TITLE: Drawing of the eight Thurston geometries?
QUESTION [60 upvotes]: Do you know of a picture, drawing, or other concise visual representation of the eight three-dimensional Thurston geometries?

I am imagining something akin to the standard picture (of a sphere, plane, and saddle) used to illustrate the three constant curvature geometries in dimension two.  Of course, it takes more doing to illustrate representative three-manifolds, and there are more choices for natural examples, but I was surprised when I couldn't find such a picture.  Another option would be to  depict or indicate some of the geometries in less direct ways, for instance via the structure of stabilizers.

REPLY [18 votes]: We have recently started working on visualizing Sol.
Sol is defined by the following metric in $\mathbb{R}^3$: $ds^2 = (e^zdx)^2 + (e^{-z}dy)^2 + dz^2$
I think it is quite easy to see what is going on there: there is a Z coordinate and moving along this Z coordinate makes your Y steps larger (double each $\log(2)$ moved on the Z axis) while making your X steps smaller (double each $-\log(2)$ moved). In a hyperbolic space both would expand together (that's how the half-space model works if you replace the Z coordinate by its logarithm).

SolvView by MagmaMcFry gives a native perspective visualization of both Sol and Nil. By native perspective I mean that the view you see here is the view you would get if you were inside the space, assuming that the light travels on geodesics.
We have added Sol as a playable geometry in the current beta of HyperRogue (viewable both in the native perspective projection and in projection of the simple model above). Here is a video of a camera rotating in Solv, looking at some surfaces of constant Z.
UPDATE: we have implemented all Thurston geometries. See the release post and the geometry page.