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 in~\cite{gluck75} of the rigidity of almost all Euclidean polyhedra to the setting of circle polyhedra, where inversive distances replace Euclidean distances and Möbius transformations replace rigid Euclidean motions.