The analysis of the rigidity and flexibility of bond-node structures and skeletal frameworks may be traced back to the 18th century mathematicians Augustin-Louis Cauchy and Leonhard Euler and their considerations of polyhedra with hinged faces. They showed in particular that convex triangulated structures, such as geodesic domes, which have rigid bars connected together at their endpoints, by universal joints, are inherently rigid. In more recent times bond-node frameworks have played a vital role in mathematical models for crystals and materials, with framework bars representing strong bonds between particular atoms or between rigid polyhedral units. In particular, material zeolites provide diverse periodic networks of corner-linked regular tetrahedra, with striking geometric and topological structure. It has been found, moreover, that the low energy excitation modes of a crystal, the so-called rigid unit modes (RUMs), or zero modes, are discernible from an infinitesimal rigidity analysis of the corresponding infinite bar and joint framework.
The main aims of the project are to develop a deeper understanding of the rigidity and flexibility of
(1) triangulated surface structures in three dimensions, including infinite triangulations and surfaces of higher genus,
(2) periodic and aperiodic bond-node frameworks of various categories.
In the first "classical" topic it is currently an open problem to determine which partial triangulations of a classical compact surface of a particular genus yield graphs with generically rigid realisations in space. A first step in this direction has been the recent characterisation by Cruickshank, Kitson and Power of generic minimal rigidity in the case of a torus with a superficial hole, or porthole. Moreover, the consideration of infinite triangulations leads to infinitely faceted structures, which are remarkably diverse even for a spherical surface, and to the completely new topic of determining the generic rigidity of (space embeddings) of compact and locally compact graphs.
The second "modern" topic aims to extend the theory of the rigid unit mode spectrum to the bond-node frameworks of bordered crystals, bicrystals and aperiodic crystals, including quasicrystals.
In these varied new directions for bond-node structures there is great potential for the enrichment of the mathematical models used in Material Science. At the same time the computation of the new invariants will benefit from the methodology of simulation and computation familiar to applied scientists.
