Math Series Part I: The Möbius Band
Some of the challenges and philosophical questions that mankind has battled with throughout history have been addressed directly by mathematics. Indeed, the study of mathematics has allowed us to pursue answers for some of the most impractical questions that are difficult or even impossible to test in reality. Although the application of some solutions are beyond what humanity is capable of at this point in time, an understanding of what they are and how they link to other mathematical and everyday principles of life grant us an extraordinary comprehension of how the world works, as well as providing insights into some of the unintuitive results of fascinating acts. This article aims to explore some of these acts and conceptions to address the remarkable ways in which mathematics and experiments can both perplex and intrigue us through unexpected and sometimes astonishing ways.
A Twisted Thought: The Möbius Band
We begin this exploration by considering the Möbius band, an interesting phenomenon whose discovery is owed to German mathematicians August Ferdinand Möbius and Johann Benedict Listing from 1858. It is a form of surface that seems to contain only one side with one boundary. A fascinating property of the Möbius strip is that it is non-orientable, meaning that, in Euclidean space, it is not possible to make a consistent choice about the surface normal at any given point on its surface.
The brilliance of this discovery is that it allows us to take a complex mathematical idea and bring it into a demonstrable object that increases layman understanding of its principles. For example, if we take a simple piece of A4-sized paper and cut lengthwise at about 4 centimeters from the edge, we can take that smaller piece of paper and create a Möbius band. By giving it one twist and then taping the two edges together in a ‘twisted’ loop, we are then able to draw a line down the middle from our connected edges that will circle the entire surface of the strip and meet back at our original starting point—without lifting our pen. And, remarkably, if we were to cut along down the center of this strip and marked line, it would produce a very unexpected result. Indeed, what we would expect to happen if we cut something in half would be for it to split equally into two equal halves. This does not happen with the Möbius band.
Instead, because we have included a half twist in the strip and have effectively reduced the surface to one boundary and one edge, the two halves form not two loops but rather one large loop of twice the original length. This also converts the Möbius band from a strip with half a twist to a strip with two full twists. This occurs because by connecting the two edges of our slice of A4 paper, we are effectively joining the two sides of the paper together to form a continuous strip that ends where it starts, which is why when cut in half it does not split in two but rather remains in this continuous form. Cutting down the middle has added a second independent edge that was on each side of the scissors as the paper was cut. If we were to draw another line down the center of this cut strip without lifting our pen, it would no longer meet at its starting position.
Although slightly similar to the Möbius band, if we were to cut down the middle of a piece of paper like this that was looped with a full twist, we would see the strip split into two equal, interlocked strips, each containing a full twist. The strip, both before and after cutting down the middle, would have two edges and two surfaces, and therefore not a continuous plane as with the Möbius band.
An equation that helps us identify the number, m, of half twists after cutting a Möbius band down the middle is m = 2n + 2, where n is the number of half twists before cutting. Thus, for our Möbius band, we will now have a strip of paper with two edges and two boundaries. If we take this larger strip and cut it down the middle, we will get two separate pieces this time that are interlocked and which both have four half twists.
What’s also interesting about the Möbius band is that if we were to cut at one third of the strip instead of the middle, we would expect to have another single strip with more twists—but the result is even more perplexing. We instead get two separate pieces this time, with the fatter piece remaining as a Möbius band with its original length—because it does not receive an extra edge from the cut—and the thinner piece having twice its original length with four half twists, two edges and two boundaries. This happens because as we draw a line down 1/3 of the band, we traverse the entire strip but we do not meet our original starting point until we have gone all the way around again. Because of this, the two strips form two separate loops that are interlocked.
These forms of strips and the way they can be created and shaped in real life allow a fantastic glimpse into the 3 dimensional aspects of an object with only one surface and edge. They grant an incredible didactic tool for understanding the challenging concepts of topology, orientability, homeomorphism and Euclidean space.