Hinged Dissections Exist

TL;DR

This paper proves that any finite collection of polygons with equal area can be hinged together to form any polygon in the collection, solving a long-standing open problem and extending to 3D polyhedra with constructive algorithms.

Contribution

It establishes the existence of common hinged dissections for polygons of equal area and extends the concept to 3D polyhedra, providing explicit algorithms and complexity bounds.

Findings

Hinged dissections exist for any finite collection of equal-area polygons.

Constructive algorithms with pseudopolynomial complexity are provided.

Extension of the result to 3D polyhedra with common dissections.

Abstract

We prove that any finite collection of polygons of equal area has a common hinged dissection. That is, for any such collection of polygons there exists a chain of polygons hinged at vertices that can be folded in the plane continuously without self-intersection to form any polygon in the collection. This result settles the open problem about the existence of hinged dissections between pairs of polygons that goes back implicitly to 1864 and has been studied extensively in the past ten years. Our result generalizes and indeed builds upon the result from 1814 that polygons have common dissections (without hinges). We also extend our common dissection result to edge-hinged dissections of solid 3D polyhedra that have a common (unhinged) dissection, as determined by Dehn's 1900 solution to Hilbert's Third Problem. Our proofs are constructive, giving explicit algorithms in all cases. For a constant number of planar polygons, both the number of pieces and running time required by our construction are pseudopolynomial. This bound is the best possible, even for unhinged dissections. Hinged dissections have possible applications to reconfigurable robotics, programmable matter, and nanomanufacturing.