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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., Competing Interests: The authors declare no conflict of interest., (Copyright © 2019 the Author(s). Published by PNAS.)

A linear stability analysis is performed for a pair of coaxial vertical chains made from permanently magnetized balls under the influence of gravity. While one chain rises from the ground, the other hangs from above, with the remaining ends separated by a gap of prescribed length. Various boundary conditions are considered, as are situations in which the magnetic dipole moments in the two chains are parallel or antiparallel. The case of a single chain attached to the ground is also discussed. The stability of the system is examined with respect to three quantities: the number of balls in each chain, the length of the gap between the chains, and a single dimensionless parameter which embodies the competition between magnetic and gravitational forces. Asymptotic scaling laws involving these parameters are provided. The Hessian matrix is computed in exact form, allowing the critical parameter values at which the system loses stability and the respective eigenmodes to be determined up to machine precision. A comparison with simple experiments for a single chain attached to the ground shows good agreement.

A Möbius band can be formed by bending a sufficiently long rectangular unstretchable material sheet and joining the two short ends after twisting by 180 ° . This process can be modelled by an isometric mapping from a rectangular region to a developable surface in three-dimensional Euclidean space. Attempts have been made to determine the equilibrium shape of a Möbius band by minimizing the bending energy in the class of mappings from the rectangular region to the collection of developable surfaces. In this work, we show that, although a surface obtained from an isometric mapping of a prescribed planar region must be developable, a mapping from a prescribed planar region to a developable surface is not necessarily isometric. Based on this, we demonstrate that the notion of a rectifying developable cannot be used to describe a pure bending of a rectangular region into a Möbius band or a generic ribbon, as has been erroneously done in many publications. Specifically, our analysis shows that the mapping from a prescribed planar region to a rectifying developable surface is isometric only if that surface is cylindrical with the midline being the generator. Towards providing solutions to this issue, we discuss several alternative modelling strategies that respect the distinction between the physical constraint of unstretchability and the geometrical notion of developability.

We study the effect of the length scales alpha and beta in the Navier-Stokes- alphabeta equations on the energy spectrum and the alignment between the vorticity and the eigenvectors of the stretching tensor in three-dimensional homogeneous and isotropic turbulent flows in a periodic cubic domain, including the limiting cases of the Navier-Stokes- alpha and Navier-Stokes equations. A significant increase in the accuracy of the energy spectrum at large wave numbers arises for beta

Lundgren's vortex model for the intermittent fine structure of high-Reynolds-number turbulence is applied to the Navier-Stokes alphabeta equations and specialized to the Navier-Stokes alpha equations. The Navier-Stokes alphabeta equations involve dispersive and dissipative length scales alpha and beta, respectively. Setting beta equal to alpha reduces the Navier-Stokes alphabeta equations to the Navier-Stokes alpha equations. For the Navier-Stokes alpha equations, the energy spectrum is found to obey Kolmogorov's -5/3 law in a range of wave numbers identical to that determined by Lundgren for the Navier-Stokes equations. For the Navier-Stokes alphabeta equations, Kolmogorov's -5/3 law is also recovered. However, granted that beta < alpha, the range of wave numbers for which this law holds is extended by a factor of alphabeta . This suggests that simulations based on the Navier-Stokes alphabeta equations may have the potential to resolve features smaller than those obtainable using the Navier-Stokes alpha equations.