|Here we join up the top and bottom surfaces, but with a half twist.|
|So we get a one sided surface with a single external boundary line.|
In the program FriCAS (described here) there is a function: moebiusBand which generates a minimal triangulation of a Möbius band.
Möbius band: One boundary, shown in red:
(2) -> moebiusBand() +1 2 3+ |2 3 4| (2) |3 4 5| |1 4 5| +1 2 5+ Type: FiniteSimplicialComplex(VertexSetAbstract)
This diagram can only be realised in 3D, to make it clearer in 2D I have duplicated nodes 3 and 4:
The blue arrows should join up, notice the head of the arrow is at the top on the left and bottom on the right.
More detail, for instance, mapping Möbius band to circle on page here.