Starting from the (apparently) elementary problem of deciding how many different topological spaces can be obtained by gluing together in pairs the faces of an octahedron, we will describe the central role played by hyperbolic geometry within three-dimensional topology. We will also point out the striking difference with the two-dimensional case, and we will review some of the results of the combinatorial and computational approach to three-manifolds developed by different mathematicians over the last several years.

This expository paper is the text of a conference given by the author to a broad audience of mathematicians. An upcoming article will contain more detailed proofs and an accurate description of the hyperbolic and non-hyperbolic orientable manifolds obtained by gluing together in pairs the faces of the octahedron. Appendix by Damian Heard and Ekaterina Pervova