  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 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 $\pi$, so that the group
$\Gamma$ generated by reflections in the faces is a discrete group of
isometries of hyperbolic space. The quotient $\mathbb{H}^3/\Gamma$ is hence a
compact hyperbolic 3-orbifold, of which we study the hyperbolic volume and
spectrum of closed geodesic lengths using SnapPea. One consequence is a volume
estimate for a ``hyperelliptic'' manifold considered by Mednykh and Vesnin (see
references).
