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Abstract: A number of recent papers have studied when symmetry causes frameworks on agraph to become infinitesimally flexible, or stressed, and when it has noimpact. A number of other recent papers have studied special classes offrameworks on generically rigid graphs which are finite mechanisms. Here weintroduce a new tool, the orbit matrix, which connects these two areas andprovides a matrix representation for fully symmetric infinitesimal flexes, andfully symmetric stresses of symmetric frameworks. The orbit matrix is a trueanalog of the standard rigidity matrix for general frameworks, and its analysisgives important insights into questions about the flexibility and rigidity ofclasses of symmetric frameworks, in all dimensions.With this narrower focus on fully symmetric infinitesimal motions, comes thepower to predict symmetry-preserving finite mechanisms - giving a simplifiedanalysis which covers a wide range of the known mechanisms, and generalizes theclasses of known mechanisms. This initial exploration of the properties of theorbit matrix also opens up a number of new questions and possible extensions ofthe previous results, including transfer of symmetry based results fromEuclidean space to spherical, hyperbolic, and some other metrics with sharedsymmetry groups and underlying projective geometry.

Autor: Bernd Schulze, Walter Whiteley


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