Here is a nice cyclic ordering of the eight geometries:
HxR, SxR, E^3, Sol, Nil, S^3, PSL, H^3
derived from staring at Peter Scott's table of Seifert fibered geometries. The table is organized by Euler characteristic of the base 2-orbifold and Euler class of the bundle. (See his BAMS article.) The cyclic ordering also has a bit of antipodal symmetry.
I didn't come up with geometric pictures of the eight geometrics but I have thought about "icons" to represent them. Here are my suggestions - I'm interested to hear what other people think/suggest.
- HxR -- triangular prism (where the triangle is slim ie ideal)
- SxR -- cylinder
- E^3 -- cube
- Sol -- tetrahedron with one pair of opposite edges truncated
- Nil -- annulus with a segment of a spiral (representing a Dehn twist)
- S^3 -- circle
- PSL -- trefoil knot
- H^3 -- figure eight knot (or possibly a slim tetrahedron)
I think it is also reasonable to ask for a "prototypical" three-manifold for each of the eight geometries. Here is an attempt:
- HxR -- punctured torus cross circle
- SxR -- two-sphere cross circle
- E^3 -- three-torus
- Sol -- mapping cylinder of [[2,1],[1,1]]
- Nil -- mapping cylinder of [[1,1],[0,1]]
- S^3 -- three-sphere
- PSL -- trefoil complement
- H^3 -- figure eight complement
Notice that all of the examples are either surface bundles over circles or circle bundles over surfaces, or both (ie products).