Loopy Love

Last week we suggested a Valentine’s Day gift that is both romantic and mathematical: a pearl necklace that can be arranged as a beautiful 3,4,5 right triangle. A great present, true, but perhaps you’re not ready to shell out thousands of dollars. So, this week we’d like to offer an impressive and very cheap alternative: Möbius paper magic.

You may already be familiar with Professor Möbius’s intriguing discovery. To make a Möbius strip we take a rectangular strip of paper, give it a half-twist of 180 degrees, and then staple the ends together.

Möbius strips are eye-catching, and it doesn’t take long to discover some of their intriguing properties. Most famously, a Möbius strip is one-sided: if we imagine a flat QED cat travelling along the paper strip, he will wind up on the other side of the paper (and upside down).

Next, you can cut your Möbius strip “in half”, by cutting along the waist of the loop. Your romantic friend will probably be amazed when the strip doesn’t fall into two pieces, but remains as one long, twisty loop.

There are many similar and slightly more intricate tricks you can perform. For example, you can begin by giving the rectangular strip two or three half-twists, rather than just one, before stapling. Is the resulting loop still a Möbius strip? See what you now get when you cut down the middle of your loop. 

That’s all fun, but we want to get to the really romantic bit. To do that, we’ll first take a step backwards and consider plain old cylinders.

Not very thrilling, but what if you staple two cylinders together at right angles, and then cut along the middles of the cylinders? (To ensure the cylinders won’t just fall part, use four staples to attach them, as shown, and avoid the staples when cutting.) What do you get?

Hardly anybody guesses that the end result will be a perfect square:

Not very romantic, but what if you begin with a cylinder stapled to a Möbius strip? Surprisingly, you’ll still just end up with a square. And now comes the really cool bit.

Let’s begin with two Möbius strips stapled together. What will you get now? It depends.

There are actually two types of Möbius strips. The type you have depends upon whether you’ve given the paper strip a left-twist or right-twist before stapling. Pictured above, we have one Möbius strip of each type stapled together.

And what happens when you cut this Möbiuscombo down the middles? You get two hearts, eternally linked! Awww!

We’ll leave you to discover what happens if you begin with two Möbius strips of the same type stapled together. And there is plenty more Möbius magic beyond. In a future column, we’ll even take you on an excursion into 3D Möbius worlds.

These paper games are very impressive. But of course, for a longer term romance you’ll probably still be opting for jewellery. Fortunately, Möbius strips are available in many beautiful materials. So, if platinum and diamonds are more your style, you can still engage in some truly stunning Möbius magic. 

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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