The five Platonic solids are some of the oldest known objects of mathematical study. In this talk (in joint work with David Aulicino and Pat Hooper) we discuss how understanding the flat geometry of Platonic solids leads to very interesting 20th and 21st century mathematics. In particular, we use the geometry of Riemann surfaces to construct a closed geodesic on the dodecahedron that passes through exactly one vertex, solving a long-standing open question.