Construction of a 3/4-ideal hyperbolic tetrahedron out of ideal tetrahedra (Q701780)

Since the volume of a hyperbolic \(3\)-manifold is a topological invariant, it is of interest to compute the volumes of the hyperbolic tetrahedra of a triangulation. Every finite tetrahedron can be constructed from an ideal tetrahedron and three 3/4-ideal tetrahedra (i.e., tetrahedra with only one finite vertex). The author expresses a 3/4-ideal tetrahedron in terms of three bilaterally symmetric 3/4-ideal tetrahedra, then expresses each bilaterally symmetric 3/4-ideal tetrahedron as a \`\` supplementary\'\'\ 3/4-ideal tetrahedron and an ideal terahedron, and then decomposes a \`\` supplementary\'\'\ 3/4-ideal tetrahedron into two ideal tetrahedra. Thus he shows how to construct any 3/4-ideal tetrahedron out of \(10\) ideal tetrahedra.