Möbius strips and more

I have taken a short break from cleaning up The Cornish Boy Quartet, which is Australian YA historical fiction, because I wanted to share something on this topic, after it came up on a librarian list which does not allow attachments. It is a selection from my Playwiths, available from Amazon or through Polymoth Books. The apparent supplier is really just me, trading as Polymoth Books, but I set the firm up so I can supply booksellers and libraries more cheaply (note that some conditions apply).

That said, this next part is free, and teachers looking for ideas can get those from the e-book, available at the Amazon link, for $5.

Cut a 5 cm strip lengthwise from paper (an old newspaper will do). Holding the strip out straight, give one end a half twist (180º) and glue or tape the two ends together. Your piece of paper is now a Möbius strip. When you twisted your strip, the inside and the outside became one continuous surface. There is also only one edge.

Take a pen and carefully draw a line along the centre of a new uncut strip. Where do you end up? Is the line drawn on the inside or outside of the paper? Now cut the strip along the line you drew. How many pieces do you get? It may help if you use the picture below to make an ant-covered Möbius strip: here is a link to a PDF that you can download and print.

Möbius ants!

You can use the PDF (this is recommended), or blow the above image up on a photocopier, so the chain of ants is 23 cm long then join two copies, as shown below, and do back-to-back photocopies. You need to experiment to get the ants on opposite sides of the page, going in opposite directions.

I had a bit of trouble following my own instructions, so here’s a step-by-step set of photos: remember that you need to print both sides

(1) PDF on-screen; (2) printed out; (3) cut up; (4) trimmed; (5) joined; and (6) a finished Möbius strip.

Cutting the Möbius strip in two different places.

Next, take the photocopied or printed sheets and cut two strips, 23 cm x 7 cm, and join them, so all the ants are in columns, and make a Möbius strip which you can cut, either straight down the centre (see left, above), or off to one side, as shown in the right-hand picture.

Try this again. But this time, give the paper a full twist. Then try one and a half twists, and see what happens. Last of all, see what you can discover about Klein bottles.

Notes

The pictures below show what you get when you cut the strip. The first picture shows that a cut down the middle gives a single loop, but there is a surprising result when you test for Möbiusness (my own word). The test is simple: draw a pen line along one side until you get back to the start: If the paper is still a Möbius strip, the line will be on both sides, but in the picture below, that doesn’t happen:

Now in the last picture, there are two interlinked loops. I cut off the big ants, and something odd happened: the little ants are isolated on a Möbius strip, but the big ants are on a non-Möbius strip.

Playwiths is full of STEAM ideas, and arose from a website of the same name. No publisher would take it on, even though the site drew more than 4 million visits over 20 years. That is why it is self-published.

Sometimes, you wonder about these people!