The Rigidity of Spherical Frameworks: Swapping Blocks and Holes

TL;DR

This paper investigates the rigidity properties of spherical frameworks with blocks and holes, establishing a correspondence between stresses and motions when swapping these features, and using projective geometry for clarity.

Contribution

It introduces a novel stress-motion correspondence in block and hole structures, demonstrating that swapping blocks and holes preserves geometric isostaticity.

Findings

Swapped structures remain isostatic when original structures are isostatic.

A projective geometric framework clarifies the stress-motion relationship.

The work generalizes the understanding of rigidity in spherical frameworks.

Abstract

A significant range of geometric structures whose rigidity is explored for both practical and theoretical purposes are formed by modifying generically isostatic triangulated spheres. In the block and hole structures (P, p), some edges are removed to make holes, and other edges are added to create rigid sub-structures called blocks. Previous work noted a combinatorial analogy in which blocks and holes played equivalent roles. In this paper, we connect stresses in such a structure (P, p) to first-order motions in a swapped structure (P', p), where holes become blocks and blocks become holes. When the initial structure is geometrically isostatic, this shows that the swapped structure is also geometrically isostatic, giving the strongest possible correspondence. We use a projective geometric presentation of the statics and the motions, to make the key underlying correspondences transparent.