Maxwell-Cremona meets the flat torus

Abstract: We consider three classes of geodesic embeddings of graphs on the plane and the Euclidean flat torus: graphs having a positive equilibrium stress, reciprocal graphs (for which there is an orthogonal embedding of the dual graph), and weighted Delaunay complexes. The classical Maxwell-Cremona correspondence and the well-known correspondence between convex hulls and weighted Delaunay triangulations imply that these three concepts are essentially equivalent for plane graphs. However, this three-way equivalence does not extend directly to geodesic graphs on the torus. Reciprocal and Delaunay graphs are equivalent, and every reciprocal graph is in positive equilibrium, but not every positive equilibrium graph is reciprocal. We establish a weaker correspondence: Every positive equilibrium graph on any flat torus is equivalent to a reciprocal/Delaunay graph on some flat torus. These results appeared in SoCG '20

combinatorics

Audience: researchers in the topic

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Virtual seminar on algebraic matroids and rigidity theory

Series comments: The COVID-19 pandemic is forcing us all to stay home, foregoing conferences and departmental seminars for the next few months. This weekly virtual seminar is an attempt to patch that departmental-seminar-sized void in our lives until it is safe to resume our more traditional forms of professional networking. Since geographic location matters a lot less for a virtual seminar than for an in-person seminar, this virtual seminar will be defined purely by its mathematical theme, algebraic matroids and rigidity theory, and not any particular department nor region.