dg.differential geometry - Drawing of the eight Thurston geometries?

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).