How to see 3-manifolds

There have been great strides made over the past 20 years in the understanding of three-dimensional topology, by translating topology into geometry. Even though a lot remains to be done, we already have an excellent working understanding of 3-manifolds. Our spatial imagination, aided by computers, is a critical tool, for the human mind is surprisingly well equipped with a bit of training and suggestion, to 'see' the kinds of geometry that are needed for 3-manifold topology. This paper is not about the theory but instead about the phenomenology of 3-manifolds, addressing the question 'What are 3-manifolds like? rather than 'What facts can currently be proven about 3-manifolds?' The best currently available experimental tool for exploring 3-manifolds is Jeff Weeks' program SnapPea. Experiments with SnapPea suggest that there may be an overall structure for the totality of 3-manifolds whose backbone is made of lattices contained in PSL(2, Q).