The Rigidity of Spherical Frameworks: Swapping Blocks and Holes

A significant range of geometric structures whose rigidity is explored for both practical and theoretical purposes are formed by modifying generically isostatic triangulated spheres. In the block and hole structures (P, p), some edges are removed to make holes, and other edges are added to create rigid sub-structures called blocks. Previous work noted a combinatorial analogy in which blocks and holes played equivalent roles. In this paper, we connect stresses in such a structure (P, p) to first-order motions in a swapped structure (P', p), where holes become blocks and blocks become holes. When the initial structure is geometrically isostatic, this shows that the swapped structure is also geometrically isostatic, giving the strongest possible correspondence. We use a projective geometric presentation of the statics and the motions, to make the key underlying correspondences transparent.Comment: 36 pages, 9 figure

Finite motions from periodic frameworks with added symmetry

Recent work from authors across disciplines has made substantial contributions to counting rules (Maxwell type theorems) which predict when an infinite periodic structure would be rigid or flexible while preserving the periodic pattern, as an engineering type framework, or equivalently, as an idealized molecular framework. Other work has shown that for finite frameworks, introducing symmetry modifies the previous general counts, and under some circumstances this symmetrized Maxwell type count can predict added finite flexibility in the structure. In this paper we combine these approaches to present new Maxwell type counts for the columns and rows of a modified orbit matrix for structures that have both a periodic structure and additional symmetry within the periodic cells. In a number of cases, this count for the combined group of symmetry operations demonstrates there is added finite flexibility in what would have been rigid when realized without the symmetry. Given that many crystal structures have these added symmetries, and that their flexibility may be key to their physical and chemical properties, we present a summary of the results as a way to generate further developments of both a practical and theoretic interest.Comment: 45 pages, 13 figure

La Division de Sommet dans les Charpentes Isostatiques

On démontre que des divisions de sommet le long de 0, 1 ou 2 arêtes d'une charpente de barres et de joints dans I'espace tridimensionnel respectent I'indépendance pour p resque toutes les positions du nouveau sommet. On démontre, en corollaire, que les divisions de sommet le long de 2 arêtes d'une charpente dans I'espace tridimensionnel préservent la rigidité stati que pour presque toutes les positions du nouveau sommet. On applique cette technique à la rigidité générique des surfaces triangu lées dans I'espace tridimensionnel, incluant toutes les sphères et les plans projectifs. Des analogues sont énoncés pour les espaces à n dimensionsWe show that vertex splits on 0, 1, or 2 edges of a bar and joint framework in 3-space preserve independence for almost all positions of the new vertex. As a corollary, we show that vertex splits on 2 edges of a framework in 3-space preserve static rigidity for almost all positions of the new vertex. This technique is applied to the generic rigidity of triangulated surfaces in 3-space, including all spheres and projective planes. Analogues for n-space are given for all n.Peer Reviewe

Symmetry adapted Assur decompositions

Assur graphs are a tool originally developed by mechanical engineers to decompose mechanisms for simpler analysis and synthesis. Recent work has connected these graphs to strongly directed graphs, and decompositions of the pinned rigidity matrix. Many mechanisms have initial configurations which are symmetric, and other recent work has exploited the orbit matrix as a symmetry adapted form of the rigidity matrix. This paper explores how the decomposition and analysis of symmetric frameworks and their symmetric motions can be supported by the new symmetry adapted tools.Comment: 40 pages, 22 figure

Rigidity of frameworks on expanding spheres

A rigidity theory is developed for bar-joint frameworks in $\mathbb{R}^{d+1}$ whose vertices are constrained to lie on concentric $d$-spheres with independently variable radii. In particular, combinatorial characterisations are established for the rigidity of generic frameworks for $d=1$ with an arbitrary number of independently variable radii, and for $d=2$ with at most two variable radii. This includes a characterisation of the rigidity or flexibility of uniformly expanding spherical frameworks in $\mathbb{R}^{3}$. Due to the equivalence of the generic rigidity between Euclidean space and spherical space, these results interpolate between rigidity in 1D and 2D and to some extent between rigidity in 2D and 3D. Symmetry-adapted counts for the detection of symmetry-induced continuous flexibility in frameworks on spheres with variable radii are also provided.Comment: 22 pages, 2 figures, updated reference

Coning, symmetry and spherical frameworks

In this paper, we combine separate works on (a) the transfer of infinitesimal rigidity results from an Euclidean space to the next higher dimension by coning, (b) the further transfer of these results to spherical space via associated rigidity matrices, and (c) the prediction of finite motions from symmetric infinitesimal motions at regular points of the symmetry-derived orbit rigidity matrix. Each of these techniques is reworked and simplified to apply across several metrics, including the Minkowskian metric \M^{d} and the hyperbolic metric \H^{d}. This leads to a set of new results transferring infinitesimal and finite motions associated with corresponding symmetric frameworks among \E^{d}, cones in $E^{d+1}$, \SS^{d}, \M^{d}, and \H^{d}. We also consider the further extensions associated with the other Cayley-Klein geometries overlaid on the shared underlying projective geometry.Comment: 38 pages, 7 figure