Combinatorial and geometric methods in topology

"How many different objects can be obtained by gluing together in pairs the faces of an octahedron? After deciding what ""object"" and ""different"" mean, this is an apparently very elementary question, but the answer is not quite so. Before facing it I will go one dimension down and consider the gluings of the edges of a polygon, discussing surface topology and showing that there are extremely few surfaces one can get from a given polygon compared to the number of inequivalent gluing patterns. Then I will introduce the notions of curvature and hyperbolic geometry in two and three dimensions, I will discuss rigidity and I will sketch how this applies to the original question, yielding the fact that the number of different results is indeed quite big."