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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.

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Hyperbolic space: (skip to about 4:00) youtube.com/watch?v=MKwAS5omW_w –  S. Carnahan May 14 '10 at 5:38
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2 Answers

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).

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I gave a talk describing some of the geometries, which has some figures picturing the geometries. These are mostly based on the descriptions in Thurston's book, which has some nice pictures. The shape of space also has nice pictures, but I don't think it describes all 8 geometries. In some sense, all but hyperbolic geometry may be pictured as 1-dimensional bundles over surfaces, or surface bundles over the circle. Hyperbolic geometry may be thought of as glass with varying index of refraction, and spherical geometry may also be thought of this way (I computed the conformal factor once, but I don't know it off the cuff).

I don't know of a figure that collates pictures of the geometries into one.

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Thanks! I especially like the hyperbolic glass image. –  cdouglas May 23 '10 at 2:01
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