A Toroidal Maxwell-Cremona-Delaunay Correspondence

We consider three classes of geodesic embeddings of graphs on Euclidean flat tori:

A torus graph G is equilibrium if it is possible to place positive weights on the edges, such that the weighted edge vectors incident to each vertex of G sum to zero.
A torus graph G is reciprocal if there is a geodesic embedding of the dual graph G* on the same flat torus, where each edge of G is orthogonal to the corresponding dual edge in G*.
A torus graph G is coherent if it is possible to assign weights to the vertices, so that G is the (intrinsic) weighted Delaunay graph of its vertices.

The classical Maxwell-Cremona correspondence and the well-known correspondence between convex hulls and weighted Delaunay triangulations imply that the analogous concepts for plane graphs (with convex outer faces) are equivalent. Indeed, all three conditions are equivalent to G being the projection of the 1-skeleton of the lower convex hull of points in R³. However, this three-way equivalence does not extend directly to geodesic graphs on flat tori. On any flat torus, reciprocal and coherent graphs are equivalent, and every reciprocal graph is equilibrium, but not every equilibrium graph is reciprocal. We establish a weaker correspondence: Every equilibrium graph on any flat torus is affinely equivalent to a reciprocal/coherent graph on some flat torus.

A toroidal Maxwell-Cremona-Delaunay correspondence