Hinged Dissections Exist

Timothy G. Abbott; Zachary  Abel; David  Charlton; Erik D. Demaine; Martin L. Demaine; Scott Duke Kominers

Article | Discrete & Computational Geometry

Hinged Dissections Exist

by Timothy G. Abbott, Zachary Abel, David Charlton, Erik D. Demaine, Martin L. Demaine and Scott Duke Kominers

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). Our proofs are constructive, giving explicit algorithms in all cases. For two planar polygons whose vertices lie on a rational grid, both the number of pieces and the 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.

Keywords: Folding; Reconfiguration; Hinge; Plygon; Refinement; Mathematical Methods;

Format: Print

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Citation:

Abbott, Timothy G., Zachary Abel, David Charlton, Erik D. Demaine, Martin L. Demaine, and Scott Duke Kominers. "Hinged Dissections Exist." Discrete & Computational Geometry 47, no. 1 (January 2012): 150–186.

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Hinged Dissections Exist

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