Mobius Loops That Zip Together

If somebody had told me 10 years ago that having kids would result in my liking maths, I wouldn’t have believed them.

In fact, I think I would have laughed in that person’s face.

Just as having a transport mad child means that I know more about buses than I would care to, having a maths mad child means that I now love Mobius loops almost as much as mathematicians do.

What the Heck is Mobius Loop?

Mobius loops are the things that infinity scarves are.

To make a Mobius loop, you take a strip of something (probably paper), bend it round into a circle, but before you stick the ends together, twist it. Then you stick the ends together.

By doing this, you take something thing that has 2 sides and 4 edges and turn it into something that has 1 side and 1 edge.

If you then cut it in half, lengthways, you would expect to get 2 circles. But that doesn’t happen! I’m not going to tell you what does happen because you can try yourself ;). It’s weird. If you twist it more times, different things happen. Mathematicians love them!

Why Would I Want to Zip 2 Mobius Loops Together?

Joining 2 Mobius loops together along their edges will result in a shape that has 1 side and no edges. This is a Klein bottle.

If you make a Klein bottle (like this hat I crocheted for my little mathematician!), where the handle thing joins the to the inside, it will have to pass through the outside.

In 4 dimensions, it wouldn’t have to do that. It would join onto the inside bit without having to pass through the outside.

Sticking paper Mobius loops together with sellotape doesn’t work very well. I tried it the other day and the tape came unstuck and I got in a muddle.

Zipping them together means that they will stay together, although you will have to leave a gap for the outer bit to pass through to the inside.

How to Make Mobius Loops That Zip Together

You Will Need

Pieces of cotton fabric, 2 zips the same length as your fabric, usual sewing supplies.

Open ended zips are best. I made the mistake of buying closed ended zips, which had to be carefully separated with a pair of scissors.

How to Make the Mobius Loops

1. From your fabric, cut 4 pieces 15 cm x 42 cm. I was using up stuff I had already, so I cut 4 pieces from different fabric.

2. Take 2 of the pieces and pin them with the wrong sides together. Sew down the middle, leaving a couple of cm free at either end.

3. On one of the long edges, fold the raw edges inwards about a cm. Take one half of one of the zips and slip it in between the folded edges. Pin it, then sew the zip in place.

4. Take the other zip and divide it into 2. Sew this zip half into the other long edge in the same way. Make sure that both the zips are the same way round. It’ll avoid confusion later!

5. At one end, fold the raw edges inwards. Then bend the strip round to make a circle. Twist it once so that the inside is facing upwards, then slip the unfolded end in between the folded ends. Pin, then sew.

6. Once you’ve done this, check the zips. You might need to sew over the ends in again, especially if you had to unpick them a bit to fold the raw edges in.

Make a second Mobius loop in the same way, except when you twist it, turn the twist the other way. So on my first Mobius loop, I twisted it towards the right, and on the second one, I twisted it towards the left.

Turning the Mobius Loops into a Klein Bottle

To make the Mobius loops into a Klein bottle, you’ll need to zip them together.

Because my zips kept coming apart at the bottom, I put a couple of stitches at the bottom of each one.

Turning them into a Klein bottle is a bit fiddly. You’ll have to leave a gap (you wouldn’t need to in 4 dimensions but unfortunately you have to in 3!) and it helps if you know what a Klein bottle looks like. The key thing is to try to get the handle shape.

These would make a fun gift for a mathematician! Or just for yourself if you like that kind of thing 🙂

If you’d like to find out more about Mobius loops and Klein bottles, my boys love this Numberphile video featuring Cliff Stoll.

<iframe width=”560″ height=”315″ src=”https://www.youtube.com/embed/AAsICMPwGPY” frameborder=”0″ allowfullscreen></iframe>