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I know from this answer that hexagonal space is topologically equivalent to a torus (and therefore to square space): Wrapped hexagon topology

My question is, if we allow a regular hexagon to exist in four-dimensional space, can we fold it up so opposite sides touch, and angles, distances, and areas are not distorted on the hexagon?

Perry Ainsworth
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    As Alex' answer points out, it's not possible to do this for a trivial reason. But if we glue opposite edges, we get a Riemann surface isometric to the flat torus, and this surface can be isometrically embedded in $\Bbb R^4$. – Travis Willse Dec 22 '18 at 04:54

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No. For instance, if the opposite sides touch, then distance between a point close to the midpoint of one side and a point close to the midpoint of the other side collapses.

Alex Ravsky
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