Non-convexity of the space of dihedral angles of hyperbolic polyhedra

Díaz, Raquel

doi:10.1016/s0764-4442(97)89092-1

Cited by 13 publications

(11 citation statements)

References 5 publications

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“…By contrast, if we allow obtuse angles things are considerably more difficult. It is unknown whether dihedral angles determine a polyhedron; furthermore the space of dihedral angles is never convex [4].…”

Section: Related Resultsmentioning

confidence: 99%

The maximum volume of hyperbolic polyhedra

Belletti¹

2020

Preprint

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We study the supremum of the volume of hyperbolic polyhedra with some fixed combinatorics and with vertices of any kind (real, ideal or hyperideal). We find that the supremum is always equal to the volume of the rectification of the 1-skeleton.The theorem is proved by applying a sort of volume-increasing flow to any hyperbolic polyhedron. Singularities may arise in the flow because some strata of the polyhedron may degenerate to lowerdimensional objects; when this occurs, we need to study carefully the combinatorics of the resulting polyhedron and continue with the flow, until eventually we get a rectified polyhedron. Contents 1 Introduction 2 Generalized hyperbolic polyhedra 3 The space of proper polyhedra and the volume function 3.1 The Bao-Bonahon existence and uniqueness theorem for hyperideal polyhedra .

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Section: Related Resultsmentioning

confidence: 99%

The maximum volume of hyperbolic polyhedra

Belletti¹

2020

Preprint

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“…Andreev's restriction to non-obtuse dihedral angles is emphatically necessary to ensure that A C be convex. Without this restriction, the corresponding space of dihedral angles, ∆ C , of compact (or finite volume) hyperbolic polyhedra realizing a given C is not convex [12]. In fact, the recent work by D íaz [13] provides a detailed analysis of this space of dihedral angles ∆ C for the class of abstract polyhedra C obtained from the tetrahedron by successively truncating vertices.…”

Section: Theorem 14 Andreev's Theoremmentioning

confidence: 99%

Andreev’s Theorem on hyperbolic polyhedra

Roeder¹,

Hubbard²,

Dunbar³

2007

Ann. inst. Fourier

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In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra (other than tetrahedra) having non-obtuse dihedral angles [4,5]. Given a combinatorial description of a polyhedron, C, Andreev's Theorem provides five classes of linear inequalities, depending on C, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing C with the assigned dihedral angles. Andreev's Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry.Andreev's Theorem is both an interesting statement about the geometry of hyperbolic 3dimensional space, as well as a fundamental tool used in the proof for Thurston's Hyperbolization Theorem for 3-dimensional Haken manifolds. It is also remarkable to what level the proof of Andreev's Theorem resembles (in a simpler way) the proof of Thurston.We correct a fundamental error in Andreev's proof of existence and also provide a readable new proof of the other parts of the proof of Andreev's Theorem, because Andreev's paper has the reputation of being "unreadable". Résumé E. M. Andreev a publié en 1970 une classification des polyèdres hyperboliques compacts de dimension trois (autrement que les tétraèdres) dont les angles dièdres sont non-obtus [4, 5]. Etant donné une description combinatoire d'un polyèdre C, le Théorème d'Andreev dit que les angles dièdres possibles sont exactement décrits par cinq classes d'inégalités linéaires. Le Théorème d'Andreev démontre également que le polyèdre résultant est alors unique à isométrie hyperbolique près. D'une part, le Théorème de Andreev est évidemment un énoncé intéressant de la géométrie de l'espace hyperbolique en dimension 3; d'autre part c'est un outil essentiel dans la preuve du Théorème d'Hyperbolization de Thurston pour les variétés Haken de dimension 3. Il est d'ailleurs remarquable à quel point la démonstration d'Andreev rappelle (en plus simple) la démonstration de Thurston. La démonstration d'Andreev contient une erreur importante. Nous corrigeons ici cette erreur et nous fournissons aussi une nouvelle preuve lisible des autres parties de la preuve, car le papier d'Andreev a la réputation d'être "illisible".

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“…(10) (11) Continuing from the last page, sub-figures (22)- (29) are another W h (6,13), W h(6, 9), W h(3, 6), W h(9, 12), W h (6,15), W h(6, 16), W h (6,17), W h(6, 7), W h (6,8), W h(6, 4), W h(8, 18), W h (7,9), W h (9,14), W h(9, 15), W h(3, 18), W h (7,8)…”

Section: Construction Of a "Difficult" Simple Polyhedronmentioning

confidence: 99%

Constructing Hyperbolic Polyhedra Using Newton's Method

Roeder

2007

Experimental Mathematics

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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 [8,9] provides as necessary and sufficient conditions five classes of linear inequalities for the dihedral angles of a compact hyperbolic polyhedron realizing a given combinatorial structure C. Andreev's Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Our construction uses Newton's method and a homotopy to explicitly follow the existence proof presented by Andreev, providing both a very clear illustration of proof of Andreev's Theorem as well as a convenient way to construct three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles.As an application, we construct compact hyperbolic polyhedra having dihedral angles that are (proper) integer sub-multiples of π, so that the group Γ generated by reflections in the faces is a discrete group of isometries of hyperbolic space. The quotient H 3 /Γ is hence a compact hyperbolic 3-orbifold, of which we study the hyperbolic volume and spectrum of closed geodesic lengths using SnapPea [58]. One consequence is a volume estimate for a "hyperelliptic" manifold considered in [35].

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Non-convexity of the space of dihedral angles of hyperbolic polyhedra

Cited by 13 publications

References 5 publications

The maximum volume of hyperbolic polyhedra

The maximum volume of hyperbolic polyhedra

Andreev’s Theorem on hyperbolic polyhedra

Constructing Hyperbolic Polyhedra Using Newton's Method

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