Affine Equivalent Classes of Parallelohedra

  • Nikolai Dolbilin
  • Jin-ichi Itoh
  • Chie Nara
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7033)

Abstract

In the paper the affine equivalence relation in the set of parallelohedra is studied. One proves the uniqueness theorem for a wide class of d-dimensional parallelohedra. From here it follows that for every d ( ≥ 2) the space of affine equivalent classes of d-dimensional primitive parallelohedra has dimension d(d + 1)/2 − 1.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Fedorov, E.S.: An introduction to the Theory of Figures, St. Petersburg (1885) (in Russian)Google Scholar
  2. 2.
    Michel, L., Ryshkov, S.S., Senechal, M.: An extension of Voronoï’s theorem on primitive parallelotopes. Europ. J. Combinatorics 16, 59–63 (1995)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Minkowski, H.: Allgemeine Lerätze über die convexen Polyeder. Gött. Nachr., 198–219 (1897)Google Scholar
  4. 4.
    Venkov, B.A.: On a class of Euclidean Polyhedra. Vestn. Leningr. Univ., Ser. Mat. Fiz. 9, 11–31 (1954)MathSciNetGoogle Scholar
  5. 5.
    Voronoi, G.: Nouvelles applications des paramètres continus á la théorie des formes quadratiques Deuxiéme meḿoire: Recherches sur les paralléloédres primitifs. J. Reine Angew. Math. 134, 198–287 (1908); 136, 67–178 (1909)MATHCrossRefGoogle Scholar
  6. 6.
    Zhitomirskii, O.K.: Verschärfung eines Satzes von Voronoi. Zh. Leningr. Fiz.-Mat. Obshch. 2, 131–151 (1929)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Nikolai Dolbilin
    • 1
  • Jin-ichi Itoh
    • 2
  • Chie Nara
    • 3
  1. 1.Institute of MathematicsRussian Academy of ScienceMoscowRussia
  2. 2.Faculty of EducationKumamoto UniversityJapan
  3. 3.Liberal Arts Education CenterAso Campus, Tokai UniversityAsoJapan

Personalised recommendations