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Consider the 4 × 4 grid graph:

enter image description here

Now torify it, i.e. connect its opposing vertices:

enter image description here

How can one tell the difference between a “meridian” and an “equator”?

The difference seems clear when looking at a “continuous” torus embedded in $\mathbb{R}^3$:


enter image description here

vs. equators

enter image description here

In the discrete case there seems to be no difference at all.

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The only reason the difference seems clear in the continuous case is that you made a choice of which pair of opposing sides of a square to connect first. – Gerry Myerson Dec 3 '12 at 23:22
But why can you make this choice in the continuous case - yielding a difference - but not in the discrete case? This is my question. – Hans Stricker Dec 3 '12 at 23:25
The embedding into $R^3$ preserves distances in one direction but not in the perpendicular direction. If you consider the continuous torus as a subset of $R^4$, the asymmetry between parallels and meridians disappears and the structure is completely symmetric. For example, an equation for a torus as a subset of $R^4$ is $x^2 + y^2 = R^2, z^2 + t^2 = r^2$, where $R$ and $r$ are the two radii. Similarly, the topological structure of the torus is just $S^1\times S^1$, where one $S^1$ is the bundle of parallels and one is the bundle of meridians, but there's no way to decide which is which. – MJD Dec 3 '12 at 23:27
@MJD: Why do you call "equators" "parallels" opposed to "meridians"? Do you refer to some literature? (I'd really like to know, because meridians seem to be as parallel as equators.) – Hans Stricker Dec 3 '12 at 23:35
In the continuous case, you are gluing things together. In the discrete, you are just drawing lines to indicate which things are adjacent, rather than identifying things that were previously distinct. If you really torify a grid, if you actually glue pairs of vertices, you'll fold it up into a torus with a grid on it, and you'll have the same meridian/equator distinction as continuously. – Gerry Myerson Dec 4 '12 at 0:27

The difference between an equator and a meridian is extrinsic to the torus, not intrinsic. A 2D being living on the surface of the torus might be able to distinguish between these two using distance measurements, but from a purely topological point of view, both directions are equal. Only in the embedding into 3D you can describe the difference as whether or not the loop goes around the hole in the torus. As your graph combinatorics does not carry sufficient information for a 3D embedding, you cannot tell the difference from these.

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