Pairing Symmetries for Euclidean and Spherical Frameworks

We consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in $${\mathbb {R}}^d$$ R d . In particular, for a graph $$G=(V,E)$$ G = ( V , E ) and a framework ( G ,  p ), we show that, under forced or incidental symmetry, infinitesimal rigidity for spherical frameworks with vertices in some subset $$X\subset V$$ X ⊂ V realised on the equator and point-hyperplane frameworks with the vertices in X representing hyperplanes are equivalent. We then show, again under forced or incidental symmetry, that infinitesimal rigidity properties under certain symmetry groups can be paired, or clustered, under inversion on the sphere so that infinitesimal rigidity with a given group is equivalent to infinitesimal rigidity under a paired group. The fundamental basic example is that mirror symmetric rigidity is equivalent to half-turn symmetric rigidity on the 2-sphere. With these results in hand we also deduce some combinatorial consequences for the rigidity of symmetric bar-joint and point-line frameworks.