A vertex-transitive map X is a map on a surface on which the automorphism group of X acts transitively on the set of vertices of X. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. In particular, there are semi-equivelar maps on the torus, on the Klein bottle and on the surfaces of Euler characteristics −1&−2
Nearby in the stack
which are not vertex-transitive. It is known that the boundaries of Platonic solids, Archimedean solids, regular prisms and antiprisms are vertex-transitive maps on
S2
. Here we show that there is exactly one semi-equivelar map on
S2
which is not vertex-transitive. More precisely, we show that a semi-equivelar map on
S2
is the boundary of a Platonic solid, an Archimedean solid, a regular prism, an antiprism or the pseudorhombicuboctahedron. As a consequence, we show that all the semi-equivelar maps on
RP2
are vertex-transitive. Moreover, every semi-equivelar map on
S2
can be geometrized, i.e., every semi-equivelar map on
S2
is isomorphic to a semi-regular tiling of
S2
. In the course of the proof of our main result, we present a combinatorial characterization in terms of an inequality of all the types of semi-equivelar maps on
S2
. Here, we present self-contained combinatorial proofs of all our results.