Hinged Dissections Exist
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Demaine_Hinged dissections.pdf
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Author(s)
Abbott, Timothy G.
Abel, Zachary Ryan
Charlton, David
Demaine, Erik D.
Demaine, Martin L.
Kominers, Scott Duke
Date Issued
January 2012
Journal
Discrete and Computational Geometry
Publisher
Springer-Verlag
Citation
Abbott, Timothy G. et al. "Hinged Dissections Exist." Discrete & Computational Geometry, 47.1 (2012, p.150-186.
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
http://dx.doi.org/10.1007/s00454-010-9305-9
https://doi.org/10.1007/s00454-010-9305-9
