  Every finite, self-dual, regular (or chiral) 4-polytope of type {3,q,3} has a
trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping
self-duality, we obtain a construction for semisymmetric trivalent graphs
(which are edge- but not vertex-transitive). In particular, the Gray graph
arises as the medial layer graph of a certain universal locally toroidal
regular 4-polytope.
