Constructing Hyperbolic Polyhedra Using Newton's Method

Abstract

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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Projective deformations of hyperbolic Coxeter 3-orbifolds

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