# Andreev’s Theorem on hyperbolic polyhedra

@article{Roeder2006AndreevsTO,
title={Andreev’s Theorem on hyperbolic polyhedra},
author={Roland K. W. Roeder and John H. Hubbard and William D. Dunbar},
journal={Annales de l'Institut Fourier},
year={2006},
volume={57},
pages={825-882}
}
• Published 7 January 2006
• Mathematics
• Annales de l'Institut Fourier
E.M. Andreev a publie en 1970 une classification des polyedres hyperboliques compacts de dimension 3 (autre que les tetraedres) dont les angles diedres sont non-obtus. Etant donne une description combinatoire d'un polyedre C, le theoreme d'Andreev dit que les angles diedres possibles sont exactement decrits par cinq classes d'inegalites lineaires. Le theoreme d'Andreev demontre egalement que le polyedre resultant est alors unique a isometrie hyperbolique pres. D'une part, le theoreme d'Andreev…
Réalisation de métriques sur les surfaces compactes
Un polyedre fuchsien de l'espace hyperbolique est une surface polyedrale invariante sous l'action d'un groupe fuchsien d'isometries (c.a.d. un groupe d'isometries qui laissent globalement invariante
G T ] 2 3 M ar 2 00 6 Constructing hyperbolic polyhedra using Newton ’ s Method
We demonstrate how to construct three-dimensional compact hyperbolic polyhedra using Newton’s Method. Under the restriction that the dihedral angles are non-obtuse, Andreev’s Theorem [5, 6] provides
Constructing Hyperbolic Polyhedra Using Newton's Method
This construction uses Newton's method and a homotopy to explicitly follow the existence proof presented by Andreev, providing both a very clear illustration of a proof of AndreeV's theorem and a convenient way to construct three-dimensional compact hyperbolic polyhedra having nonobtuse dihedral angles.
A proof of the Koebe-Andre'ev-Thurston theorem via flow from tangency packings
Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two
Compact hyperbolic tetrahedra with non-obtuse dihedral angles
Given a combinatorial description $C$ of a polyhedron having $E$ edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize $C$ is generally not a convex subset of
Deformations of hyperbolic convex polyhedra and cone-3-manifolds
The Stoker problem, first formulated in Stoker (Commun. Pure Appl. Math. 21:119–168, 1968), consists in understanding to what extent a convex polyhedron is determined by its dihedral angles. By means
Espace des modules de certains polyèdres projectifs miroirs
A projective mirror polyhedron is a projective polyhedron endowed with reflections across its faces. We construct an explicit diffeomorphism between the moduli space of a mirror projective polyhedron
Fuchsian polyhedra in Lorentzian space-forms
Let S be a compact surface of genus > 1, and g be a metric on S of constant curvature $${K\in\{-1,0,1\}}$$ with conical singularities of negative singular curvature. When K = 1 we add the condition
Compact hyperbolic coxeter thin cubes
• Mathematics
• 2014
Andreev’s Theorem provides a complete characterization of 3-dimensional compact hyperbolic combinatorial polytope having non-obtuse dihedral angles. Cube is one of such polytope. In this article,
COMBINATORIAL CHARACTERIZATION OF RIGHT-ANGLED HYPERBOLICITY OF 3-ORBIFOLDS
We study the right-angled hyperbolicity of a class of 3-handlebodies with simple facial structures, each of which possesses the property that its nerve is a triangulation of its boundary. We show

## References

SHOWING 1-10 OF 54 REFERENCES
Hyperideal polyhedra in hyperbolic 3-space
• Mathematics
• 2002
A hyperideal polyhedron is a non-compact polyhedron in the hyperbolic 3-space 3 which, in the projective model for 3 ⊂ 3 , is just the intersection of 3 with a projective polyhedron whose vertices
Uniformisation en dimension trois
© Association des collaborateurs de Nicolas Bourbaki, 1998-1999, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. ens.fr/) implique l’accord avec les conditions
On an Elementary Proof of Rivin's Characterization of Convex Ideal Hyperbolic Polyhedra by their Dihedral Angles
In 1832, Jakob Steiner (Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander, Reimer (Berlin)) asked for a characterization of those planar graphs which are combinatorially
Compact hyperbolic tetrahedra with non-obtuse dihedral angles
Given a combinatorial description $C$ of a polyhedron having $E$ edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize $C$ is generally not a convex subset of
A characterization of ideal polyhedra in hyperbolic $3$-space
The goals of this paper are to provide a characterization of dihedral angles of convex ideal (those with all vertices on the sphere at infinity) polyhedra in H3, and also of those convex polyhedra
Proofs from THE BOOK
• Art
• 1998
This revised and enlarged fifth edition features four new chapters, which contain highly original and delightful proofs for classics such as the spectral theorem from linear algebra, some more recent
A BRANCHED ANDREEV-THURSTON THEOREM FOR CIRCLE PACKINGS OF THE SPHERE
• Mathematics
• 1996
ANDREEV-THURSTON THEOREM. Let r be a triangulation of S which is not simplicially equivalent to a tetrahedron and let <£: r—»[0, \n\ be given, where T denotes the set of edges of x. Suppose that (A)
Lectures on Hyperbolic Geometry
• Mathematics
• 1992
Focussing on the geometry of hyperbolic manifolds, the aim here is to provide an exposition of some fundamental results, while being as self-contained, complete, detailed and unified as possible.
Proofs from "The Book"
Proofs from THE BOOK, by Martin Aigner and Giinter M. Ziegler. Pp.199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag). I would like to begin with the authors' words from the Preface of this book: