Paper by Erik D. Demaine

Reference:

Timothy G. Abbott, Zachary Abel, David Charlton, Erik D. Demaine, Martin L. Demaine, and Scott Duke Kominers, “Hinged Dissections Exist”, Discrete & Computational Geometry, volume 47, number 1, 2012, pages 150–186.

BibTeX

@Article{HingedDissections_DCG,
  AUTHOR        = {Timothy G. Abbott and Zachary Abel and David Charlton and
                   Erik D. Demaine and Martin L. Demaine and
                   Scott Duke Kominers},
  TITLE         = {Hinged Dissections Exist},
  JOURNAL       = {Discrete \& Computational Geometry},
  journalurl    = {http://link.springer.de/link/service/journals/00454/},
  volume        = 47,
  number        = 1,
  year          = 2012,
  pages         = {150--186},

  doi           = {https://dx.doi.org/10.1007/s00454-010-9305-9},
  dblp          = {https://dblp.org/rec/journals/dcg/AbbottACDDK12},
  comments      = {This paper is also available from <A HREF="http://dx.doi.org/10.1007/s00454-010-9305-9">SpringerLink</A> and as <A HREF="https://arXiv.org/abs/0712.2094">arXiv:0712.2094</A>.},
  length        = {33 pages},
  withstudent   = 1,
  replaces      = {HingedDissections_SoCG2008},
  papers        = {HingedDissections_SoCG2008},
}

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.

Comments:

This paper is also available from SpringerLink and as arXiv:0712.2094.

Length:

The paper is 33 pages.

Availability:

The paper is available in PostScript (2058k), gzipped PostScript (770k), and PDF (648k).

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Related papers:

HingedDissections_SoCG2008 (Hinged Dissections Exist)