Symmetry, Automorphisms, and Self-duality of Infinite Planar Graphs and Tilings

We consider tilings of the plane whose graphs are locally finite and 3-connected. We show that if the automorphism group of the graph has finitely many orbits, then there is an isomorphic tiling in either the Euclidean or hyperbolic plane such that the group of automorphisms acts as a group of isometries. We apply this fact to the classification of self-dual doubly periodic tilings via Coxeter’s 2-color groups. 1. Definitions By a tiling, T = (G, p) we mean a tame embedding p of an infinite, locally finite three–connected graph G into the plane whose geometric dual G∗ is also locally finite (and necessarily three-connected.) We say an embedding of a graph is tame if it is piecewise linear and the image of the set of vertices has no accumulation points. Each such tiling realizes the plane as a regular CW–complex whose vertices, edges and faces we will refer to indiscriminately as cells. It can be shown [9] that all such graphs can be embedded so that the edges are straight lines and the faces are convex, however we make no such requirements. The boundary ∂X of a subcomplex X is the set of all cells which belong both to a cell contained in X as well as a cell not contained in X. Given a CW-complex X, the barycentric subdivision of X, B(X), is the simplicial complex induced by the flags of X, however it is sufficient for our purposes to form B(X) adding one new vertex to the interior of every edge and face and joining them as in Figure 1. If X is a plane tiling and each face is the convex hull of its vertices, then we can