Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths

TL;DR

This paper characterizes when a plane graph with specified angles and edge lengths can be folded flat without crossings, providing efficient algorithms for testing foldability and counting folded states, thus generalizing origami theory.

Contribution

It proves a conjecture from 2001 by establishing necessary and sufficient conditions for flat foldability of plane graphs with prescribed angles and lengths.

Findings

Characterization of flat foldability conditions

Linear-time algorithm for foldability testing

Polynomial-time algorithm for counting folded states

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.