# Not the Nauru graph

The orthogonal polyhedron shown below has an interesting combination of properties: it is orthogonally convex (any axis-parallel line intersects it in a point, an interval, or the empty set) but not simply-connected.

My first thought on completing this drawing was: its skeleton is a highly symmetric cubic torus graph with 24 vertices...must be the Nauru graph! But it isn't. It's vertex-transitive but, unlike the Nauru graph, not edge-transitive: the edges that wrap the short way around the torus each belong to only two six-cycles (the two faces on either side of the edge) but the edges that wrap the long way around belong to three (the two faces and their equatorial cycle).