Möbius bands obtained by isometrically deforming circular helicoids

Title: Möbius bands obtained by isometrically deforming circular helicoids
 

Speaker: Professor Eliot Fried ν

Abstract. We formulate and numerically solve a variational problem for determining an isometric, chirality preserving deformation from a circular helicoid to a Möbius band. The problem involves a single dimensionless parameter, the number ν of turns in a given circular helicoid. We determine a critical value of ν below which no solutions can be found and above which one or more stable solutions exist. The Möbius band corresponding to that choice of ν has three half twists, three-fold rotational symmetry, and coincides with the limit surface obtained as number of links/hinges of a Möbius kaleidocycle is taken to infinity. We also find values of ν that give rise to stable solutions with \(\small{n = 2k + 1, k\geq 2} \) half twists and \(n\) -fold rotational symmetry. Each such Möbius band has the lowest energy of any stable competitor with the same number of half twists, none of which exhibit rotational symmetry. We also find that certain choices of ν give rise to two stable Mobius bands with the same energy but different numbers of half twists. Applications, especially to the synthesis of nonorientable monocyclic hydrocarbon compounds, will also be mentioned.

This work is joint with OIST postdoctoral scholar Vikash Chaurasia.