Home > Research > Departments & Centres > Geometric Rigidity

## Geometric Rigidity

### Organisation profile

Geometric Rigidity theory is concerned with the rigidity of structures which are defined by geometric constraints (fixed lengths, fixed areas, fixed directions, etc.) on a set of rigid objects (points, line segments, polygons, etc.). The origins reach back to 1813 and Cauchy's rigidity theorem for a convex triangulated polyhedron with rigid edges and flexible joints. The theory is also concerned with the infinitesimal flexibility and continuous flexibility of structures, ranging from Victorian mechanisms, to robot arms and the crystal networks of zeolites.

‌The mathematics of rigidity has both combinatorial and geometric aspects and draws on techniques from a wide range of mathematical areas, including graph theory, matroid theory, positive semidefinite programming, representation theory of symmetry groups and algebraic and projective geometry. When the structures are infinite, as in a mathematical model for a periodic crystal for example, methods from functional analysis and operator theory become relevant.

The rigidity and flexibility properties of a structure — either man-made, such as a building, bridge or mechanical linkage, or found in nature, such as a biomolecule, protein or crystal — are critical to its form, behaviour and functioning. Thus rigidity theory has many practical applications in fields such as engineering, robotics, materials science, medicine and biochemistry. The methodology of rigidity and configuration uniqueness also has applications in computer-aided design, network sensoring and NMR reconstruction.

The mathematical theory embraces on the one hand the analysis of generic structures, in which the combinatorics of connectivity play the dominant role, and, on the other hand, structures with symmetries and regularities where symmetry group actions are significant. Indeed the rigidity analysis of symmetric structures is an important area of research which has seen a dramatic increase in interest over the last few years, both in mathematics and in the applied sciences.

(114)

## Uniquely realisable graphs in analytic normed planes

Dewar, S., Hewetson, J. & Nixon, A., 22/07/2024, (E-pub ahead of print) In: International Mathematics Research Notices.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

• Forthcoming

## Universal rigidity of ladders on the line

Chen, B., Connelly, R., Gortler, S., Nixon, A. & Theran, L., 30/06/2024, (Accepted/In press) In: Studia Scientiarum Mathematicarum Hungarica: Combinatorics, Geometry and Topology.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

• Published

## Maximum likelihood thresholds via graph rigidity

Bernstein, D., Dewar, S., Gortler, S., Nixon, A., Sitharam, M. & Theran, L., 13/06/2024, In: Annals of Applied Probability. 34, 3, p. 3288-3319 32 p.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

(32)
• ## American Institute of Mathematics: Rigidity and flexibility of microstructures

Activity: Participating in or organising an event typesParticipation in workshop, seminar, course

• ## Galway Mathematics Seminar

Activity: Talk or presentation typesInvited talk

• ## SIAM conference on applied algebraic geometry

Activity: Participating in or organising an event typesParticipation in conference -Mixed Audience