"Rigidity, flexibility, and symmetry of structures"

Walter Whiteley
York University

Monday, September 21, 2009
4:00 pm, 845 Patterson Office Tower

Abstract:

Over the last 30 years, a substantial body of results on first-order (infinitesimal) and finite rigidity has been developed. The results have both a geometric side (which frameworks G(p) are rigid) and a combinatorial side (which graphs G make rigid frameworks at almost all configurations p). In this work, the underlying projective geometry of the configuration is central to whether a framework G(p) looses the generic rigidity of the graph G. Because of this projective invariance, all the basic theory has applied to all the metrics coming from the shared projective geometry: Euclidean, Spherical, Hyperbolic, de Sitter, ... . We will present a quick overview of these results, with physical models, figures, and some applications (such as biomolecules).

With this background, we will also describe some recent results characterizing when symmetry changes the rigidity of a structure. Because the geometric rigidity (and combinatorial rigidity) are presented through matrices, the characters of the group representations for the symmetry group become key to predicting when symmetry has no impact, and when there is a loss of rigidity. With some models and diagrams, we present an overview of recent results of a number of researchers, including the results of the 2009 Thesis of Bernd Schulze at York. Among other things, these symmetry results predict many of the classical mechanisms.