Department of Engineering

In the last thirty years, there have been parallel interests in the rigidity of structures from the points of view of engineering and mathematics, especially discrete geometry. In engineering, one is interested in the behavior of a structure under some collection of possible loads, and from the point of view of geometry, there are several very basic rigidity questions relating points and distance constraints among the points. For example, in 1813, Cauchy showed that any bar-and-joint framework coming from a convex triangulated surface was rigid, where the vertices were nodes and the edges were bars. More recently, there are several geometric results that are inspired from some of the models used in engineering. For example, there are methods to compute the rigidity of structures, built with cables and struts called tensegrities, which can be well described as a collection of points with upper and lower bounds on certain pairs of distances between the points. When a tensegrity has some symmetry, the representation theory of finite groups can be quite useful in calculating its rigidity. Packings of granular material, such as frictionless spherical balls, can be described as points with lower bounds on the pairwise distances. The statics of an appropriately jammed packing can be quite useful in describing its behavior. And there is much, much more.

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