Hinged Dissections Exist
Author(s)
Abbott, Timothy G.; Abel, Zachary Ryan; Charlton, David; Demaine, Erik D.; Demaine, Martin L.; Kominers, Scott Duke; ... Show more Show less
DownloadDemaine_Hinged dissections.pdf (648.5Kb)
OPEN_ACCESS_POLICY
Open Access Policy
Creative Commons Attribution-Noncommercial-Share Alike
Terms of use
Metadata
Show full item recordAbstract
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.
Date issued
2012-01Department
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 MathematicsJournal
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
ISSN
0179-5376
1432-0444