Invariant dual mechanics of tensegrity and origami

Statics and kinematics are often viewed as intertwined branches of mechanics. The principle of virtual work indicates this interconnection: For a system in static equilibrium, the total work performed by all forces during any virtual displacement must equal zero. In this work, we make the interplay between statics and kinematics more explicit by focusing on two engineering structures-tensegrity and origami. Specifically, we demonstrate a quantitative duality relationship between the states of self-stress of tensegrity and the infinitesimal mechanisms of origami. More importantly, we show that this duality remains invariant under nondegenerate linear transformations applied to the tensegrity and origami configurations. Furthermore, we establish that the stability property of tensegrity, particularly superstability, is preserved under such transformations. We apply the invariant duality theory to tensegrity and origami structures with prismatic and polyhedral geometries, illustrating its broad applicability. Such duality is also applicable to the fast generation of irregular, three-dimensional architected materials and structures.