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I want to ask a question about the Three-torus model of the universe, as described in wikipedia:

https://en.wikipedia.org/wiki/Three-torus_model_of_the_universe

"is a proposed model describing the shape of the universe as a three-dimensional torus."

I'm not sure i understand the maths of three-dimensional tori, but I want to ask:

Does this model posit that the shape of the universe is that of a ring doughnut? If so, is the universe contained inside the ring doughnut, or is it supposed to exist only on the surface?

Or does the model say that the universe has the shape of a higher dimensional version of a ring doughnut? If so, does it exist inside the higher dimensional ring doughnut, or on its surface?

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  • $\begingroup$ Note that this model is inconsistent with the Cosmological Principle, since it's negatively curved "on the inside", and positively curved "on the outside". $\endgroup$
    – pela
    Apr 17, 2016 at 7:52
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    $\begingroup$ @pela - The only notion of curvature that's physically relevant in general relativity is intrinsic curvature, not the extrinsic curvature of a surface relative to some higher-dimensional embedding space, and the intrinsic curvature of a torus is zero everywhere, just like the intrinsic curvature of a cylinder (see the discussion on pages 2-3 here). $\endgroup$
    – Hypnosifl
    Apr 17, 2016 at 13:55
  • $\begingroup$ @Hypnosifl: Hmm, I can't visualize this, but I believe you. Thanks! $\endgroup$
    – pela
    Apr 17, 2016 at 18:00

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A three-torus is the boundary of the solid three torus, just like the two-torus is the surface of a solid donut. You can imagine it as a cubical room where each wall/ceiling/floor is a portal to the opposite-facing wall (i.e. the wall to your right is a portal that sends you to your left), but preserves orientation (when you walk out of the portal, your heart is still on your left).

You can also think of it as a world where your position is described by three coordinates (like x, y, and z in Euclidean space), but each coordinate corresponds to an angle on the unit circle. If you go far enough in one coordinate, you loop back to where you started (360 degrees is the same as 0 on a circle).

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  • $\begingroup$ The graphics (and accompanying explanation) in this article and this one may help with visualization. And for anyone who's willing to sit through an eleven-minute animated educational video, this is also very helpful. $\endgroup$
    – Hypnosifl
    Apr 17, 2016 at 14:12

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