Filling a Hole in a Crease Pattern: Isometric Mapping from Prescribed Boundary Folding

TL;DR

This paper proves that for polygonal boundary foldings, an interior isometric folding exists and can be computed efficiently, addressing a key problem in origami mathematics.

Contribution

It establishes the existence and polynomial-time computability of isometric interior foldings given boundary foldings for polygonal sheets.

Findings

Existence of isometric interior foldings proven for polygonal boundaries

Algorithm for computing such foldings runs in polynomial time

Applicable to boundary foldings with finitely many points

Abstract

Given a sheet of paper and a prescribed folding of its boundary, is there a way to fold the paper's interior without stretching so that the boundary lines up with the prescribed boundary folding? For polygonal boundaries nonexpansively folded at finitely many points, we prove that a consistent isometric mapping of the polygon interior always exists and is computable in polynomial time.