RIGIDITY OF SPHERICAL FRAMEWORKS: SWAPPING BLOCKS AND HOLES

A significant range of geometric structures whose rigidity has been explored, for both practical and theoretical purposes, are formed by modifying generically isostatic triangulated spheres. In block and hole structures (P, p), some edges are removed (to make holes) and others are added (to create rigid subparts called blocks). Previous work noted a combinatorial analogy, in which blocks and holes played equivalent roles-so that they might be interchanged. In this paper, we geometrically connect stresses in such structures (P, p) to first-order motions in a swapped structure ((P) over bar, 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 an affine presentation of the statics and the motions to make the key underlying correspondences transparent.