Hyperbanana Graphs

A bar-and-joint framework is a finite set of points to-gether with specified distances between selected pairs. In rigidity theory we seek to understand when the re-maining pairwise distances are also fixed. If there exists a pair of points which move relative to one another while maintaining the given distance constraints, the frame-work is flexible; otherwise, it is rigid. Counting conditions due to Maxwell give a neces-sary combinatorial criterion for generic minimal bar-and-joint rigidity in all dimensions. Laman showed that these conditions are also sufficient for frameworks in R2. However, the flexible “double banana ” shows that Maxwell’s conditions are not sufficient to guaran-tee rigidity in R3. We present a generalization of the double banana to a family of hyperbananas. In dimen-sions 3 and higher, these are (infinitesimally) flexible, providing counterexamples to the natural generalization of Laman’s theorem.