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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    $\begingroup$ Hyperbolic space: (skip to about 4:00) youtube.com/watch?v=MKwAS5omW_w $\endgroup$
    – S. Carnahan
    May 14, 2010 at 5:38
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    $\begingroup$ Although 3-manifold topology is much richer (as Sam Nead's answer shows), the 8 geometries can be divided between hyperbolic, spherical, and collapsing (you can find a sequence of Riemannian metrics with bounded curvature where the injectivity radius everywhere converges to zero). This parallels the 2-dimensional geometries consisting of hyperbolic (surfaces of genus 2 or greater), spherical, and collapsing (flat torus). $\endgroup$
    – Deane Yang
    Oct 31, 2018 at 14:20
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    $\begingroup$ Found this 3-dimensional.space, I think is the same people mentioned at the end of Sam Nead answer $\endgroup$
    – Dabed
    Apr 24, 2021 at 2:23

4 Answers 4


Here is a nice cyclic ordering of the eight geometries:

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

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. (The original intent was to find eight pictures or objects suitable for a teething ring.) 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
  • $\Bbb E^3$ -- 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
  • $\Bbb E^3$ -- 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).

EDIT: In addition to the beautiful work of Zeno Rogue and Pierre Berger (linked to in the other answers) the reader may be interested in the visualisation efforts of Remi Coulon, Brian Day, Sabetta Matsumoto, Henry Segerman, and Steve Trettel. You can interact with seven of the eight here. Finally, here is a snap-shot of Sol geometry.

Flying through Sol geometry


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.

  • $\begingroup$ Dear Ian: your link doesn't work anymore and I couldn't find the talk on your homepage - would you mind posting another link? Thanks! $\endgroup$ Jun 5, 2014 at 3:14
  • $\begingroup$ @MariusKempe: I changed the link to a dropbox folder containing the slides - let me know if you can't access it. $\endgroup$
    – Ian Agol
    Jun 5, 2014 at 5:44

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).

enter image description here 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.

  • $\begingroup$ Thurston doesn't specify a Riemannian metric, but a Lie group acting on the manifold. What would Solv look like if, for geodesics, you used the Lie exponential instead of the Riemann exponential? It's easier to compute, and it's bijective (you won't see the same point in different directions): $$\exp\begin{bmatrix}w&0&u\\0&-w&v\\0&0&0\end{bmatrix}=\begin{bmatrix}e^w&0&\frac{e^w-1}wu\\0&e^{-w}&\frac{1-e^{-w}}wv\\0&0&1\end{bmatrix}=\begin{bmatrix}z&0&x\\0&1/z&y\\0&0&1\end{bmatrix},\quad z>0$$ $\endgroup$
    – mr_e_man
    Apr 17, 2023 at 4:51
  • $\begingroup$ @mr_e_man The visualization based on Lie exponentials is also implemented in our RogueViz/HyperRogue (change "projection" from "native perspective" to "Lie perspective"). [twitter.com/ZenoRogue/status/1526610527149035521](Here) is a Twitter thread with some examples. $\endgroup$
    – Zeno Rogue
    Apr 18, 2023 at 20:44
  • $\begingroup$ @mr_e_man (thanks for your comment about Thurston not specifying a Riemannian metric -- although, if I see correctly, assuming that the Lie group operation is an isometry, there is not much freedom: possible Riemannian metrics differ only by scale (on the 'z' axis) and by the angle between the 'x' and 'y' axes. ) $\endgroup$
    – Zeno Rogue
    Apr 18, 2023 at 23:19

Not only could you see the eight geometries at ihp's exposition esthetopies last summer, but you could also hear them. The exposition is now over but the pictures are on the site of Pierre Berger, the author of the exposition. Here is a picture of SOL (copyright P. Berger)

SOL geometry

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    $\begingroup$ Last year, I was at Universum, the science museum in Mexico City, and was lucky enough to get a "back-stage tour" of Imaginario Matemático, the math section that was under renovation at the time. They were working on an (I think permanent) exhibit just like what you describe. It may not have opened yet because I'm not finding anything about it online, but if you look carefully at the 2nd-to-last picture in the "Galería" on this page universum.unam.mx/exposiciones/imaginario-matematico you can see it in the background. $\endgroup$
    – j0equ1nn
    Feb 19, 2019 at 17:01

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