Hyperbanana Graphs

TL;DR

This paper introduces hyperbanana graphs, a family of flexible frameworks in three or more dimensions, demonstrating that Maxwell's counting conditions are not sufficient for rigidity beyond two dimensions.

Contribution

It generalizes the double banana framework to hyperbananas, providing counterexamples to the extension of Laman's theorem in higher dimensions.

Findings

Hyperbanana graphs are flexible in dimensions 3 and higher.

They serve as counterexamples to the sufficiency of Maxwell's conditions.

The work extends understanding of rigidity and flexibility in high-dimensional frameworks.

Abstract

A bar-and-joint framework is a finite set of points together with specified distances between selected pairs. In rigidity theory we seek to understand when the remaining 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 framework is flexible; otherwise, it is rigid. Counting conditions due to Maxwell give a necessary combinatorial criterion for generic minimal bar-and-joint rigidity in all dimensions. Laman showed that these conditions are also sufficient for frameworks in R^2. However, the flexible "double banana" shows that Maxwell's conditions are not sufficient to guarantee rigidity in R^3. We present a generalization of the double banana to a family of hyperbananas. In dimensions 3 and higher, these are (infinitesimally) flexible, providing counterexamples to the natural generalization of Laman's theorem.