Guillemin-Pollack: “Moebius strip is not orientable”

Guillemin and Pollack define an orientation on a $k$-dimensional manifold $X\subset \mathbb R^N$ with boundary is a smooth choice of orientations for all tangent spaces $T_x(X)$. (The smoothness conditions means this: around each $x\in X$ there must exists a local parametrization $h: U\to X$ such that $dh_u: \mathbb R^k\to T_{h(u)}(X)$ preserves orientation at each $u\in U\subset H^k=\{(x_1,\dots,x_k): x_k \ge 0\}$.

They then claim that the Moebius strip isn't orientable. "You are invited to make a paper model and "prove" pictorially that no smooth orientation of the Moebius strip exists. The difficulty is that if you walk around a transparent strip pitching pennies heads up, eventually you return to the starting point to find tails up!"

I have no idea how their "argument" with heads and tails is connected with the definition they gave, and what it means mathematically that "we walk around a strip".

Another (actually, I guess the same) argument I've seen is that one take a normal vector, "translates" it around the strip, and notes that the direction is changed. But again I don't understand what such argument contradicts to (and again what it means to translate the vector).

• They assume you can see through the paper and that the paper has no thickness. Their metaphor and intuition are not consistent. – The Count Mar 28 '18 at 0:29