Two Rings Crosswise

What’s going on

Make two equal paper rings. Glue them together crosswise, at right angles, so they meet in a small glued square, two layers thick. Now cut each ring along its middle line, all the way round, through the glued square too.

Press Cut both with Two plain rings chosen above. What falls out is a square frame, flat on the table.

Where do the right angles come from?

Each cut splits a ring into two half-rings, so there are four half-rings in all. The glued square is cut by both lines, crosswise, so it falls into four small squares, one in each corner.

Each small square still holds the end of one half-ring from the first ring and the end of one half-ring from the second, and the two rings met at a right angle. So every small square is a right-angled corner, and the four half-rings join through the four corners into one closed frame with four right angles: a square.

Now with Möbius strips

Do the same with two Möbius strips. Friends doing this at the same table can get different results. Try both options in the scene:

• Twisted the same way: two separate loops, each with two right-angle corners. Laid flat, each looks like a heart.
• Twisted opposite ways: two hearts again, but this time hooked together. You cannot pull them apart without tearing.

What decides it is the direction of the half-twist. One end of a strip can be turned clockwise or anticlockwise before gluing, and the two strips you get are mirror images of each other. You cannot turn one into the other without ungluing. With paper, check which way you twisted before you glue.

The scene does not draw the answer by hand: it follows the four half-rings through the corners, then measures how the resulting loops wind around each other. That number, the linking number, is 0 for the separate hearts and 1 for the hooked ones.

Try it yourself

1. Count the pieces first

Before you cut two plain rings crosswise: how many small squares will the glued patch fall into, and how many half-rings will there be?

Answer

Four small squares, because two cuts cross the patch at right angles. And four half-rings, two from each ring. Each half-ring has two ends, and each small square holds two ends, so everything joins up into loops through the corners.

2. Which way did you twist?

Make two Möbius strips. How can you tell, before cutting, whether they are twisted the same way or opposite ways?

Answer

Hold each strip with the glued seam towards you and follow one edge round: it climbs either like a right-handed screw or like a left-handed one. Two strips with the same handedness are twisted the same way. A strip and its reflection in a mirror always have opposite handedness.

3. The strip with a slit

Cut a slit along the middle of a strip near one end, pass the other end through the slit, give it half a turn and glue. Then continue the slit along the whole band. What comes out?

Answer

A single ring twisted through 720°, two full turns. Try it with paper: it is the same band the middle cut of a Möbius strip gives in the previous gem, made a different way.

Go deeper

• This crosswise-rings experiment is a favourite of maths clubs because it only needs paper, but the result depends on a hidden property, the handedness of a twist, that many people never notice.
• Chemists build molecules shaped like twisted rings, and whether such a molecule is left- or right-handed changes how it behaves: the same idea of mirror images that decides linked or separate hearts here.