on the whole space if the two following conditions are satisfied: for each (non-ideal) vertex in the gluing the sum of the solid angles of the polyhedra to which it belongs is equal to ; for each edge in the gluing the sum of the dihedral angles of the polyhedra to which it belongs is equal to . A notable example of this construction is the Seifert–Weber space which is obtained by gluing opposite faces of a regular dodecahedron. A variation on this construction is by using hyperbolic Coxeter polytopes (polytopes whose dihedral angles are of the form ). Such a polytope gives rise to a Kleinian reflection group, which is a discrete subgroup of isometries of hyperbolic space. Taking a torsion-free finite-index subgroup one obtains a hyperbolic manifold (which can be recovered by the previous construction, gluing copies of the original Coxeter polytope in a manner prescribed by an appropriate Schreier coset graph). Gluing ideal tetrahedra and hyperbolic Dehn surgery In the previous construction the manifolds obtained are always compact. To obtain manifolds with cusps one has to use polytopes which have ideal vertices (i.e. vertices which lie on the sphere at infinity). In this setting the gluing construction does not always yield a complete manifold. Completeness is detected by a system of equations involving the dihedral angles around the edges adjacent to an ideal vertex, which are commonly called Thurston's gluing equations. In case the gluing is complete the ideal vertices become cusps in the manifold. An example of a noncompact, finite volume hyperbolic manifold obtained in this way is the Gieseking manifold which is constructed by gluing faces of a regular ideal hyperbolic tetrahedron together. It is also possible to construct a finite-volume, complete hyperbolic manifold when the gluing is not complete. In this case the completion of the metric space obtained is a manifold with a torus boundary and under some (not generic) conditions it is possible to glue a hyperbolic solid torus on each boundary component so that the resulting space has a complete hyperbolic metric. Topologically, the manifold is obtained by hyperbolic Dehn surgery on the complete hyperbolic manifold which would result from a complete gluing. It is not known whether all hyperbolic 3–manifolds of finite volume can be constructed in this way. In practice however this is how computational software (such as SnapPea or Regina) stores hyperbolic manifolds. Arithmetic constructions The construction of