Bar-joint Frameworks Research Articles

Coincident-point rigidity in normed planes

A bar-joint framework (G,p) is the combination of a graph G and a map p assigning positions, in some space, to the vertices of G. The framework is rigid if every edge-length-preserving continuous motion of the vertices arises from an isometry of the space. We will analyse rigidity when the space is a (non-Euclidean) normed plane and two designated vertices are mapped to the same position. This non-genericity assumption leads us to a count matroid first introduced by Jackson, Kaszanitsky and the third author. We show that independence in this matroid is equivalent to independence as a suitably regular bar-joint framework in a normed plane with two coincident points; this characterises when a regular normed plane coincident-point framework is rigid and allows us to deduce a delete-contract characterisation. We then apply this result to show that an important construction operation (generalised vertex splitting) preserves the stronger property of global rigidity in normed planes and use this to construct rich families of globally rigid graphs when the normed plane is analytic.

Equilibrium stresses and rigidity for infinite tensegrities and frameworks

Asymptotic equilibrium stresses are defined for countably infinite tensegrities and generalisations of the Roth-Whiteley characterisation of first-order rigidity are obtained. Generalisations of prestress stability and second order rigidity are given for countably infinite bar-joint frameworks and are shown to give sufficient conditions for continuous rigidity relative to certain prescribed motions. The proofs are based on a new short proof for finite frameworks that prestress stability ensures continuous rigidity.

Corrigendum to ‘flexible circuits in the d $d$‐dimensional rigidity matroid’

A bar-joint framework $(G,p)$ in $\mathbb{R}^d$ is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of $\mathbb{R}^d$. It is known that, when $(G,p)$ is generic, its rigidity depends only on the underlying graph $G$, and is determined by the rank of the edge set of $G$ in the generic $d$-dimensional rigidity matroid $\mathcal{R}_d$. Complete combinatorial descriptions of the rank function of this matroid are known when $d=1,2$, and imply that all circuits in $\mathcal{R}_d$ are generically rigid in $\mathbb{R}^d$ when $d=1,2$. Determining the rank function of $\mathcal{R}_d$ is a long standing open problem when $d\geq 3$, and the existence of non-rigid circuits in $\mathcal{R}_d$ for $d\geq 3$ is a major contributing factor to why this problem is so difficult. We begin a study of non-rigid circuits by characterising the non-rigid circuits in $\mathcal{R}_d$ which have at most $d+6$ vertices.

Frameworks with Coordinated Edge Motions

We develop a rigidity theory for bar-joint frameworks in Euclidean $d$-space in which specified classes of edges are allowed to change length in a coordinated fashion that requires differences of lengths to be preserved within each class. Rigidity for these coordinated frameworks is a generic property, and we characterize the rigid graphs in terms of redundant rigidity in the standard $d$-dimensional rigidity matroid. We also interpret our main results in terms of matroid unions.

Coincident Rigidity of 2-Dimensional Frameworks

Fekete, Jord\'an and Kaszanitzky [4] characterised the graphs which can be realised as 2-dimensional, infinitesimally rigid, bar-joint frameworks in which two given vertices are coincident. We formulate a conjecture which would extend their characterisation to an arbitrary set T of vertices and verify our conjecture when |T| = 3.

Linear zero mode spectra for quasicrystals

A converse is given to the well-known fact that a hyperplane localised zero mode of a crystallographic bar-joint framework gives rise to a line or lines in the zero mode (RUM) spectrum. These connections motivate definitions of linear zero mode spectra for an aperiodic bar-joint framework G that are based on relatively dense sets of linearly localised flexes. For a Delone framework in the plane the limit spectrumLlim(G,a_) is defined in this way, as a subset of the reciprocal space for a reference basis a_ of the ambient space. A smaller spectrum, the slippage spectrumLslip(G,a_), is also defined. For quasicrystal parallelogram frameworks associated with regular multi-grids, in the sense of de Bruijn and Beenker, these spectra coincide and are determined in terms of the geometry of G.

Generalised rigid body motions in non-Euclidean planes with applications to global rigidity

A bar-joint framework (G,p) in a (non-Euclidean) real normed plane X is the combination of a finite, simple graph G and a placement p of the vertices in X. A framework (G,p) is globally rigid in X if every other framework (G,q) in X with the same edge lengths as (G,p) arises from an isometry of X. The weaker property of local rigidity in normed planes (where only (G,q) within a neighbourhood of (G,p) are considered) has been studied by several researchers over the last 5 years after being introduced by Kitson and Power for ℓp-norms. However global rigidity is an unexplored area for general normed spaces, despite being intensely studied in the Euclidean context by many groups over the last 40 years. In order to understand global rigidity in X, we introduce new generalised rigid body motions in normed planes where the norm is determined by an analytic function. This theory allows us to deduce several geometric and combinatorial results concerning the global rigidity of bar-joint frameworks in X.

On rigid origami III: local rigidity analysis.

In this article, rigid origami is examined from the perspective of rigidity theory. First- and second-order rigidity are defined from local differential analysis of the consistency constraint; while the static rigidity and prestress stability are defined after finding the form of internal force and load. We will show the hierarchical relation among these local rigidities with examples representing different levels. The development of theory here follows the same path as the conventional rigidity theory for bar-joint frameworks, but starts with different high-order rotational constraints. We also bring new interpretation to the internal force and geometric error of constraints associated with energy. Examining the different aspects of the rigidity of origami might give a novel perspective for the development of new folding patterns, or for the design of origami structures where some rigidity is required.

Rigidity through a Projective Lens

In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar–joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas.

Crystal flex bases and the RUM spectrum

A theory of infinite spanning sets and bases is developed for the first-order flex space of an infinite bar-joint framework, together with space group symmetric versions for a crystallographic bar-joint framework ${{\mathcal {C}}}$. The existence of a crystal flex basis for ${{\mathcal {C}}}$ is shown to be closely related to the spectral analysis of the rigid unit mode (RUM) spectrum of ${{\mathcal {C}}}$ and an associated geometric flex spectrum. Additionally, infinite spanning sets and bases are computed for a range of fundamental crystallographic bar-joint frameworks, including the honeycomb (graphene) framework, the octahedron (perovskite) framework and the 2D and 3D kagome frameworks.

Symbol functions for symmetric frameworks

We prove a variant of the well-known result that intertwiners for the bilateral shift on ℓ2(Z) are unitarily equivalent to multiplication operators on L2(T). This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts.

Parallelogram frameworks and flexible quasicrystals

The first-order flex space of the bar-joint framework $G_P$ of a parallelogram tiling $P$ is determined in terms of an explicit free basis. Applications are given to braced parallelogram frameworks and to quasicrystal frameworks associated with multigrids in the sense of de Bruijn and Beenker. In particular we characterise rigid bracing patterns, identify quasicrystal frameworks with finite dimensional flex spaces, and define a zero mode spectrum.

Global rigidity of periodic graphs under fixed-lattice representations

In [9] Hendrickson proved that (d+1)-connectivity and redundant rigidity are necessary conditions for a generic (non-complete) bar-joint framework to be globally rigid in Rd. Jackson and Jordán [10] confirmed that these conditions are also sufficient in R2, giving a combinatorial characterization of graphs whose generic realizations in R2 are globally rigid. In this paper, we establish analogues of these results for infinite periodic frameworks under fixed lattice representations. Our combinatorial characterization of globally rigid generic periodic frameworks in R2 in particular implies toroidal and cylindrical counterparts of the theorem by Jackson and Jordán.

Rigidity of symmetric frameworks in normed spaces

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 p≠2). Complete combinatorial characterisations are obtained for half-turn rotation in the ℓ1 and ℓ∞-plane. As a key tool, a new Henneberg-type inductive construction is developed for the matroidal class of (2,2,0)-gain-tight gain graphs.

Graph rigidity for unitarily invariant matrix norms

A rigidity theory is developed for bar-joint frameworks in linear matrix spaces endowed with a unitarily invariant matrix norm. Analogues of Maxwell's counting criteria are obtained and minimally rigid matrix frameworks are shown to belong to the matroidal class of (k,l)-sparse graphs for suitable k and l. An edge-colouring technique is developed to characterise infinitesimal rigidity for product norms and then applied to show that the graph of a minimally rigid bar-joint framework in the space of 2×2 symmetric (respectively, hermitian) matrices with the trace norm admits an edge-disjoint packing consisting of a (Euclidean) rigid graph and a spanning tree.

Constructing Isostatic Frameworks for the $\ell^1$ and $\ell^\infty$ Plane

We use a new coloured multi-graph constructive method to prove that if the edge-set of a graph $G=(V,E)$ has a partition into two spanning trees $T_1$ and $T_2$ then there is a map $p:V\to \mathbb{R}^2$, $p(v)=(p_1(v),p_2(v))$, such that $|p_i(u)-p_i(v)| \geqslant |p_{3-i}(u)-p_{3-i}(v)|$ for every edge $uv$ in $T_i$ ($i=1,2$). As a consequence, we solve an open problem on the realisability of minimally rigid bar-joint frameworks in the $\ell^1$ or $\ell^\infty$-plane. We also show how to adapt this technique to incorporate symmetry and indicate several related open problems on rigidity, redundant rigidity and forced symmetric rigidity in normed spaces.

Pairing Symmetries for Euclidean and Spherical Frameworks

We consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in {mathbb {R}}^d. In particular, for a graph G=(V,E) and a framework (G, p), we show that, under forced or incidental symmetry, infinitesimal rigidity for spherical frameworks with vertices in some subset Xsubset V realised on the equator and point-hyperplane frameworks with the vertices in X representing hyperplanes are equivalent. We then show, again under forced or incidental symmetry, that infinitesimal rigidity properties under certain symmetry groups can be paired, or clustered, under inversion on the sphere so that infinitesimal rigidity with a given group is equivalent to infinitesimal rigidity under a paired group. The fundamental basic example is that mirror symmetric rigidity is equivalent to half-turn symmetric rigidity on the 2-sphere. With these results in hand we also deduce some combinatorial consequences for the rigidity of symmetric bar-joint and point-line frameworks.

Double-Distance Frameworks and Mixed Sparsity Graphs

A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint frameworks in a variety of such contexts. The main results are combinatorial characterisations for (i) frameworks restricted to surfaces with both Euclidean and geodesic distance constraints, (ii) frameworks in the plane with Euclidean and non-Euclidean distance constraints, and (iii) direction-length frameworks in the non-Euclidean plane.

Algebraic spectral synthesis and crystal rigidity

A spectral synthesis property is obtained for closed shift-invariant subspaces of vector-valued functions on the lattice Zd. This result generalises Marcel Lefranc's 1958 theorem for scalar-valued functions. Applications are given to homogeneous systems of multivariable vector-valued discrete difference equations and to the first-order flexibility of crystallographic bar-joint frameworks.

Create a Rigid and Safe Grid-Like Structure

The failure of the building is not the consequence of the not strong enough elements of the structure in the majority of the cases, but the bracing elements inappropriate places. We consider a 4 × 4 and 4 × 5 braced frames to understand the connections between the lateral stiffnesses, and bracing graph to achieve the stiffest and the more safety design. In our consideration, we study those relationships that based on our frame using their finite element analyses and some new result in optimizations of structural design. We offer some conclusions, including perspectives and future developments in the rigidity of scaffolds and tall building as symmetrical and grid-like bar-joint frameworks.