Paper by Martin L. Demaine
- Timothy G. Abbott, Zachary Abel, David Charlton, Erik D. Demaine, Martin L. Demaine, and Scott D. Kominers, “Hinged Dissections Exist”, in Proceedings of the 24th Annual ACM Symposium on Computational Geometry (SoCG 2008), College Park, Maryland, June 9–11, 2008, pages 110–119.
We prove that any finite collection of polygons of equal area has a
common hinged dissection, that is, 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). We also extend our result to edge-hinged dissections of solid
3D polyhedra that have a common (unhinged) dissection,
as determined by Dehn's 1900 solution to Hilbert's Third Problem.
Our proofs are constructive, giving explicit algorithms in all cases.
For a constant number of planar polygons, both the number of pieces and
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.
- The paper is available in PostScript (1931k), gzipped PostScript (719k), and PDF (580k).
- See information on file formats.
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- Related papers:
- HingedDissections_DCG (Hinged Dissections Exist)
See also other papers by Martin Demaine.
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