One Side, One Edge

What’s going on

Take a long strip of paper and colour its corners: red at the top, green at the bottom, at both ends. Now glue the ends together.

If you glue them straight, red meets green. You get an ordinary ring, like a bracelet, with an inside and an outside.

If you turn one end half a turn, 180°, before gluing, red meets red and green meets green. Press Glue it above and watch the corners. This twisted ring is the Möbius strip.

How many sides?

Any piece of paper has two sides, a front and a back. A plain ring too: the inside and the outside. Surely the Möbius strip has two sides as well?

Press Draw the middle line. The pen starts halfway between the edges and keeps to the middle, never lifting off the paper.

• After one lap it is not back where it started. The line has come round on the back of the paper.
• Only after two laps does the line close. By then it has covered both faces of the original strip, and it never crossed an edge.

So the “front” and the “back” are one and the same surface. The Möbius strip has one side. Switch to No turn and draw again: on a plain ring the pen is home after one lap, and the inside stays blank. Two sides.

How many edges?

A strip of paper has two long edges, top and bottom. Press Send the caterpillar: it crawls along the top edge.

On the Möbius strip it also goes round twice. On the second lap it walks along what looked like the bottom edge, then arrives back at its starting point. It has walked every bit of edge there is. The Möbius strip has one edge.

This is why the half-twist matters and the size does not. Stretch the strip, make it wider, bend it any way you like without tearing or gluing: it keeps one side and one edge. Properties like these, which survive any stretching, are what topology studies.

Make one

You need a strip of paper about 3 cm wide and 30 cm long, glue or sticky tape, and a pen. Make three or four Möbius strips at once: you will want them for cutting in the next gem. Then draw along the middle of one with a pen, exactly as in the scene, and see the line come back on the other face.

Try it yourself

1. A full turn

Turn one end a whole turn, 360°, before gluing. How many sides and how many edges does this ring have?

Answer

Two sides and two edges, like a plain ring. A whole turn brings the end back the right way up, so red meets green again. What matters is whether the number of half-turns is odd or even: odd gives one side, even gives two.

2. Three half-turns

And with three half-turns, 540°?

Answer

One side and one edge, like the Möbius strip. After three half-turns the end is upside down, just as after one, so red meets red and the pen again needs two laps.

3. Paint it

You want to paint a Möbius strip blue on one side and yellow on the other. How do you do it?

Answer

You can't. There is only one side: start painting blue anywhere and keep going without crossing the edge, and you will paint the whole strip blue. The pen in the scene does exactly that.

Go deeper

• The strip was discovered twice in 1858, by two German scientists working independently: August Ferdinand Möbius, a mathematician and astronomer in Leipzig, and Johann Benedict Listing, a mathematician and physicist in Göttingen. Listing also coined the word topology.
• Some conveyor belts are made as Möbius strips. With a single side, the whole surface wears evenly, so the belt lasts longer.
• The recycling symbol, three arrows chasing each other round a triangle, is drawn as a Möbius strip.