On Reversibility among Parallelohedra

  • Jin Akiyama
  • Ikuro Sato
  • Hyunwoo Seong
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7579)


Given two convex polyhedra α and β, we say that α and β are a reversible pair if α has a dissection into a finite number of pieces which can be rearranged to form β in such a way that no face of the dissection of α includes any part of an edge of α, no face of the dissection of β includes any part of an edge of β, the pieces are hinged on some of their edges so that the pieces of the dissection are connected as in a tree-structure, all of the exterior surface of α is in the interior of β, and all of the exterior surface of β comes from the interior of α. Let \(\mathfrak{P}_{i}\) denote one of the five families of parallelohedra (see Section 2 for the corresponding definitions). In this paper, it is shown that given an arbitrary canonical parallelohedron P, there exists a canonical parallelohedron \(Q \in \mathfrak{P}_{i}\) such that the pair P and Q is reversible for each \(\mathfrak{P}_{i}\).


Line Segment Voronoi Cell Convex Polyhedron Exterior Surface Canonical Paral 
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  1. 1.
    Conway, J.H., Sloane, N.J.A.: Sphere packings, lattices and groups, 3rd edn. A Series of Comprehensive Studies in Mathematics, vol. 290. Springer (1999)Google Scholar
  2. 2.
    Boltyanskii, V.G.: Equivalent and Equidecomposable Figures. D. C. Health and Co. (1963), Translated and adapted from the first Russian edition (1956) by Henn, A.K., Watts, C.E.Google Scholar
  3. 3.
    Boltyanskii, V.G.: Hilbert’s third problem. V. H. Winton & Sons (1978), Translated by Silverman, R.A.Google Scholar
  4. 4.
    Frederickson, G.N.: Dissections: Plane and Fancy. Cambridge University Press, New York (1997)CrossRefzbMATHGoogle Scholar
  5. 5.
    Frederickson, G.N.: Hinged Dissections: Swinging and Twisting. Cambridge University Press, New York (2002)zbMATHGoogle Scholar
  6. 6.
    Frederickson, G.N.: Piano-Hinged Dissections: Time to Fold. Cambridge University Press, New York (2006)zbMATHGoogle Scholar
  7. 7.
    Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, New York (2007)CrossRefzbMATHGoogle Scholar
  8. 8.
    Akiyama, J., Nakamura, G.: Dudeney Dissection of Polygons. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 14–29. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    Akiyama, J., Nakamura, G.: Congruent Dudeney dissections of triangles and convex quadrilaterals – all hinge points interior to the sides of the polygons. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry. Algorithms and Combinatorics, vol. 25, pp. 43–63 (2003)Google Scholar
  10. 10.
    Fedorov, E.S.: An introduction to the theory of figures. In: Notices of the Imperial Mineralogical Society (St. Petersburg) Ser. 2, vol. 21, pp. 1–279 (1885); Republished with comments by Akad. Nauk. SSSR, Moscow (1953) (in Russian)Google Scholar
  11. 11.
    Alexandrov, A.D.: Convex Polyhedra. Springer Monographs in Mathematics (2005)Google Scholar
  12. 12.
    Dolbilin, N., Itoh, J., Nara, C.: Geometric realization on affine equivalent 3-parallelohedra (to be published)Google Scholar
  13. 13.
    Akiyama, J., Seong, H.: On the reversibilities among quasi-parallelogons (to be published)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jin Akiyama
    • 1
  • Ikuro Sato
    • 2
  • Hyunwoo Seong
    • 3
  1. 1.Research Institute for Mathematics EducationTokyo University of ScienceShinjukuJapan
  2. 2.Department of Pathology, Research InstituteMiyagi Cancer CenterNatori-cityJapan
  3. 3.The University of TokyoJapan

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