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
MetadataShow full item record
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.
DepartmentMassachusetts 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
Discrete and Computational Geometry
Abbott, Timothy G. et al. "Hinged Dissections Exist." Discrete & Computational Geometry, 47.1 (2012, p.150-186.
Author's final manuscript