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On Reversibility among Parallelohedra

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

Abstract

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 \(\mathfrak{P}_{i}\).

Keywords

Line Segment Voronoi Cell Convex Polyhedron Exterior Surface Canonical Paral 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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