Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths

Abstract

When can a plane graph with prescribed edge lengths and prescribed angles (from among $\{0,180^{\circ },360^{\circ }$\}) be folded flat to lie in an infinitesimally thin line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to $360^{\circ }$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.

21 pages, 10 figures

References in corpus (1)

• A Generalized Carpenter's Rule Theorem for Self-Touching Linkages