We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial matroids and associated linear representation results that may be interesting in their own right. The same techniques immediately yield a Maxwell-Laman-type combinatorial characterization for frameworks embedded in 2-dimensional cones that arise as quotients of the plane by a finite order rotation.
@cite_18: We regard a graph G as a set 1,...,v) together with a nonempty set E of two-element subsets of (1, . . ., v). Letp = (Pi, . . . &) be an element of RIv representing v points in RI. Consider the figure G(p) in RI consisting of the line segments [pi, pj in RI for i,j) E E. The figure G (p) is said to be rigid in RI if every continuous path in RI', beginning atp and preserving the edge lengths of G(p), terminates at a point q E RIV which is the image (Tp1, . . ., Tpv) of p under an isometry T of RW. Otherwise, G (p) is flexible in RW. Our main result establishes a formula for determining whether G (p) is rigid in RI for almost all locations p of the vertices. Applications of the formula are made to complete graphs, planar graphs, convex polyhedra in R3, and other related matters.
@cite_21: 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.
@cite_24: 
@cite_2: 
@cite_15: Abstract We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks . The characterization is a true analogue of the Maxwell–Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorithms. To prove our rigidity theorem we introduce and develop periodic direction networks and Z 2 - graded-sparse colored graphs .
@cite_16: A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connects these two areas and provides a matrix representation for fully symmetric infinitesimal flexes, and fully symmetric stresses of symmetric frameworks. The orbit matrix is a true analog of the standard rigidity matrix for general frameworks, and its analysis gives important insights into questions about the flexibility and rigidity of classes of symmetric frameworks, in all dimensions. With this narrower focus on fully symmetric infinitesimal motions, comes the power to predict symmetry-preserving finite mechanisms—giving a simplified analysis which covers a wide range of the known mechanisms, and generalizes the classes of known mechanisms. This initial exploration of the properties of the orbit matrix also opens up a number of new questions and possible extensions of the previous results, including transfer of symmetry based results from Euclidean space to spherical, hyperbolic, and some other metrics with shared symmetry groups and underlying projective geometry.
@cite_13: Recent work has shown that if an isostatic bar-and-joint framework possesses nontrivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are “fixed” by various symmetry operations of the framework.
@cite_11: 
The general area of rigidity with symmetry has been somewhat active in the past few years, but the results here are independent of much of it. For completeness, we review some work along similar lines. A specialization of our [Theorem A] MT10 is due to Ross @cite_2 @cite_18 . Schulze @cite_13 @cite_11 and Schulze and Whiteley @cite_16 studied the question of when incidental'' symmetry induces non-generic behaviors in finite frameworks, which is a different setting than the forced'' symmetry we consider here and in @cite_15 . Ross, Schulze, and Whiteley @cite_21 have studied the present problems, but they do not give any combinatorial characterizations. Borcea and Streinu @cite_24 have proposed a kind of doubly generic'' periodic rigidity, where the combinatorial model does not include the colors on the quotient graph.