[Math Lair] The Möbius Strip

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Any strip of paper joined at the ends to form a continuous round band has two edges and, as one would expect, two surfaces: an exterior surface and an interior surface. However, giving the strip of paper a half-twist before joining the ends produces a band with a single surface and a single side, known as a Möbius strip. This strange phenomenon was first described by the nineteenth-century German mathematician August Ferdinand Möbius, for whom the strip has been named.

A Möbius strip, cut out from a piece of paperYou can try this yourself; cut a strip of paper about 1–2 inches wide (2.5–5 cm) and 1–2 feet long (30–60 cm), give one side a half-twist, and then tape the two ends together. If you do so, it is easy to show that the strip only has one side by drawing a pencil line down the middle of the band without lifting the pencil from the paper. The result is that "both" surfaces have a continuous pencil line running around them.

The result of cutting a Möbius strip in half through the middle of the strip.Cutting a normal loop of paper along the centre of the loop, the result is two loops of paper. So, if one were to cut along the middle of a Möbius strip, one might perhaps expect two Möbius strips. However, this is not the case. Rather, one larger band results. This new strip has four half-twists (or two full twists) instead of the original one half-twist, so it now has two surfaces and two edges. So, it is no longer a Möbius strip. However, cutting this new, larger strip down the middle results in two interlocking bands, each of which also has four half-twists.

The result of the second cut along the middle.

You may want to experiment with other possibilities. What happens if you cut the Möbius strip 1/3 of the way from the edge, instead of in the middle? What if you create strips with two half-twists, or three half-twists?