Advertisement

Journal of Mathematical Sciences

, Volume 239, Issue 5, pp 644–653 | Cite as

On the Volume Formula for a Hyperbolic Octahedron with mm2-Symmetry

  • V. A. KrasnovEmail author
  • E. Sh. Khisyametdinova
Article
  • 6 Downloads

Abstract

In this paper, explicit integral volume formulas for arbitrary compact hyperbolic octahedra with mm2-symmetry are obtained in terms of dihedral angles. Also, we provide an algorithm to compute the volume of such octahedra in spherical spaces.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. V. Abrosimov, “On volumes of polyhedra in a space of constant curvature,” Bull. Kemerovo State Univ., 3/1, 7–13 (2011).Google Scholar
  2. 2.
    N. V. Abrosimov and G. A. Baygonakova, “Hyperbolic octahedron with mmm-symmetry,” Sib. È lectron. Math. Izv., 10, 123–140 (2013).MathSciNetzbMATHGoogle Scholar
  3. 3.
    N. V. Abrosimov, M. Godoy-Molina, and A.D. Mednykh, “On the volume of a spherical octahedron with symmetries,” Sovrem. Mat. Fundam. Napravl., 60, 3–12 (2008).zbMATHGoogle Scholar
  4. 4.
    D. V. Alekseevskiy, E.B. Vinberg, and A. S. Solodovnikov, “Geometry of spaces of constant curvature,” Encycl. Math. Sci., 29, 1–138 (1988).MathSciNetzbMATHGoogle Scholar
  5. 5.
    G. A. Baygonakova, M. Godoy-Molina, and A.D. Mednykh, “On geometric properties of hyperbolic octahedron with mmm-symmetry,” Bull. Kemerovo State Univ., 3/1, 13–18 (2011).Google Scholar
  6. 6.
    J. Bolyai, “Appendix. The Theory of Space,” Elsevier, Amsterdam (1987).zbMATHGoogle Scholar
  7. 7.
    Yu. Cho and H. Kim, “On the volume formula for hyperbolic tetrahedra,” Discrete Comput. Geom., 22, 347–366 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D. A. Derevnin and A. D. Mednykh, “A formula for the volume of a hyperbolic tetrahedron,” Russian Math. Surveys, 60, 346–348 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    R. V. Galiulin, S.N. Mikhalev, and I.Kh. Sabitov, “Some applications of the volume formula for an octahedron,” Mat. Zametki, 1, 27–43 (2004).CrossRefzbMATHGoogle Scholar
  10. 10.
    V. A. Krasnov, “On integral expressions for volumes of hyperbolic tetrahedra,” Sovrem. Mat. Fundam. Napravl., 49, 89–99 (2013).Google Scholar
  11. 11.
    V. A. Krasnov, “On the volume of hyperbolic octahedra with nontrivial symmetry,” Sovrem. Mat. Fundam. Napravl., 51, 74–87 (2013).Google Scholar
  12. 12.
    H. Kneser, “Der Simplexinhalt in der nichteuklidischen Geometrie,” Deutsche Math., 1, 337–340 (1936).zbMATHGoogle Scholar
  13. 13.
    G. Leibon, “The symmetries of hyperbolic volume,” preprint (2002).Google Scholar
  14. 14.
    N. I. Lobachevskiy, “Imaginary Geometry” [in Russian], OGIZ–GITTL, Moscow–Leningrad (1949).Google Scholar
  15. 15.
    J. Milnor, “Hyperbolic geometry: the first 150 years,” Bull. Amer. Math. Soc., 6, No. 1, 307–332 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Y. Mohanty, “The Regge symmetry is a scissors congruence in hyperbolic space,” Algebr. Geom. Topol., 3, 1–31 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    J. Murakami, “The volume formulas for a spherical tetrahedron”, Arxiv: 1011.2584v4 2012.Google Scholar
  18. 18.
    J. Murakami and A. Ushijima, “A volume formula for hyperbolic tetrahedra in terms of edge lengths,” J. Geom., 83, No. 1-2, 153–163 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    J. Murakami and M. Yano, “On the volume of a hyperbolic and spherical tetrahedron,” Comm. Anal. Geom., 13, 379–400 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    L. Schläfli, “Theorie der vielfachen Kontinuität,” Gesammelte mathematische Abhandlungen, Birkhäuser, Basel, 167–387 (1950).CrossRefGoogle Scholar
  21. 21.
    G. Sforza, “Spazi metrico-proiettivi,” Ric. Esten. Differ. Ser., 8, No. 3, 3–66 (1906).Google Scholar
  22. 22.
    A. Ushijima, “A volume formula for generalized hyperbolic tetrahedra,” Non-Euclid. Geom., 581, 249–265 (2006).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.RUDN UniversityMoscowRussia

Personalised recommendations