Infinite bond-node frameworks

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 particular three new notions, namely the geometric spectrum, crystal flex complexity, and the mean flexibility dimension (generalising the RUM dimension), provide promising new invariants and signatures for a deeper understanding of these structures.

Additionally the project will consider mathematical limits of 3-periodic bond-node structures which lie in a new category of constraint systems introduced recently by Power and Schulze, namely "string-node meshes". These are bond-node networks which are "ultranano" in the sense that the set of nodes is dense in space. For these abstract fibred structures both strain-free rigidity phenomena and finite motion phenomena are possible, even in critically coordinated cases, and the project will tackle the analysis of the prediction and quantification of such phenomena.

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.

Combinatorial and Geometric Rigidity theory combines a rich mathematical theory and diverse applications in science and technology. As such the research proposal provides an opportunity to deliver impact across a broad spectrum. The main impact pathways are (i) New applications of Discrete Mathematics and Analysis, (ii) Methodological impacts in Engineering, Chemistry and Materials Science, and (iii) Postdoctoral Career development.

The research will pursue investigations in Combinatorics and Analysis necessary for a deeper understanding of the flexibility and rigidity of discrete bond-node structures. This impact pathway will be effectively realised by collaborations between the PI, the co-investigator Derek Kitson, a postdoctoral research associate and consultations and collaborations with pure and applied mathematicians.

Bond-node models are ubiquitous in the sciences for modelling the rigidity and flexibility of structures, from engineered and micro-engineered structures down to the small scale of zeolites and molecular crystals. The research project has the potential to impact on these methodologies in the longer term. The establishment of the Materials Science Institute at Lancaster in 2016 has already opened up potential collaborations and in the Lancaster workshops of the proposal we intend to invite a range of applied scientists, including engineers, chemists and condensed matter scientists. In Materials Science the crystallographic methodologies which rest on Maxwell counting analysis in critically coordinated bond-node structures will be enriched both through functional analytic methods and through new paradigm models and invariants.

A key impact of the project will be the training of a PDRA in applications of Combinatorics and Analysis. He or she would be involved in international collaboration and cross-discipline activities and career-development will enhanced by participation in national and international workshops.