Almost all circle polyhedra are rigid

Abstract

We verify the infinitesimal inversive rigidity of almost all triangulated circle polyhedra in the Euclidean plane $${\mathbb {E}}^{2}$$ , as well as the infinitesimal inversive rigidity of tangency circle packings on the 2-sphere $${\mathbb {S}}^{2}$$ . From this the rigidity of almost all triangulated circle polyhedra follows. The proof adapts Gluck’s proof (Geometric Topology, volume 238 of Lecture Notes in Mathematics, pp 225–239, 1975) of the rigidity of almost all Euclidean polyhedra to the setting of circle polyhedra, where inversive distances replace Euclidean distances and Mobius transformations replace rigid Euclidean motions.