Journal of the Mathematical Society of Japan

A generalization of Andreev's Theorem

Raquel DÍAZ

Full-text: Open access


Andreev's Theorem studies the existence of compact hyperbolic polyhedra of a given combinatorial type and given dihedral angles, all of them acute. In this paper we consider the same problem but without any restriction on the dihedral angles. We solve it for the descendants of the tetrahedron, i.e. those polyhedra that can be obtained from the tetrahedron by successively truncating vertices; for instance, the first of them is the triangular prism.

Article information

J. Math. Soc. Japan Volume 58, Number 2 (2006), 333-349.

First available in Project Euclid: 1 June 2006

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Mathematical Reviews number (MathSciNet)

Primary: 51M10: Hyperbolic and elliptic geometries (general) and generalizations
Secondary: 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15] 52B10: Three-dimensional polytopes

hyperbolic polyhedra dihedral angles Andreev's Theorem


DÍAZ, Raquel. A generalization of Andreev's Theorem. J. Math. Soc. Japan 58 (2006), no. 2, 333--349. doi:10.2969/jmsj/1149166778.

Export citation


  • E. M. Andreev, On convex polyhedra in Lobachevskii spaces, Math. USSR Sb., 10 (1970), 413–440.
  • E. M. Andreev, Convex polyhedra of finite volume in Lobachevskii space, Math. USSR Sb., 12 (1970), 255–259.
  • R. Benedetti and J. J. Risler, Real Algebraic and Semialgebraic Geometry, Actualités Math., Hermann, 1990.
  • A. L. Cauchy, Sur les polygones et polyèdres, J. Ec. Polytechnique, 16 (1813), 87–99.
  • R. D\y az, Non-convexity of the space of dihedral angles of hyperbolic polyhedra, C. R. Acad. Sci. Paris Série I, 325 (1997), 993–998.
  • R. D\y az, A Characterization of Gram Matrices of Polytopes, Discrete Comput. Geom., 21 (1999), 581–601.
  • F. R. Gantmacher, The theory of matrices, Vol.,I, Chelsea Publishing Company, New York, 1959.
  • B. Iversen, Hyperbolic Geometry, Cambridge Univ. Press, 1992.
  • J. Milnor, The Schläfli differential equality, In: Collected papers Vol.,1: Geometry, Houston, Publish or Perish Inc., 1994.
  • I. Rivin and C. D. Hodgson, A characterization of compact convex polyhedra in hyperbolic 3-space, Invent. Math., 111 (1993), 77–111.
  • E. B. Vinberg, Hyperbolic reflection groups, Russian Math. Surveys, 40 (1985), no.,1, p.,31–75.