= Möbius band
{c}
{wiki}
= Möbius strip
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{synonym}
The Möbius band is the <topological surface> obtained by gluing the ends of a rectangle with a half-turn. Precisely, it is the <quotient space>
$$
M=([0,1]\times[-1,1])/\bigl((0,v)\sim(1,-v)\bigr).
$$
The twist identifies the two apparent long edges into one <boundary> circle. Following the <core circle of a Möbius band> reverses local handedness, so this surface is not <orientable>. An untwisted <untwisted band> has two <boundary> circles and is <orientable>: the distinction survives stretching and bending because <orientability> is a <topological invariant>.
To explore the construction with paper, mark the two long edges in different colours, join the short ends after a half-turn, and trace an edge. The trace passes from one marked edge to the other without ever leaving the <boundary>. Separately, trace the centre line and track a local transverse arrow: it comes back reversed after one circuit. The <boundary> closes after twice the travel around the centre, while the transverse arrow needs two circuits to return to its original direction.
\Image[/mobius-band-normal-reversal.png]
{title=One boundary and reversal of a continuously followed normal on a Möbius band}
This original mathematical model has central radius $R=2$ and half-width $a=0.55$:
$$
F(\theta,w)=\bigl((R+w\cos(\theta/2))\cos\theta,\ (R+w\cos(\theta/2))\sin\theta,\ w\sin(\theta/2)\bigr),\qquad 0\le\theta\le2\pi,\quad -a\le w\le a.
$$
Its seam obeys $F(2\pi,w)=F(0,-w)$. The dotted <circle> is the <core circle of a Möbius band>; the teal curve follows the single <boundary> through $0\le\theta\le4\pi$. At the same point on the core, the arrows for $\theta=0$ and $\theta=2\pi$ are opposite. The graph compares a continuously followed <unit normal> with its starting value in the ambient space; the endpoint reversal, rather than any intermediate zero of the graph, establishes the obstruction. It is not a statement about <parallel transport>.
The <Möbius band> has the <homotopy type> of a <circle>: a <deformation retraction> shrinks each width coordinate to zero. Nevertheless its <boundary> represents twice a generator of its <fundamental group>. The <boundary of a Möbius band winds twice around its core>, so with compatible generator choices the boundary generator maps to $c^2$ in $\pi_1(M)\cong\mathbb Z$.
The historical account by Julyan H. E. Cartwright and Diego L. González places the mathematical discovery by August Möbius and Johann Listing in 1858, while investigating representations of twisted bands predating that work. A familiar-looking picture and a recognition of the underlying <topological invariant> are different kinds of evidence. https://arxiv.org/abs/1609.07779[Cartwright and González, Möbius strips before Möbius, 2016]{ref}
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