Bohr Jakob, Markvorsen Steen
Department of Micro and Nanotechnology, Technical University of Denmark, Kongens Lyngby, Denmark.
PLoS One. 2013 Oct 3;8(10):e74932. doi: 10.1371/journal.pone.0074932. eCollection 2013.
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.
当对直的丝带进行扭转时，会自发形成一种重复的晶体状图案。该图案类似于用等腰三角形进行的镶嵌，可以很容易地用从透明投影片上剪下的丝带演示出来。我们使用一种直纹程序对可展丝带进行了一般性描述，其中丝带由两个生成函数唯一描述。这种构造定义了一个没有奇点的可微标架，即丝带标架，从而避免了弗伦内 - 塞雷标架的缺点。利用平面三角形和圆柱弧对观察到的自发图案进行建模，并且表明丝带结构源于丝带端到端长度的最大化，即源于对丝带长度的最优利用。从不兼容的内在几何形状和长程有序的出现这两个角度对该现象进行了讨论。
