Polygonal ribbons in two and three dimensions

Abstract A polygonal ribbon is a finite sequence of polygons (not necessarily coplanar) such that each pair of successive polygons intersect exactly in a common side. The study of such ribbons can be regarded as an initial step toward the study of polygonally approximated surfaces. In this paper we investigate various geometric and topological properties of polygonal ribbons in two and three dimensions, including properties such as nonselfintersection, orientability, and twist. When the polygons are all of simple types—for example, when they are all triangles or all rectangles—they can be represented compactly in terms of such quantities as vertex coordinates, side lengths and dihedral angles. For nonselfintersecting (NSI) ribbons, we establish basic connectivity properties of the ribbon and its border. Finally, we briefly treat the special case of planar ribbons.