“Thin sheets have long been known to experience an increase in stiffness when they are bent, buckled, or assembled into smaller interlocking structures. We introduce a unique orientation for coupling rigidly foldable origami tubes in a “zipper” fashion that substantially increases the system stiffness and permits only one flexible deformation mode through which the structure can deploy. The flexible deployment of the tubular structures is permitted by localized bending of the origami along prescribed fold lines. All other deformation modes, such as global bending and twisting of the structural system, are substantially stiffer because the tubular assemblages are overconstrained and the thin sheets become engaged in tension and compression. The zipper-coupled tubes yield an unusually large eigenvalue bandgap that represents the unique difference in stiffness between deformation modes. Furthermore, we couple compatible origami tubes into a variety of cellular assemblages that can enhance mechanical characteristics and geometric versatility, leading to a potential design paradigm for structures and metamaterials that can be deployed, stiffened, and tuned. The enhanced mechanical properties, versatility, and adaptivity of these thin sheet systems can provide practical solutions of varying geometric scales in science and engineering.”