# What surface do we get by joining the opposite edges of a hexagon?

We know that torus can be obtained from a square by joining the opposite sides.

Can we do the same thing with a hexagon?

If it’s not possible to ‘fold’ hexagon in 3D Euclidean space, may be it’s possible in highter dimensions?

Of course, like in the example with a square, stretching is allowed.

I intentionaly don’t use the correct topology terms, since I don’t know topology. I probably should’ve used terms such as manifold and homeomorphism.

I hope the question is clear enough in layman terms.

Related question: What are all topological spaces obtained by gluing the edges of a triangle?

Edit

A useful link from Fredrik Meyer which probably answers the question. And the image from the link which should help with orienation:

we see that the surface is the torus.

Says the link. If it’s true, can we show how folding a hexagon results in a torus?

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How are you orienting the pairs of edges for the join? The result would be different depending on those choices. (E.g., should the top edge going left-to-right be identified with the bottom edge left-to-right, or with the bottom edge right-to-left? And then the same question for the other pairs of edges.)
– Mark Dickinson
2 days ago

@MarkDickinson, I thought the picture was clear enough. The arrows show which sides should be joined.
– Yuriy S
2 days ago

3

– Fredrik Meyer
2 days ago

@MarkDickinson, now I get it. They shouldn’t be reflected. For example, the square above could be folded into a Mobius strip instead of torus. I mean nice bending without twist
– Yuriy S
2 days ago

@FredrikMeyer, thanks. The link says it’s the torus again, but I don’t see how it could be obtained
– Yuriy S
2 days ago

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