Möbius Strips

The Möbius strip, also called the twisted cylinder, is a one-sided surface with no boundaries. It looks like an infinite loop. Like a normal loop, an ant crawling along it would never reach an end, but in a normal loop, an ant could only crawl along either the top or the bottom.

A Möbius strip has only one side, so an ant crawling along it would wind along both the bottom and the top in a single stretch. A Möbius strip can be constructed by taking a strip of paper, giving it a half twist, then joining the ends together.

Möbius strips can be any size and shape, some of which are easily visualizable in Euclidean space, and others of which are are not easy to visualize. The topology of Möbius strips make it a rare Euclidean representation of the infinite, and mathematicians have expanded on this and generalized it in the form of Klein bottles.

Here is how to make a Möbius strip:

1. Cut a long strip of paper. The strip should be several centimeters across, and the length $l$ should be much longer than the width $w$.
2. Bring the ends together to make a simple loop.
3. Before attaching the ends together, add a single half-twist to one side of the strip (as in the image to the right).

Enjoy your Möbius strip!

The Möbius strip is one-sided, which can be demonstrated by drawing a line down the center of the Möbius strip. By following this line with your finger without lifting your finger from the surface, when your finger has traveled the length $l$ of the strip, it is on the other side of the piece of paper from the starting position. Continuing to trace the center line, your finger will return to the starting position after traveling a total distance of $2l$. By this property, for any two points in the Möbius strip, it is possible to draw a path between the two points without lifting your pencil from the piece of paper or crossing the edge.

The Möbius strip also has only one boundary, which can be demonstrated by tracing the edge of the Möbius strip with your finger. Just as with the line down the center, by following the boundary line with your finger, when your finger has traveled the length $l$ of the band, it will be on the boundary edge of the Möbius strip directly opposite from the starting point, and by continuing to trace the boundary edge, your finger will return to the starting position after traveling a total distance of $2l$. The Möbius strip has Euler characteristic $\chi=0$

Consider a cylindrical shell, which is the shape of a tin can with top and bottom removed. This object is obtained by taking a rectangle and identifying two of the edges with the same orientation.

Now, what happens if we flip one of the orientations of the arrows in the above diagram? The resulting identification of directed boundaries gives a Möbius strip.

Think about how the gluing of these edges would happen in 2-dimensional Euclidean space. It turns out it is not possible to glue these edges without self-intersections in 2 dimensions. However, by allowing another dimension, we can allow the "twist" to live in the third dimension and, as shown in the above construction, a Möbius strip in 3-dimensional Euclidean space is non-self-intersecting. Cylinders and Möbius strips are examples of 2-manifolds with boundary, topological objects such that the neighborhood of each point looks locally like Euclidean space, i.e. a plane.

Continuing this idea, what happens if we identify all four edges in the rectangle to each other? Note that as with the cylindrical shell and Möbius strip, we have two choices for the orientations of edges, either in the same direction or different directions.

The Möbius strip has played a prominent role in mathematics, art, magic, and literature, such as in the works of M.C. Escher, "Möbius Strip I" and "Möbius Strip II (Red Ants)."

How about a well-work Möbius strip joke?

Q: "Why did the chicken cross the Möbius strip?"

A: "To get to the same side!"

What happens when a Möbius strip is cut down the center line?

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You should first try this experiment yourself!

Instead of getting two strips, the result is a single strip with one full twist $(360^\circ).$ $_\square$

• Can you see why this is the case by considering the directed edge identification diagram for the Möbius strip?

Now, what happens if you draw a center line down this resulting figure? Try cutting the strip down the center line a second time and see what you get.

• Can you explain the result?

Now, instead of drawing a line down the center of the Möbius strip, draw a line with distance $\frac{1}{3}w$ from the edge. What happens when a Möbius strip is cut down this line? Is it the same as the above example?

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You should first try this experiment yourself!

The result is two strips:

• one Möbius strip of width $\frac{1}{3}w$ and length $l$ and
• one longer strip of width $\frac{1}{3}w$ and length $2l$ with one full $(360^\circ)$ twist.

This is different than in the first example because the cut along the middle of a Möbius strip returns to the starting point after the length $l$ of the strip. However, by cutting from one-third of the way from the edge, the return to the starting point occurs at a total distance of $2l$. $_\square$

Can you manipulate the interlocked rings (without creating any sharp creases) to stack on top of each other to form a stack of 3 Möbius strips? (This is shown in Martin Gardner's Mathematical Magic Show.)

Take a piece of paper and make a $k$ half twists of $180^\circ$ before identifying the edges $($where $k=1$ half twist gives a Möbius strip$).$ How many boundaries and how many sides does the resulting figure contain?

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You should first try this experiment yourself!

If $k$ is even, the result has two boundaries and two edges. A special case is $k=0$, which gives the cylindrical shell as shown in the edge identification diagram. For $k$ even, if the strip is cut along the center line, it will separate into two rings, and each ring will contain $k$ half twists.

If $k$ is odd, the result has one boundary and one side. If this strip is cut along the center line, the result is a single strip with $2k$ half twists. $_\square$