We study a formula, due to Roger Penrose, which gives the number T(G) of edge-3-colorings of a cubic connected plane graph G in terms of certain families of cycles of G. We present an equivalent formula which gives T(G) in terms of the set of embeddings of G on orientable surfaces. Then, using the theory of bicycles and leftright paths of Rosenstiehl and Shank, we obtain another equivalent formula, which refers only to algebraic properties of G. Finally, we prove the new formula for all cubic connected graphs, thus generalizing Penrose's formula to non-planar graphs.