Euclid's Elements concludes with the construction and classification of the so-called Platonic solids, which are the “regular” polyhedra.  While this climax to his famous text is an amazing achievement, it is far from the end of the story.  Missed by mathematicians over the following two thousand years, Euler found a unifying principle that applies to all polyhedra and that resolves, at long last, Plato's dismay at the shortcomings of the understudied branch of solid geometry.  In this talk we will look at Euclid's classification and Euler's theorem, and see how it simplifies Euclid's work and does much more.