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A vertex-transitive map $X$ is a map on a surface on which the automorphismgroup of $X$ acts transitively on the set of vertices of $X$. If theface-cycles at all the vertices in a map are of same type then the map iscalled a semi-equivelar map. Clearly, a vertex-transitive map issemi-equivelar. Converse of this is not true in general. In particular, thereare semi-equivelar maps on the torus, on the Klein bottle and on the surfacesof Euler characteristics $-1$ $\&$ $-2$ 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 $\mathbb{S}^2$.Here we show that there is exactly one semi-equivelar map on $\mathbb{S}^2$which is not vertex-transitive. More precisely, we show that a semi-equivelarmap on $\mathbb{S}^2$ is the boundary of a Platonic solid, an Archimedeansolid, a regular prism, an antiprism or the pseudorhombicuboctahedron. As aconsequence, we show that all the semi-equivelar maps on $\mathbb{RP}^2$ arevertex-transitive. Moreover, every semi-equivelar map on $\mathbb{S}^2$ can begeometrized, i.e., every semi-equivelar map on $\mathbb{S}^2$ is isomorphic toa semi-regular tiling of $\mathbb{S}^2$. In the course of the proof of our mainresult, we present a combinatorial characterization in terms of an inequalityof all the types of semi-equivelar maps on $\mathbb{S}^2$. Here, we presentself-contained combinatorial proofs of all our results.

Platonic solids, Archimedean solids and semi-equivelar maps on the sphere