We pose and numerically solve a variational problem in finding isometric, chirality preserving deformations from circular helicoid to Mobius bands. The formulation involves a single dimensionless parameter: the prescribed number ν of turns of a referential helicoid. Below a critical threshold of ν, no solutions are found; above it, one or more stable configurations exist. The Mobius band corresponding to the threshold has three half twists, and three-fold rotational symmetry. Specific values of ν give rise to stable solutions with n = 2k + 1, k ≥ 2, half twists and n-fold rotational symmetry. Each such Mobius band has the lowest energy of any stable competitor with the same number of half twists, but lacking rotational symmetry. By identifying and characterizing these deformations, we establish a framework that could guide the synthesis of uniformly twisted, structurally adjustable, and topologically switchable molecular assemblies. In addition to exhibiting static stability, the M¨obius bands we describe admit periodic everting motions—each preserving distances and bending energy instantaneously—that might be used to guide the design of adaptive materials and soft robotic systems capable of reversible, energy-efficient shape transformation.
Bio: Eliot Fried is a Professor at the Okinawa Institute of Science and Technology (OIST), where he leads the Mathematics, Mechanics, and Materials Unit. He received his Ph.D. and M.S. in Applied Mechanics from the California Institute of Technology in 1991 and 1989, respectively. His research focuses on the formulation and analysis of mathematical models for complex systems in mechanics and materials science, employing tools from differential and algebraic geometry, asymptotic analysis, bifurcation theory, and scientific computing. Professor Fried was the 2011–12 Midwest Mechanics Seminar speaker and currently serves as Editor-in-Chief of the Journal of Elasticity and Editor for Mechanics of Materials.
Host: Professor Nikhil Admal