|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 Moebius 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.