Here is a nice *cyclic* ordering of the eight geometries:

$$\Bbb H^2\times \Bbb R,\quad \Bbb S^2\times \Bbb R,\quad E^3,\quad \mathsf{Sol},\quad \mathsf{Nil},\quad \Bbb S^3,\quad \mathsf{PSL},\quad \Bbb H^3$$

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 BLMS 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.

- $\Bbb H^2\times \Bbb R$ -- triangular prism (where the triangle is slim ie ideal)
- $\Bbb S^2\times \Bbb R$ -- cylinder
- $E^3$ -- cube
- $\mathsf{Sol}$ -- tetrahedron with one pair of opposite edges truncated
- $\mathsf{Nil}$ -- annulus with a segment of a spiral (representing a Dehn twist)
- $\Bbb S^3$ -- circle
- $\mathsf{PSL}$ -- trefoil knot
- $\Bbb H^3$ -- 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:

- $\Bbb H^2\times \Bbb R$ -- punctured torus cross circle
- $\Bbb S^2\times \Bbb R$ -- two-sphere cross circle
- $E^3$ -- three-torus
- $\mathsf{Sol}$ -- mapping cylinder of $[[2,1],[1,1]]$
- $\mathsf{Nil}$ -- mapping cylinder of $[[1,1],[0,1]]$
- $\Bbb S^3$ -- three-sphere
- $\mathsf{PSL}$ -- trefoil complement
- $\Bbb H^3$ -- figure eight complement

Notice that all of the examples are either surface bundles over circles or circle bundles over surfaces, or both (i. e. products).