Conference Agenda

The minisymposium will combine the classical rigidity theory of linkages in discrete and computational geometry with the theory of circle packing, and patterns, on surfaces that arose from the study of 2- and 3-manifolds in geometry and topology. The aim being to facilitate interaction between these two areas. The classical theory of rigidity goes back to work by Euler and Cauchy on triangulated Euclidean polyhedra. The general area is concerned with the problem of determining the nature of the configuration space of geometric objects. In the modern theory the objects are geometric graphs (bar-joint structures) and the graph is rigid if the configuration space is finite (up to isometries). More generally one can consider tensegrity structures where distance constraints between points can be replaced by inequality constraints. The theory of (circle, disk and sphere) packings is vast and well known, with numerous practical applications. Of particular relevance here are conditions that result in the packing being non-deformable (jammed) as well as recent work on inversive distance packings. These inversive distance circle packings generalised the much studied tangency and overlapping packings by allowing ``adjacent'' circles to be disjoint, but with the control of an inversive distance parameter that measures the separation of the circles. The potential for overlap between these areas can be easily seen by modelling a packing of disks in the plane by a tensegrity structure where each disk is replaced by a point at its centre and the constraint that the disks cannot overlap becomes the constraint that the points cannot get closer together.

A d-dimensional framework is a pair (G, p), where G=(V, E) is a graph and p is a map from V to the d-dimensional Euclidean space. An infinitesimal motion of (G, p) is another map from V to R^d such that moving each point of the framework in that direction does not change the distances corresponding to edges in the first order. The framework is infinitesimally rigid if all of its infinitesimal motions correspond to some isometries of R^d.

Laman (1970) characterized the infinitesimal rigidity of bar-joint frameworks in the plane when the framework is in generic position, that is, when the coordinates of the points are algebraically independent over the field of rationals. Adiprasito and Nevo (2018) recently asked the following question: Which graph classes have infinitesimally rigid realizations for each of its members on a fixed constant number of points in R^d. They showed that triangulated planar graphs have such realizations on 76 points in R^3, however, for each constant c and for d>1, there always exists a graph in the class of generically rigid graphs in R^d that cannot be realized as an infinitesimally rigid bar-joint framework on any c points in R^d.

Based on the above results, it is a natural question whether planar graphs which are generically rigid in the plane have an infinitesimally rigid realization on a constant number of points of the plane. The main result of my talk is that every planar graph which is generically rigid in the plane has an infinitesimally rigid realization on 26 points of the plane. Moreover, given any set of 26 points in the plane such that the coordinates of the points are algebraically independent over the field of rationals, one can find an infinitesimally rigid realization of any rigid planar graph on that set.

A (bar-joint) framework (G,p) in R^d is the combination of a graph G and a map p assigning positions to the vertices of G. A framework is rigid if the only edge-length-preserving continuous motions of the vertices arise from isometries of R^d. The framework is globally rigid if every other framework with the same edge lengths arises from isometries of R^d. Both rigidity and global rigidity, generically, are well understood when d=2. A linearly constrained framework in R^d is a generalisation of framework in which some vertices are constrained to lie on one or more given hyperplanes. Streinu and Theran characterised rigid linearly constrained generic frameworks in R^2 in 2010. In this talk I will discuss an analogous result for the global rigidity of linearly constrained generic frameworks. This is joint work with Hakan Guler and Bill Jackson.

We will explain how Rivin's realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric is equivalent to a discrete conformal uniformization theorem for spheres, and how both can be proved in a constructive way using a convex variational principle.

We develop a combinatorial rigidity theory for symmetric bar-joint frameworks in a general finite dimensional normed space. In the case of rotational symmetry, matroidal Maxwell-type sparsity counts are identified for a large class of d-dimensional normed spaces (including all lp spaces with pnot=2). Complete combinatorial characterisations are obtained for half-turn rotation in the l1 and l-infinity plane. As a key tool, a new Henneberg-type inductive construction is developed for the matroidal class of (2,2,0)-gain-tight graphs. This is joint work with Anthony Nixon and Bernd Schulze.