講題：Some exceptional linkages, their smooth limit, and isometric deformations from helicoids to ruled Möbius bands
Abstract:Closed kinematic chains of interconnected links, know also as linkages, serve to induce motion or transmit force in mechanical systems. We introduce a family of linkages consisting of n ≥ 7 identical links connected by revolute hinges. Each such linkage exhibits just one internal degree-of-freedom, manifested by an everting motion. From the standpoint of the Chebyshev–Grübler–Kutzbach mobility criterion, any linkage in this family with n ≥ 8 links is underconstrained and, thus, is deemed exceptional. In the limit as n → ∞, these linkages converge to a smooth, ruled Möbius band with three half twists and three-fold rotational symmetry. The rulings of this surface are aligned with the unit binormal of its midline, which is a geodesic and has uniform torsion. We ﬁnd that this Möbius band can also be obtained by a stable isometric deformation of a helicoid with a certain number of turns. Also, helicoids with more turns can be isometrically deformed into stable Möbius bands with more half twists. Among all stable Möbius bands with k ≥ 3 half twists that can be so obtained, the one with k-fold rotational symmetry has the least bending energy. While knotted Möbius bands can also be produced from helicoids, we ﬁnd that they saddle points of the bending energy. Returning to the family of linkages mentioned above, we present various consequences of relaxing the requirement that the constituent links be identical, subject to a particular proportionality rule.
講者：Eliot Fried（Okinawa Institute of Science and Technology Graduate University）