Can you really construct a Möbius strip from this model?

Solution 1:

Note that if you cut a disk out of your lens shape, the remaining annulus is embedded in $\mathbb{R}^3$ without any "twists". If this could be "glued" to itself to form a Möbius band in $\mathbb{R}^3$, then the (external) boundary of your lens-shape region would be mapped to the "soul" of the Möbius band; namely, the curve running along the middle of the Möbius band for one of the standard embeddings.

However, it is easy to check that if one cuts the Möbius band open along its "soul", one gets a strip with two twists as you make full circle around it, rather than the strip with no twists that would result if one removed a disk from the "lens". Thus such a construction is impossible. Note that the use of the term "soul" is consistent with the term used in Riemannian geometry for manifolds of nonnegative curvature.

Solution 2:

You could cut out a very large disk so only a small (half-)strip along the edge of the diagram is left. You then get the problem that this punctured but flat diagram is not twisted in the right amount to be glued together along the edge into a physical Möbius strip.

Alternatively, however, cut out a disk that spans the edge of the diagram. What is left is now ready to be twisted and glued together in the completely standard construction of a Möbius strip. That's not much extra fun, of course.