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Discrete & Computational Geometry

, Volume 47, Issue 1, pp 150–186 | Cite as

Hinged Dissections Exist

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

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 Polygon Refinement 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Timothy G. Abbott
    • 1
  • Zachary Abel
    • 2
  • David Charlton
    • 3
  • Erik D. Demaine
    • 1
  • Martin L. Demaine
    • 1
  • Scott Duke Kominers
    • 4
  1. 1.MIT Computer Science and Artificial Intelligence LaboratoryCambridgeUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Department of Computer ScienceBoston UniversityBostonUSA
  4. 4.Department of EconomicsHarvard University, and Harvard Business SchoolBostonUSA

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