# Discrete Mathematics and Geometric Rigidity

‌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 sensor development 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.

# People

Discrete Mathematics and Geometric Rigidity

Analysis and Probability, Discrete Mathematics and Geometric Rigidity

Analysis and Probability, Discrete Mathematics and Geometric Rigidity

+44 (0)1524 594914 B06a, B - Floor, Fylde College

Algebra and Geometry, Discrete Mathematics and Geometric Rigidity

+44 (0)1524 594695 B43, B - Floor, Fylde College

Analysis and Probability, Discrete Mathematics and Geometric Rigidity

+44 (0)1524 593958

Algebra and Geometry, Discrete Mathematics and Geometric Rigidity

+44 (0)1524 592173 B11, B - Floor, Fylde College

Discrete Mathematics and Geometric Rigidity

# Projects

Infinite bond-node frameworks
31/07/2017 → 31/05/2019
Research

Comibinatorial Rigidity, Symmetric Geometric Constraint Systems and Applications
01/06/2015 → 31/08/2016
Research

Crystal Frameworks, Operator Theory and Combinatorics
01/09/2012 → 31/08/2014
Research

## Workshops

2018: Bond-node structures: rigidity, combinatorics and chemistry

http://www.lancaster.ac.uk/maths/bond-node-structures-2018/

2017: Bond-node structures: rigidity, combinatorics and materials science

http://www.lancaster.ac.uk/maths/bond-node-structures/

2016: Workshop on Geometric Rigidity

http://www.lancaster.ac.uk/maths/news-and-events/

2015: Workshop on Geometric Rigidity

http://www.lancaster.ac.uk/maths/news-and-events/

2010: Rigidity of Frameworks and Applications

http://www.maths.lancs.ac.uk/~power/LancRigidFrameworks.htm