Flexible circuits in the d ‐dimensional rigidity matroid

A bar‐joint framework ( G , p ) in R d is rigid if the only edge‐length preserving continuous motions of the vertices arise from isometries of R d . It is known that, when ( G , p ) is generic, its rigidity depends only on the underlying graph G , and is determined by the rank of the edge set of G in the generic d ‐dimensional rigidity matroid ℛ d . Complete combinatorial descriptions of the rank function of this matroid are known when d = 1 , 2 , and imply that all circuits in ℛ d are generically rigid in R d when d = 1 , 2 . Determining the rank function of ℛ d is a long standing open problem when d ≥ 3 , and the existence of nonrigid circuits in ℛ d for d ≥ 3 is a major contributing factor to why this problem is so difficult. We begin a study of nonrigid circuits by characterising the nonrigid circuits in ℛ d which have at most d + 6 vertices.