One brick at a time: a survey of inductive constructions in rigidity theory

We present a survey of results concerning the use of inductive constructions to study the rigidity of frameworks. By inductive constructions we mean simple graph moves which can be shown to preserve the rigidity of the corresponding framework. We describe a number of cases in which characterisations of rigidity were proved by inductive constructions. That is, by identifying recursive operations that preserved rigidity and proving that these operations were sufficient to generate all such frameworks. We also outline the use of inductive constructions in some recent areas of particularly active interest, namely symmetric and periodic frameworks, frameworks on surfaces, and body-bar frameworks. We summarize the key outstanding open problems related to inductions.Comment: 24 pages, 12 figures, final versio

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

Mobility of a class of perforated polyhedra

A class of over-braced but typically flexible body-hinge frameworks is described. They are based on polyhedra with rigid faces where an independent subset of faces has been replaced by a set of holes. The contact polyhedron C describing the bodies (vertices of C) and their connecting joints (edges of C) is derived by subdivision of the edges of an underlying cubic polyhedron. Symmetry calculations detect flexibility not revealed by counting alone. A generic symmetry-extended version of the Grübler-Kutzbach mobility counting rule accounts for the net mobilities of infinite families of this type (based on subdivisions of prisms, wedges, barrels, and some general inflations of a parent polyhedron). The prisms with all faces even and all barrels are found to generate flexible perforated polyhedra under the subdivision construction. The investigation was inspired by a question raised by Walter Whiteley about a perforated polyhedron with a unique mechanism reducing octahedral to tetrahedral symmetry. It turns out that the perforated polyhedron with highest (OhOh) point-group symmetry based on subdivision of the cube is mechanically equivalent to the Hoberman Switch-Pitch toy. Both objects exhibit an exactly similar mechanism that preserves TdTd subgroup symmetry over a finite range; this mechanism survives in two variants suggested by Bob Connelly and Barbara Heys that have the same contact graph, but lower initial maximum symmetry.Supported by EPSRC First Grant EP/M013642/1.This is the final version of the article. It first appeared from Elsevier via https://doi.org/10.1016/j.ijsolstr.2016.02.00

Projective plane graphs and 3-rigidity

A P-graph is a simple graph G which is embeddable in the real projective plane P. A (3,6)-tight P-graph is shown to be constructible from one of 8 uncontractible P-graphs by a sequence of vertex splitting moves. Also it is shown that a P-graph is minimally generically 3-rigid if and only if it is (3,6)-tight. In particular this characterisation holds for graphs that are embeddable in the M\"{o}bius strip.Comment: 21 pages, 21 Figure

Partial triangulations of surfaces with girth constraints

Barnette and Edelson have shown that there are finitely many minimal triangulations of a connected compact 2-manifold M. A similar finiteness result is obtained for cellular partial triangulations that satisfy the Maxwell count 3v-e=6 and girth inequality constraints for certain embedded cycles. Also a characterisation of cellular M-embedded (3,6)-tight graphs is given in terms of the satisfaction of higher genus girth inequalities.Comment: 20 pages, 5 figure

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

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