Learn what Möbius strips are, how to make them, and why they are such strange and interesting objects.
Do you know how to turn an everyday two-sided and four-edged piece of paper into an object that has only one side and one edge? If you think about that question for a minute you might convince yourself that it’s completely crazy. An object with one side and one edge sounds impossible, right? But, by the end of this article, you’ll see that it really is possible—especially since we’re going to actually make one of these objects called a Möbius strip.
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What Do Two-Sided Objects Look Like?
Just to make sure you understand how strange and interesting the one-sided object we’re going to talk about is, let’s start our discussion by taking a look at two-sided objects. So, what do two-sided objects look like? Well, there are lots of them. For example, you can think of a thin sheet of paper as a two-sided object. It has a front and back, which is why every sheet of paper in a book can be used to show two pages worth of text.
But two-sided objects don’t have to be flat and simple like a sheet of paper, they can also be more complex like the surface of a basketball. If you were a tiny ant, you could crawl around on the exterior surface of the ball. But, if you could get inside, you could also crawl around on the inside surface of the ball. So an object with a closed curved shape like a basketball is also two-sided.
Preparing to Make a Möbius Strip
Now that we understand what it means for an object to be two-sided, let’s think about what a one-sided object would look like. Actually, instead of just thinking about what a one-sided object would look like, let’s make one. That’s right—it’s time for a little hands-on mathematically-inspired art project.
All you need for this project is a normal sheet of paper, a pair of scissors, a pen or pencil, and a piece of tape. You’ll want to actually follow along and do this project—both because it’s the best way to learn and understand this stuff, and because it’s fun—which means that if you’re reading this article on the go, you may want to pause at this point and come back to it when you have your paper, scissors, pen or pencil, and tape ready.
How to Make a Möbius Strip
Okay, now that you have all of your supplies gathered, let’s make a one-sided shape called a Möbius strip. Here’s what you need to do:
Cut about a two-inch wide strip of paper from your full sheet and lay it out in front of you so that the long side of the paper is laying horizontally.
Write the letter “A” at the top-left corner of this strip, the letter “B” at the bottom-left corner, the letter “C” at the top-right corner, and the letter “D” at the bottom-right corner.
Hold the strip of paper in front of you. Now twist it one-half a turn so that the letters “A” and “B” on the left still face you but the letters “C” and “D” on the right now face away from you.
Bring the two short edges of your twisted strip together and tape them to make one long twisted loop. Corner “B” should match up with corner “C” and corner “A” should match up with corner “D”.
That’s it! The object in front of you is a Möbius strip…and it has only one side and one edge too!
Why Is a Möbius Strip One-Sided?
What exactly do I mean when I say that this object has only one side? Well, try taking your pen or pencil and drawing a line around the center of the entire strip. In other words, pick any starting location along your Möbius strip and then draw a line all the way around it along the middle of the strip. What happens? Go ahead and try it and see!
If you hadn’t put the half-twist in the strip before taping the ends together, your line down the middle would have covered only one of the two sides. In other words, if you had started on the outside, your line would have gone around the outside of the loop until it returned back to the starting point. Or, if you had started on the inside, the line would have gone completely around the inside.
[[AdMiddle]But that’s not what happens with a Möbius strip! (At least it shouldn’t have happened for you!) When you draw a line around the center of a Möbius strip, you’ll find that the line eventually reconnects back to the original line…but only after that line has covered the entire Möbius strip. Which means that this object has only one side—there is no separate front and back!
Why Is a Möbius Strip One-Edged?
You can also play a similar game by tracing your finger along one edge of the Möbius strip (just don’t give yourself a paper cut). You’ll find that if you keep your finger moving along an edge, you’ll end up touching every edge of the object and end up right back where you started—which means that this object has only one edge.
What Happens When You Cut a Möbius Strip?
Before we finish up for today, here are a few more experiments for you to try with your Möbius strip:
Experiment 1: Now that you’ve drawn a line all the way around the center of your strip, try using your scissors to cut along this line. Before you do it, be sure to take a minute to contemplate what you think is going to happen. So, what happens when you cut a Möbius strip “in half”?
Experiment 2: Make another Möbius strip. This time, instead of cutting along the center line, try starting about one-third of the way from an edge and cutting all the way around the strip at that distance from the edge. What do you think will happen? What actually does happen?
After you’ve done these experiments, be sure to check out the Math Dude’s Facebook page where we’ll be talking about the results.
Finally, you might be a little surprised that what we’ve talked about today is actually “math”—perhaps it seems more like arts and crafts! Well, it turns out that there are many types of math, and one of these types is all about studying the properties and shapes of interesting and bizarre objects like the Möbius strip. We’ll talk some more about this kind of math—called topology—in future articles.
Please email your math questions and comments to firstname.lastname@example.org. You can get updates about the Math Dude podcast, the “Video Extra!” episodes on YouTube, and all my other musings about math, science, and life in general by following me on Twitter. And don’t forget to join our great community of social networking math fans by becoming a fan of the Math Dude on Facebook.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading math fans!