Ribbon Crystals

Jakob Bohr,Steen Markvorsen,Chris Lorenz

doi:10.1371/journal.pone.0074932

Abstract

A repetitive crystal-like pattern is spontaneously formed upon the twisting of straight ribbons. The pattern is akin to a tessellation with isosceles triangles, and it can easily be demonstrated with ribbons cut from an overhead transparency. We give a general description of developable ribbons using a ruled procedure where ribbons are uniquely described by two generating functions. This construction defines a differentiable frame, the ribbon frame, which does not have singular points, whereby we avoid the shortcomings of the Frenet–Serret frame. The observed spontaneous pattern is modeled using planar triangles and cylindrical arcs, and the ribbon structure is shown to arise from a maximization of the end-to-end length of the ribbon, i.e. from an optimal use of ribbon length. The phenomenon is discussed in the perspectives of incompatible intrinsic geometries and of the emergence of long-range order.

Highlights

Korte, Starostin, and van der Heijden reported on the fascinating occurrence of triangular buckling patterns in ribbons [1]
Following the method used in the study of the equilibrium shape of the Mobius strip [2], they elegantly built an analysis of the pattern on the triangular region of the Mobius strip [2] and used it to form a repetitive structure of triangular regions for the twisted ribbons [1]
The tessellation phenomenon can be explained to result from a geometrical optimization of the end-to-end length, Lee

Summary

Introduction

Korte, Starostin, and van der Heijden reported on the fascinating occurrence of triangular buckling patterns in ribbons [1]. In order for the ribbon to be a regular surface for u[{b,b1⁄2 we must impose the following condition on the instantaneous direction of the field A(s), i.e. h(s), the derivative h_ (s), and the half-width of the ribbon b: The normal curvature of the center line is, according to Euler’s theorem: kn(s)~k1(s) sin2 (h(s))zk2(s) cos2 (h(s))~{w(s) sin (h(s)): ð15Þ

Results

Conclusion