Abstract
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 .
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