explicitly in "The Elements".) Why not look it up in the Sci Li and give a brief report? Maybe I'm wrong and there are several British and French writers who should have been given credit for staking a claim in the polytope rush.

When somebody tells you that, "The fourth dimension is time, isn't it?" you can answer that "Time is A fourth dimension, not The fourth dimension," and see how they react to that.

I'm not sure who first caught on to the duality idea. I would say it was around by the time of Kepler but I'm fairly sure it was around earlier. It became more important in the nineteenth century when people began formalizing the concept of symmetry groups. Perhaps you would like to try a bit of bibliographical sleuthing and see how far back you can go?