  We study the translation surfaces obtained by considering the unfoldings of
the surfaces of Platonic solids. We show that they are all lattice surfaces and
we compute the topology of the associated Teichm\"uller curves. Using an
algorithm that can be used generally to compute Teichm\"uller curves of
translation covers of primitive lattice surfaces, we show that the
Teichm\"uller curve of the unfolded dodecahedron has genus 131 with 19 cone
singularities and 362 cusps. We provide both theoretical and rigorous
computer-assisted proofs that there are no closed saddle connections on the
surfaces associated to the tetrahedron, octahedron, cube, and icosahedron. We
show that there are exactly 31 equivalence classes of closed saddle connections
on the dodecahedron, where equivalence is defined up to affine automorphisms of
the translation cover. Techniques established here apply more generally to
Platonic surfaces and even more generally to translation covers of primitive
lattice surfaces and their Euclidean cone surface and billiard table quotients.
