There have been great strides made over the past 20 years in the under
standing of three-dimensional topology, by translating topology into g
eometry. Even though a lot remains to be done, we already have an exce
llent working understanding of 3-manifolds. Our spatial imagination, a
ided by computers, is a critical tool, for the human mind is surprisin
gly 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-manif
olds, addressing the question 'What are 3-manifolds like? rather than
'What facts can currently be proven about 3-manifolds?' The best curre
ntly available experimental tool for exploring 3-manifolds is Jeff Wee
ks' program SnapPea. Experiments with SnapPea suggest that there may b
e an overall structure for the totality of 3-manifolds whose backbone
is made of lattices contained in PSL(2, Q).