Euclid's monumental treatise The Elements culminates in the proof that there are exactly five regular polyhedra. In fact, it has been asserted that this theorem was the very purpose of the work. (There's a philosophical side to it—the four elements of nature and the universe, no less.) It is relatively simple to prove that there can be none other than these five, but to construct them is tricky for the icosahedron and, even more so, for the dodecahedron. In the fourth century AD, the great Greek geometer Pappus (who lives on in modern theorems called Pappus-Pascal and Pappus-Guldin) found a simpler construction than Euclid's. He used four parallel circles on the sphere, each one through five vertices of the dodecahedron. Here is the accompanying figure, as given by Thomas Heath in The thirteen books of Euclid's Elements, Vol. III, 1908.
The three pages containing Pappus's proof as given by Heath are here or here. Heinrich Dörrie, 100 Great Problems of Elementary Mathematics (Dover, 1965) used spherical trigonometry to simplify Pappus's solution (relevant pages here or here). There was room for further improvement though, and here (also here) we present a proof requiring nothing beyond two applications of the spherical rule of cosines. It's an exercise in spherical trigonometry quite comparable to the well known elementary problem to determine the spherical distance between two places on earth. (To make the text self-contained, we included some stuff on the golden ratio.) This is our figure, adapted from Dörrie.
