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- Statement
Does every pair of equal-area polygons have a hinged dissection? A

*dissection*of one polygon A to another B is a partition of A into a finite number of pieces that may be reassembled to form B. A*hinged dissection*is a dissection where the pieces are hinged at vertices and the reassembling is achieved by rotating the pieces about their hinges in the plane of the polygons.- Origin
- Status/Conjectures
Now settled: Hinged dissections exist [AAC+08]. Update to this entry soon.

- Partial and Related Results
There are two main partial results. First, any two

*polyominoes*of the same area have a hinged dissection [DDE+03]. A polyomino is a polygon formed by joining unit squares at their edges; see [Kla97] and Problem 37. The polyomino result generalizes to hinged dissections of all edge-to-corresponding-edge gluings of congruent copies of any polygon. Second, any asymmetric polygon has a hinged dissection to its mirror image [Epp01]. Both of these results interpret the problem as ignoring possible intersections between the pieces as they hinge, following what Frederickson calls the “wobbly-hinged” model. This freedom may not be necessary, although this seems not to be established in the literature.Many specific examples of hinged dissections can be found in [Fre02].

- Appearances
- Categories
polygons

- Entry Revision History
J. O’Rourke, 25 Mar 2003; J. O’Rourke, 23 Jan 2009.

- [AAC+08]
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)*, pages 110–119, College Park, Maryland, June 9–11 2008.- [DDE+03]
Erik D. Demaine, Martin L. Demaine, David Eppstein, Greg N. Frederickson, and Erich Friedman. Hinged dissection of polyominoes and polyforms.

*Computational Geometry: Theory and Applications*, 31(3):237–262, 2003. arXiv:cs.CG/9907018.- [Epp01]
David Eppstein. Hinged kite mirror dissection. ACM Computing Research Repository, June 2001. arXiv:cs.CG/0106032, http://www.arXiv.org/abs/cs.CG/0106032.

- [Fre02]
Greg Frederickson.

*Hinged Dissections: Swinging & Twisting*. Cambridge University Press, 2002.- [Kla97]
David A. Klarner. Polyominoes. In Jacob E. Goodman and Joseph O’Rourke, editors,

*Handbook of Discrete and Computational Geometry*, chapter 12, pages 225–242. CRC Press LLC, Boca Raton, FL, 1997.- [O'R02b]
Joseph O’Rourke. Computational geometry column 44.

*Internat. J. Comput. Geom. Appl.*, 13(3):273–275, 2002. Also in*SIGACT News*,**34**(2):58–60 (2002), Issue 127.