The Torus and the Möbius Strip

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On the face of it, the torus and the Möbius strip look like they have very little in common. The Möbius strip, which is just a rectangular strip of paper that has been given a half twist and then glued together at the ends, is famous for being non-orientable. If you draw a line along the center of the strip, you eventually get back to where you started. When you zoom out and look at the whole strip, you see that both side of the rectangular strip are marked. Alas, the Möbius strip only has one side! When I did this exercise as a small child during a grade school activity, it left a deep impression.

On the other hand the torus is perfectly orientable. A surface being orientable just means that you can coherently assign two distinct sides to it. For example, with the torus, it is clear what we mean when we talk about its inside and its outside.

Another difference is the fact that the Möbius strip has a boundary, along the edges of the rectangular strip that aren't glued, while the torus has none.

It was thus very surprising for me to learn as a grad student that there is a simple operation that turns the torus into the Möbius strip.

To understand this operation, first notice that the points of the torus are specified by two periodic angular coordinates, which we will call \(u\) and \(v\). Moving the \(u\) coordinate move the point around a circle in the horizontal plane and moving the \(v\) coordinate moves the point around a different circle in a vertical plane.

When we glue each point \((u,v)\) to the point with coordinates flipped \((v,u)\), the resulting space is miraculously the Möbius strip! This gluing operation is also known as taking the quotient. We saw this operation when looking at the projective plane, where we glued together opposing points of the sphere to get the projective plane.

In the demonstration above, this gluing operation is indicated by showing the pair of points that are glued together. The diagonal of the torus, where the two coordinates are equal is drawn in blue. To show that the resulting space is indeed the Möbius strip, the corresponding point is drawn on the Möbius strip.

To move the point around you can change the coordinates in the panel. You can also do so by clicking and dragging on the torus and the Möbius strip. When the pair of torus points is moved onto the diagonal, they merge into a single point.

You will also notice that the diagonal of the torus maps to the boundary of the Möbius strip. When we move perpendicularly across the rectangular strip from one part of the boundary to the other corresponds to moving from one point of the torus diagonal to another point on the diagonal diametrically opposed.

If you select the flow option, you'll see several points flowing across the mobius strip and torus. One group of points on the torus satisfies the equation \[ v(t) = u(t) + \delta_i, \] for some fixed \(\delta_i\), while \(u(t)\) steadily loops from \(-\pi\) to \(\pi\). On the torus, you'll be able to identify a group of points with the same \(u\) coordinate cycling in and out of the torus, as well as the group of points they are glued to with the same \(v\) coordinate cycling around like a merry-go-round. The resulting paths on the Möbius strip are paths of constant distance from the boundary.