The orbit rigidity matrix of a symmetric framework

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.Comment: 41 pages, 12 figure

Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications

Abstract. Let A be an n by N real-valued matrix with n < N; we count the number of k-faces fk(AQ) when Q is either the standard N-dimensional hypercube IN or else the positive orthant RN +. To state results simply, consider a proportional-growth asymptotic, where for fixed δ, ρ in (0, 1), we have a sequence of matrices An,Nn and of integers kn with n/Nn → δ, kn/n → ρ as n → ∞. If each matrix An,Nn has its columns in general position, then fk(AIN)/fk(I N) tends to zero or one depending on whether ρ> min(0, 2 − δ−1) or ρ < min(0, 2 − δ−1). Also, if each An,Nn is a random draw from a distribution which is invariant under right multiplication by signed permutations, then fk(ARN +)/fk(RN +) tends almost surely to zero or one depending on whether ρ> min(0, 2 − δ−1) or ρ < min(0, 2 − δ−1). We make a variety of contrasts to related work on projections of the simplex and/or cross-polytope. These geometric face-counting results have implications for signal processing, information theory, inverse problems, and optimization. Indeed, face counting is related to conditions for uniqueness of solutions of underdetermine