Almost all circle polyhedra are rigid

John C. Bowers, Philip L. Bowers, Kevin Pratt

We verify the infinitesimal inversive rigidity of almost all triangulated circle polyhedra in the Euclidean plane E^2, as well as the infinitesimal inversive rigidity of tangency circle packings on the 2-sphere S^2. From this the rigidity of almost all triangulated circle polyhedra follows. The proof adapts Gluck's proof in [7] of the rigidity of almost all Euclidean polyhedra to the setting of circle polyhedra, where inversive distances replace Euclidean distances and Moebius transformations replace rigid Euclidean motions.