Wrapped hexagon topology

Question

When a square is wrapped at the edges, obtains a torus. Is it possible to do the same for a hexagon?

What is the shape/topology of wrapping a hexagon?

rectangle wrapped to torus hexagon

Note: I was just reading this https://github.com/aichallenge/aichallenge/wiki/Ants-Problem-Description and came up with this question. I'm a physicist.

fyi if you take a $4g$-gon and identify sides $ababcdcd...$ you get a genus $g$ surface — yoyo, Jul 24 '11 at 13:53

score 19 · Accepted Answer · edited Mar 25 '21 at 04:41

19

It's still a torus, lol - just with a 180° twist relative to the square-wrapped version:

from http://www.math.cornell.edu/~mec/Winter2009/Victor/part1.htm

edited Mar 25 '21 at 04:41

uhoh

answered Jul 24 '11 at 10:46

anon

Thanks, that is exactly what I wanted. I didn't realize that the tube should be twisted before connecting the ends. – whoplisp Jul 24 '11 at 11:03
What makes the difference between a square and a hexagonal torus? Does it e.g. affect to way how to draw $K_{3,3}$ on a torus? – draks ... Aug 28 '19 at 11:43
@draks... Since the hexagon has threefold symmetry (actually more) I would assume torus constructed from a hexagon would better exhibit how to draw the $K_{3,3}$ (I mean you get a torus either way though). – anon Dec 09 '19 at 20:43
See also What surface do we get by joining the opposite edges of a hexagon? – uhoh Apr 04 '21 at 02:50

1 Answers1