this is no doubt older than dirt, but it just popped into my head when trying to think of how to convince my beginning linear algebra class that every linear isometry of R^3 has an eigenvector: i.e. it follows from the theorem that one cannot comb the hair on a billiard ball. do you see how? it might not be totally well known, at least to students, because we so seldom mingle topology and linear algebra in the same course, unlike in "real life" i.e. research. anyway they seemed to like it, perhaps for its freshness, after all the tedious matrices. of course when i tried to ilustrate the theorem with my own head, my bald spot made it seem I was giving myself an unfair advantage.