We enumerate all the spaces obtained by gluing in pairs the faces of the octahedron in an orientation-reversing fashion. Whenever such a gluing gives rise to nonmanifold points, we remove small open neighborhoods of these points, so we actually deal with three-dimensional manifolds with (possibly empty) boundary.
There are 298 combinatorially inequivalent gluing patterns, and we show that they define 191 distinct manifolds, of which 132 are hyperbolic and 59 are not. All the 132 hyperbolic manifolds have already been considered in different contexts by other authors, and we provide here their known “names” together with their main invariants. We also give the connected sum and JSJ decompositions for the 59 nonhyperbolic examples.
Our arguments make use of tools coming from hyperbolic geometry, together with quantum invariants and more classical techniques based on essential surfaces. Many (but not all) proofs were carried out by computer.
"The 191 Orientable Octahedral Manifolds." Experiment. Math. 17 (4) 473 - 486, 2008.