Bracing frameworks consisting of parallelograms

Abstract

A rectangle in the plane can be continuously deformed preserving its edge lengths, but adding a diagonal brace prevents such a deformation. Bolker and Crapo characterized combinatorially which choices of braces make a grid of squares infinitesimally rigid using a bracing graph: a bipartite graph whose vertices are the columns and rows of the grid, and a row and column are adjacent if and only if they meet at a braced square. Duarte and Francis generalized the notion of the bracing graph to rhombic carpets, proved that the connectivity of the bracing graph implies rigidity and stated the other implication without proof. Nagy Kem gives the equivalence in the infinitesimal setting. We consider continuous deformations of braced frameworks consisting of a graph from a more general class and its placement in the plane such that every 4-cycle forms a parallelogram. We show that rigidity of such a braced framework is equivalent to the non-existence of a special edge coloring, which is in turn equivalent to the corresponding bracing graph being connected.

Flexing infinite frameworks with applications to braced Penrose tilings

“…(ii) ⇔ (iii): The proof of this for finite P-frameworks (see [12,Theorem 4.5]) extends to countably infinite P-frameworks.…”

“…To determine which braced Penrose frameworks are rigid, we shall study the much larger class of P-frameworks (Definition 4.3). First defined in [12] as the generalization of 1-skeletons of parallelogram tilings, it was shown by Grasegger and Legerský for a finite P-framework (G, ρ) with some parallelograms braced, the following properties are equivalent: (i) (G, ρ) is rigid, (ii) G has a cartesian NAC-coloring (Definition 4.7), and (iii) the bracing graph (Definition 4.5) of (G, ρ) is connected. We shall extend this result to countably infinite P-frameworks, allowing us to give a characterization for rigid braced Penrose frameworks.…”

“…We will apply the methods used in the previous section to a specific class of framework that was first introduced in [12]. We begin with the following definitions.…”

“…If G has a cartesian NAC-coloring, then (G, ρ) is flexible since the constructive method given by [12,Theorem 4.4] can be extended to countably infinite P-frameworks; see also the construction of flex ρ t in the proof of Lemma 7.4. It exploits the construction described in the proof Proposition 3.1 with carefully chosen vectors r i and b j so that the obtained flex starts at ρ.…”

“…Proof. We extend the construction introduced in [12] to countably infinite frameworks and show that it yields a C k -symmetric flex for both finite and infinite case. Let (G, ρ) be a braced C k -symmetric (countably infinite) P-framework.…”