What does a "torus obtained by identifying each of the three pairs of opposite edges of a regular hexagon" look like? What is it called?

Asked Mar 19 '21 at 00:51

Active Mar 19 '21 at 01:01

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Wikipedia's Eisenstein integer; Quotient of C by the Eisenstein integers says:

The quotient of the complex plane C by the lattice containing all Eisenstein integers is a complex torus of real dimension 2. This is one of two tori with maximal symmetry among all such complex tori.^{[citation needed]} This torus can be obtained by identifying each of the three pairs of opposite edges of a regular hexagon. (The other maximally symmetric torus is the quotient of the complex plane by the additive lattice of Gaussian integers, and can be obtained by identifying each of the two pairs of opposite sides of a square fundamental domain, such as [0,1] × [0,1].)

I understand how identifying each of the two pairs of opposite edges of a rectangle makes a "normal torus" or donut, but I can't visualize what this torus might look like.

Does this torus have a name? Can it be illustrated?

edited Mar 19 '21 at 01:01

asked Mar 19 '21 at 00:51

uhoh

1,864

The image is copied in the answer itself. – Lee Mosher Mar 19 '21 at 01:02
And there is no particular term. Any topological space homeomorphic to THE torus $S^1 \times S^1$ is called A torus. – Lee Mosher Mar 19 '21 at 01:04
@LeeMosher wait, I understand the drawing! I'm good. – uhoh Mar 19 '21 at 01:08
1

Great, good to hear! – Lee Mosher Mar 19 '21 at 01:17
@LeeMosher I've just asked Does any polygon with side number 2n >= 2 form a torus when all pairs of opposite sides are joined? (works for n=2, 3) – uhoh Apr 04 '21 at 02:48

What does a "torus obtained by identifying each of the three pairs of opposite edges of a regular hexagon" look like? What is it called?

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