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Register a new userA closed linkage mechanism having the shape of a discrete Mobius strip
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Authors
Shizuo Kaji
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A closed linkage mechanism in three-dimensional space is an object comprising rigid bodies connected with hinges in a circular form like a rosary. Such linkages include Bricard6R and Bennett4R. To design such a closed linkage, it is necessary to solve a high-degree algebraic equation, which is generally difficult. In this lecture, the author proposes a new family of closed linkage mechanisms with an arbitrary number of hinges as an extension of a certain Bricard6R. They have singular properties, such as one-dimensional degree of freedom (1-DOF), and certain energies taking a constant value regardless of the state. These linkage mechanisms can be regarded as discrete Mobius strips and may be of interest in the context of pure mathematics as well. However, many of the properties described here have been confirmed only numerically, with no rigorous mathematical proof, and should be interpreted with caution.
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Integral theorems such as Stokes and Gauss are fundamental in many parts of Physics. For instance, Faradays law allows computing the induced electric current on a closed circuit in terms of the variation of the flux of a magnetic field across the surface spanned by the circuit. The key point for applying Stokes theorem is that this surface must be orientable. Many students wonder what happens to the flux through a surface when this is not orientable, as it happens with a Mobius strip. On an orientable surface one can compute the flux of a solenoidal field using Stokes theorem in terms of the circulation of the vector potential of the field along the oriented boundary of the surface. But this cannot be done if the surface is not orientable, though in principle this quantity could be measured on a laboratory. For instance, checking the induced electric current on a circuit along the boundary of a surface if the field is a variable magnetic field. We shall see that the answer to this puzzle is simple and the problem lies in the question rather than in the answer.
A linkage mechanism consists of rigid bodies assembled by joints which can be used to translate and transfer motion from one form in one place to another. In this paper, we are particularly interested in a family of spacial linkage mechanisms which consist of $n$-copies of a rigid body joined together by hinges to form a ring. Each hinge joint has its own axis of revolution and rigid bodies joined to it can be freely rotated around the axis. The family includes the famous threefold symmetric Bricard6R linkage also known as the Kaleidocycle, which exhibits a characteristic turning over motion. We can model such a linkage as a discrete closed curve in $mathbb{R}^3$ with a constant torsion up to sign. Then, its motion is described as the deformation of the curve preserving torsion and arc length. We describe certain motions of this object that are governed by the semi-discrete mKdV equations, where infinitesimally the motion of each vertex is confined in the osculating plane.
The notions of discrete conformality on triangle meshes have rich mathematical theories and wide applications. The related notions of discrete uniformizations on triangle meshes, suggest efficient methods for computing the uniformizations of surfaces. This paper proves that the discrete uniformizations approximate the continuous uniformization for closed surfaces of genus $geq1$, when the approximating triangle meshes are reasonably good. To the best of the authors knowledge, this is the first convergence result on computing uniformizations for surfaces of genus $>1$.
Here we report the synthesis, structure and detailed characterisation of three n-membered oxovanadium rings, Na$_n$[(V=O)$_n$Na$_n$(H$_2$O)$_n$($alpha$, $beta$, or $gamma$-CD)$_2$]$m$H$_2$O (n = 6, 7, or 8), prepared by the reactions of (V=O)SO$_4$$cdot$$x$H$_2$O with $alpha$, $beta$, or $gamma$-cyclodextrins (CDs) and NaOH in water. Their alternating heterometallic vanadium/sodium cyclic core structures were sandwiched between two CD moieties such that O-Na-O groups separated neighbouring vanadyl ions. Antiferromagnetic interactions between the $S$ = 1/2 vanadyl ions led to $S$ = 0 ground states for the even-membered rings, but to two quasi-degenerate $S$ = 1/2 states for the spin-frustrated heptanuclear cluster.
We report on experiments with Mobius strip microlasers which were fabricated with high optical quality by direct laser writing. A Mobius strip looks like a ring with a twist and exhibits the fascinating property that it has only one boundary and a one-sided nonorientable surface. Hence, in contrast to conventional ring or disk resonators, a Mobius strip cavity cannot sustain Whispering Gallery Modes (WGMs). Comparison between experiments and FDTD simulations evidenced that the resonances are indeed not of whispering gallery type, but localized along periodic geodesics. This finding is supported, on the one hand, by an extension of the effective index approximation to curved layers, and on the other hand, by an algorithm to systematically identify the periodic geodesics.
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