Professor Coolidge's Test

Summary

I recall how, at a large mathematical gathering, Professor Coo1idge rose, advanced to the front of the room, and there frightened the group by announcing that he was going to give them a little mathematics test. Now mathematics professors may like to give tests, but to take one is quite another matter. To calm his audience, Professor Coolidge said he merely wanted to verify that most mathematicians know very little elementary solid geometry.

Professor Coolidge started by reviewing a few definitions, such as those of the medians and the altitudes of triangles and tetrahedra. “Now,” he said, “though, as a high school student of geometry knows, the medians of a triangle are concurrent, can the same be said of the medians of a tetrahedron?” after some hesitation, almost everyone present said that surely they must be. Professor Coolidge assured them that this is indeed the case. He next similarly asked, “Though, as an high school student of geometry knows, the altitudes of a triangle are concurrent, can the same be said of the altitudes of a tetrahedron?” Many present said that of course they are concurrent, most of the others said that they blame well ought to be, and the few remaining ones, fearing some sort of a trap, were noncommittal. Professor Coolidge then explained that the altitudes of a tetrahedron usually are not concurrent, and that concurrency occurs only in the so-called orthocentric tetrahedra, in which each edge of the tetrahedron is perpendicular in space to the opposite edge of the tetrahedron.