Response from Prof. B.

It is true that there was a spirit of competition at work in the nineteenth century phenomenon I described, but it may be that I dramatized it a bit. One of the reviewers of my book didn't care for that, and he said I was unfair to Stringham et al. because Schlaefli had published his work in such an obscure place that no one could have been expected to know about it. I don't think that that was my point.

It is true that even today there are some "hot" subjects in mathematics that attract a competitive breed of scientists. They would rather be the first to come up with something that a lot of people are trying to find than blaze a new trail by themselves. That sounds like more of a criticism than I mean it to be, since some of these fashionable subjects got that way because they are very significant, and the people who come up with new ideas to solve them are to be appreciated. But the highly competitive aspect is not that important most of the time.

Abbott was well educated up in mathematics up through calculus and differential equations, but he would not have run into higher dimensional questions in his schooling. He well might have known about some of the debates about geometry taking place in general intellectual journals, but the polytope question itself never made it into the public arena. I conjecture that the proximate source of information about the geometry of four-space came from C. H. Hinton. You can read about my theory in an article I wrote in 1990.

Thanks for the Dali web reference, although I am not that impressed by the wording there. Among other things the author gets the name of the pianting wrong, and his comments are not very well formulated. He is right in identifying the woman as Gala, Dali's wife, who appears in so many of his paintings, especially at this time. The four little boxes delineate the vertices of one of the cubes, a more or less transparent one, coming toward the viewer and partially including the figure.

In your exercises, you probably mean "dodecagon" rather than "dodecahedron"? That same error appears in the first edition of "Flatland", put there, I hypothesize, by an over-zealous copy editor.

By the way, I can't seem to download either the autobiography or the artwork in your links. I did get the regular polytope, and I was excited to think that all of Coxeter's images might be available at the Toronto site. But no. As it happens, Coxeter is a good friend and a great person, a hero in this business and still quite active.