Skip to main content

Hinged Dissections Exist

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.

References

  1. 1.

    Abbott, T.G., Demaine, E.D., Gassend, B.: A generalized Carpenter’s Rule Theorem for self-touching linkages, January 2009. arXiv:0901.1322

  2. 2.

    Akiyama, J., Nakamura, G.: Dudeney dissection of polygons. In: Revised Papers from the Japan Conference on Discrete and Computational Geometry, Tokyo, Japan. Lecture Notes in Computer Science, vol. 1763, pp. 14–29. Springer, Berlin (1998)

    Google Scholar 

  3. 3.

    Akiyama, J., Nakamura, G., Nozaki, A., Ozawa, K., Sakai, T.: The optimality of a certain purely recursive dissection for a sequentially n-divisible square. Comput. Geom., Theory Appl. 24(1), 27–39 (2003)

    MATH  MathSciNet  Google Scholar 

  4. 4.

    Burnikel, C., Funke, S., Mehlhorn, K., Schirra, S., Schmitt, S.: A separation bound for real algebraic expressions. In: Proceedings of the 9th Annual European Symposium on Algorithms, Aarhus, Denmark. Lecture Notes in Computer Science, vol. 2161, pp. 254–265. Springer, Berlin (2001)

    Google Scholar 

  5. 5.

    Bolyai, F.: Tentamen juventutem studiosam in elementa matheseos purae, elementaris ac sublimioris, methodo intuitiva, evidentiaque huic propria, introducendi. Typis Collegii Refomatorum per Josephum et Simeonem Kali, Maros Vásárhely (1832–1833)

  6. 6.

    Connelly, R., Demaine, E.D., Rote, G.: Straightening polygonal arcs and convexifying polygonal cycles. Discrete Comput. Geom. 30(2), 205–239 (2003)

    MATH  MathSciNet  Google Scholar 

  7. 7.

    Connelly, R., Demaine, E.D., Demaine, M.L., Fekete, S., Langerman, S., Mitchell, J.S.B., Ribó, A., Rote, G.: Locked and unlocked chains of planar shapes. In: Proceedings of the 22nd Annual ACM Symposium on Computational Geometry, Sedona, Arizona, pp. 61–70 (2006)

    Google Scholar 

  8. 8.

    Cohn, M.J.: Economical triangle-square dissection. Geom. Dedic. 3, 447–467 (1975)

    Article  MATH  Google Scholar 

  9. 9.

    Czyzowicz, J., Kranakis, E., Urrutia, J.: Dissections, cuts, and triangulations. In: Proceedings of the 11th Canadian Conference on Computational Geometry, Vancouver, Canada (1999). http://www.cs.ubc.ca/conferences/CCCG/elec_proc/c33.ps.gz

    Google Scholar 

  10. 10.

    Davis, J.F., Kirk, P.: Lecture Notes in Algebraic Topology. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  11. 11.

    Dehn, M.: Über den Rauminhalt. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, pp. 345–354 (1900). Later published in Math. Ann. 55, 465–478 (1902)

  12. 12.

    Demaine, E.D., Eppstein, D., Erickson, J., Hart, G.W., O’Rourke, J.: Vertex-unfolding of simplicial manifolds. In: Discrete Geometry: In Honor of W. Kuperberg’s 60th Birthday, pp. 215–228. Dekker, New York (2003)

    Google Scholar 

  13. 13.

    Demaine, E.D., Mitchell, J.S.B., O’Rourke, J.: Problem 47: Hinged dissections. In The Open Problems Project, March 2003. http://www.cs.smith.edu/~orourke/TOPP/P47.html

  14. 14.

    Demaine, E.D., Demaine, M.L., Eppstein, D., Frederickson, G.N., Friedman, E.: Hinged dissection of polyominoes and polyforms. Comput. Geom., Theory Appl. 31(3), 237–262 (2005)

    MATH  MathSciNet  Google Scholar 

  15. 15.

    Demaine, E.D., Demaine, M.L., Lindy, J.F., Souvaine, D.L.: Hinged dissection of polypolyhedra. In: Proceedings of the 9th Workshop on Algorithms and Data Structures, Waterloo, Canada. Lecture Notes in Computer Science, vol. 3608, pp. 205–217 (2005)

    Google Scholar 

  16. 16.

    Dudeney, H.E.: Puzzles and prizes. Weekly Dispatch, 1902. The puzzle appeared in the April 6 issue of this column. An unusual discussion followed on April 20, and the solution appeared on May 4

  17. 17.

    Dupont, J.L., Sah, C.-H.: Homology of Euclidean groups of motions made discrete and Euclidean scissors congruences. Acta Math. 164(1), 1–27 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  18. 18.

    Eppstein, D.: Hinged kite mirror dissection, June 2001. arXiv:cs.CG/0106032

  19. 19.

    Frederickson, G.N.: Dissections: Plane and Fancy. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  20. 20.

    Frederickson, G.N.: Hinged Dissections: Swinging & Twisting. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  21. 21.

    Gerwien, P.: Zerschneidung jeder beliebigen Anzahl von gleichen geradlinigen Figuren in dieselben Stücke. J. Reine Angew. Math. (Crelle’s J.) 10, 228–234 (1833) and Taf. III

    Article  MATH  Google Scholar 

  22. 22.

    Griffith, S.: Growing Machines. PhD thesis, Media Laboratory, Massachusetts Institute of Technology, September 2004

  23. 23.

    Grünbaum, B., Shephard, G.C.: Pick’s theorem. Am. Math. Mon. 100(2), 150–161 (1993)

    Article  MATH  Google Scholar 

  24. 24.

    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  25. 25.

    Jessen, B.: The algebra of polyhedra and the Dehn–Sydler theorem. Math. Scand. 22, 241–256 (1968)

    MATH  MathSciNet  Google Scholar 

  26. 26.

    Kelland, P.: On superposition, part II. Trans. R. Soc. Edinb. 33, 471–473 (1864) and plate XX

    Google Scholar 

  27. 27.

    Kranakis, E., Krizanc, D., Urrutia, J.: Efficient regular polygon dissections. Geom. Dedic. 80, 247–262 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  28. 28.

    Kreinovich, V.: Equidecomposability (scissors congruence) of polyhedra in ℝ3 and ℝ4 is algorithmically decidable: Hilbert’s 3rd problem revisited. Geombinatorics 18(1), 26–34 (2008)

    MATH  MathSciNet  Google Scholar 

  29. 29.

    Lemon, D.: The Illustrated Book of Puzzles. Saxon, London (1890)

    Google Scholar 

  30. 30.

    Lindgren, H.: Recreational Problems in Geometric Dissections and How to Solve Them. Dover, New York (1972). Revised and enlarged by Greg Frederickson

    MATH  Google Scholar 

  31. 31.

    Lowry, M.: Solution to question 269, [proposed] by Mr. W. Wallace. In: Leybourn, T. (ed.) Mathematical Repository, part 1, vol. 3, pp. 44–46. W. Glendinning, London (1814)

    Google Scholar 

  32. 32.

    Madachy, J.S.: Geometric dissections. In: Madachy’s Mathematical Recreations, pp. 15–33. Dover, New York (1979), Chap. 1. Reprint of Mathematics on Vacation, Scribner, 1975

    Google Scholar 

  33. 33.

    Mao, C., Thallidi, V.R., Wolfe, D.B., Whitesides, S., Whitesides, G.M.: Dissections: Self-assembled aggregates that spontaneously reconfigure their structures when their environment changes. J. Am. Chem. Soc. 124, 14508–14509 (2002)

    Article  Google Scholar 

  34. 34.

    O’Rourke, J.: Computational geometry column 44. Int. J. Comput. Geom. Appl. 13(3), 273–275 (2002)

    Article  MathSciNet  Google Scholar 

  35. 35.

    Ozanam, J.: Récréations Mathématiques et Physiques, pp. 297–302. Claude Antoine Jombert, fils, Paris (1778). According to [19], these pages were added to the book by Jean Montucla under the pseudonym M. de Chanla

    Google Scholar 

  36. 36.

    Panckoucke, A.-J.: Les Amusements Mathématiques. Chez André-Joseph Panckoucke, Lille (1749)

    Google Scholar 

  37. 37.

    Pick, G.: Geometrisches zur Zahlenlehre. Sitzungsber. Dtsch. Naturwissenschaftlich-Medicinischen Ver. Böhmen “Lotos” Prag 19, 311–319 (1900)

    Google Scholar 

  38. 38.

    Rus, D., Butler, Z., Kotay, K., Vona, M.: Self-reconfiguring robots. Commun. ACM 45(3), 39–45 (2002)

    Article  Google Scholar 

  39. 39.

    Sydler, J.-P.: Conditions nécessaires et suffisantes pour l’équivalence des polyèdres de l’espace euclidien à trois dimensions. Comment. Math. Helv. 40, 43–80 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  40. 40.

    Wallace, W. (ed.): Elements of Geometry, 8th edn. Bell & Bradfute, Edinburgh (1831)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Zachary Abel.

Additional information

T.G. Abbott’s research was partially supported by an NSF Graduate Research Fellowship and an MIT-Akamai Presidential Fellowship.

E.D. Demaine’s research was partially supported by NSF CAREER award CCF-0347776, DOE grant DE-FG02-04ER25647, and AFOSR grant FA9550-07-1-0538.

S.D. Kominers’ research was partially supported by an NSF Graduate Research Fellowship.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Abbott, T.G., Abel, Z., Charlton, D. et al. Hinged Dissections Exist. Discrete Comput Geom 47, 150–186 (2012). https://doi.org/10.1007/s00454-010-9305-9

Download citation

Keywords

  • Folding
  • Reconfiguration
  • Hinge
  • Polygon
  • Refinement