Single degree of freedom everting ring linkages with nonorientable topology.

Linkages are assemblies of rigid bodies connected through joints. They serve as the basis for force- and movement-managing devices ranging from ordinary pliers to high-precision robotic arms. Aside from planar mechanisms, like the well-known four-bar linkage, only a few linkages with a single internal degree of freedom-meaning that they can change shape in only one way and may thus be easily controlled-have been known to date. Here, we present "Möbius kaleidocycles," a previously undiscovered class of single-internal degree of freedom ring linkages containing nontrivial examples of spatially underconstrained mechanisms. A Möbius kaleidocycle is made from seven or more identical links joined by revolute hinges. These links dictate a specific twist angle between neighboring hinges, and the hinge orientations induce a nonorientable topology equivalent to the topology of a [Formula: see text]-twist Möbius band. Apart from having many technological applications, including perhaps the design of organic ring molecules with peculiar electronic properties, Möbius kaleidocycles raise fundamental questions about geometry, topology, and the limitations of mobility for closed loop linkages.