The Curious World of Möbius

Author: Cindy Hu

From: West Orange, NJ, USA

Image credit

A mathematician confided

That a Möbius band is one-sided

And you’ll get quite a laugh

If you cut it in half,

For it stays in one piece when divided.

- Classic mathematicians’ limerick


Introduction


Who said math and art were separate subjects? Or that logic and creativity had to be kept apart?


Let’s try an experiment. Go out onto the street (not forgetting to social distance) and ask someone how many sides a piece of paper has. They will most likely give you an answer of 2, along with a very strange look.


The fact that a sheet of paper has 2 sides is something we all take for granted. For example, when we’re done writing on one side of a page, we just flip it over and keep on writing. Printers are even programmed to print double-sided pages to save paper.


I’m about to prove this all wrong.


A little DIY


Take a piece of paper, some tape or glue, and some scissors if you have them. Cut or tear out a strip long enough to make a loop. Give one end a half-twist and tape or glue it to the other end. Behold your new Möbius strip!


Why I made you do the strange experiment and craft


It’s because a Möbius strip is an example of a piece of paper with only one side. Don’t believe me? Trace a line along one side of the Möbius strip. Keep going, and you’ll end up on the other “side” of the strip. Go even farther, and you’ll end up exactly where you started.


Now try this: cut the Möbius strip in half along the middle. Keep going until you get back to where you started. Since you cut it in “half” along the middle, you would expect to get to halves, right?


Apparently not, because all you get is one long loop of paper, half as thin as the original Möbius strip.


How does it work?


The curious properties of the Möbius strip lie in its odd number of half-twists. If you’d done three half-twists instead of one, you’d get a very similar result. Give that a try, and you might find that it looks vaguely familiar - it’s the universal recycling symbol! Who knew that we could look at one-side objects our whole life without realizing it?


Möbius strips ended up being very important to the field of mathematics. It played a major role in the development of a subject known as topology, which studies the properties of objects only by bending or twisting them without breaking them.


Furthermore, Möbius strips give us a glimpse into how the higher dimensions of our universe work. For a long time, humans were content with the idea that only three dimensions exist: length, width, and height. Anyone who said otherwise was scorned because it’s impossible to visualize a fourth dimension. But just because we can’t see higher dimensions doesn’t mean they aren’t there.


Consider a world in 2D, where everything is flat. Attempting to prove the existence of the third dimension to its inhabitants would likely get you admitted into a mental institution or prison. Now consider a pair of scissors located in this world, flat and made for right-handed people. (I could write a whole separate article on discrimination against lefties, but not now.) Suppose someone like me wanted to use the scissors. In 3D space, it would be easy enough to just pick it up and flip it over so my left hand could hold it (the blades will be oriented differently, but that’s beside the point.) However, in our 2D world, “up” doesn’t exist and “pick it up” has no meaning.


How do we solve this dilemma? We return back to our friend the Möbius strip! If we take the flat pair of scissors, place it on the strip, and move it all the way around so it returns to its original starting point, it will be reversed into a left-handed object. Our world’s 2D inhabitants would be quite confused, and they’d probably deem it magic. But we know otherwise.


This begs a question: if we create 3D-like effects in a 2D world, what’s to say we can’t create 4D-like effects in a 3D world? Someday in the future, we might be well on our way to investigating the mysterious fourth spatial dimension.


Conclusion and further investigations


German mathematician August Ferdinand Möbius is usually credited with the discovery of the Möbius strip, although other Johann Benedict Listing had also described it in his unpublished work. This magical piece of paper has fascinated people for centuries, and for good reason.


If you’d like to find out more about Möbius strips, the fourth spatial dimension, or topology, feel free to research the mathematicians I’ve mentioned above. Additionally, Flatland by Edwin Abbott is a fun novella about life in a hypothetical 2D world. Beyond its mathematical twist, Flatland also offers some satirical commentary on caste and social status in Victorian England. Lastly, Hyperspace by Michio Kaku offers more insight into the higher dimensions of our universe and how they connect to time travel and the four types of forces scientists have discovered. I’m reading this book right now, and it provided me with the inspiration for this post.


Just remember, if you ever wake up and find that your organs have been reversed, you may have made your way around a 3D Möbius strip in your sleep!

 

Author: Cindy Hu


Cindy is a high school junior in NJ who loves math, physics, and biology, among many other subjects. Her hobbies include reading, playing the piano, and learning to crochet.

 

Citations:

  1. The Editors of Encyclopaedia Britannica. (2020, November 13). August Ferdinand Möbius. Retrieved December 10, 2020, from https://www.britannica.com/biography/August-Ferdinand-Mobius

  2. Gunderman, D., & Gunderman, R. (2018, September 25). The Mathematical Madness of Möbius Strips and Other One-Sided Objects. Retrieved December 10, 2020, from https://www.smithsonianmag.com/science-nature/mathematical-madness-mobius-strips-and-other-one-sided-objects-180970394/

  3. Kaku, M. (1995). Hyperspace: A scientific odyssey through parallel universes, time warps, and the tenth dimension. New York, New York: Oxford University Press.

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