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Abstract
A symmetry extension of Maxwell's rule for rigidity of frames shows that every fully triangulated torus has at least six states of self-stress with a well defined set of symmetries related to rotational and translational representations in the point group. In contrast, in trivalent polyhedral toroidal frames, i.e. the duals of the deltahedra, the rigid body motions plus the mechanisms are at least equal in number to the bars, and their symmetry representation contains that of a set of vector slides along the bars.
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