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 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. An error in Andreev’s original proof of existence was found by the author while writing the computer program described in this paper and corrected by the author in a joint publication with Hubbard, and Dunbar in [12, 18]. Our construction uses Newton’s method and a homotopy method to explicitly follow the existence proof presented in [12, 18]. 1 Statement of Andreev’s Theorem Andreev’s Theorem [5, 6] provides a complete characterization of compact hyperbolic polyhedra having non-obtuse dihedral angles. In [12, 18], we explain an error in Andreev’s proof found while writing the computer program described in this paper and we provide a correction of Andreev’s proof. Other approaches to Andreev’s Theorem can be found by Rivin and Hodgson [15, 9], Thurston [24], Marden and Rodin [14], and Bowers and Stephenson [8]. In this paper we show that the proof from [12, 18] is constructive when combined with Newton’s Method for solving nonlinear equations. Combinatorial descriptions of hyperbolic polyhedra that are relevant to Andreev’s Theorem fall into three classes, simple, truncated, and compound, all defined later in this section. The proof in [12, 18] provides an explicit continuous path in the space of polyhedra deforming a given simple polyhedron P to one of two which are easily constructed by hand: the N -faced prism PrN and the N -faced split prism DN . We use Newton’s method to follow such a [email protected] Partially supported by a U.S. National Defense Science and Engineering Fellowship and by the Fields Institute as a Jerrold E. Marsden postdoctoral fellow.