Frameworks, Symmetry and Rigidity Supported by a London Mathematical Society Scheme 7 Grant

Symmetry equations are obtained for the rigidity matrix of a bar-joint framework in R d. This leads to a short proof of the Fowler-Guest symmetry group generalisation of the Calladine-Maxwell counting rules. Similar symmetry equations are obtained for the Jacobian of diverse framework systems, including the constrained point-line systems that appear in CAD, body-pin frameworks, and hybrid systems of distance constrained objects. We derive generalisations of the Fowler-Guest character formula for these and once again obtain counting rules in terms of counts of symmetry-fixed elements. Also we obtain conditions for isostaticity for asymmetric frameworks, both in the presence of symmetry in subframeworks and when symmetries occur in partition-derived frameworks.