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The Möbius strip

Infinity and strange math · level 2, lesson 1 of 6

A question to start

A sheet of paper has two sides, a front and a back. You could colour the front red and the back blue, and the two colours would never touch, except at the edges.

What if you could make a loop of paper with only one side? An ant could walk over every part of it, on "both sides", without ever crossing an edge. And if you cut it down the middle, it would not fall into two pieces.

It sounds impossible. You can make one in two minutes with a strip of paper and some tape. Get them ready: this lesson is try first, then explain.

The idea

What you need

  • A few long strips of paper, about 3 centimetres wide. Cut them from the long side of a sheet of paper, or tape two strips end to end to make them longer.
  • Sticky tape.
  • A crayon or felt pen.
  • Scissors (ask a grown-up to help if you need it).

Making the strip

Hold a strip flat. Mark the corners: A at the top left and B at the bottom left. At the other end, mark B at the top right and A at the bottom right.

AB BA give one end a half twist, then tape A to A and B to B (no twist and A to B would make an ordinary ring)

Now bring the two ends together as if to make a ring. Before you tape, turn one end over. That is called a half twist: half of a full turn. Now the A's meet and the B's meet. Tape them together, all the way across.

The loop you have made is called a Möbius strip (say "MER-bee-us"). It is named after August Möbius, a German mathematician. He and another German mathematician, Johann Listing, each found it in 1858.

Make an ordinary ring too, from another strip, with no twist. You will need it to compare.

Test 1: how many sides?

Take the ordinary ring. Put your crayon on the outside, in the middle of the strip, and draw a line along it without lifting the crayon. You go round once and meet your line. The inside is still clean. The ring has two sides, the inside and the outside, and you only coloured one.

Now do the same on the Möbius strip. Put the crayon in the middle and draw, turning the strip as you go. Keep going. Do not lift the crayon and do not cross an edge.

What happens? You go round once and you are not back at the start: your crayon is on the "other side" of the paper, just under where you began. Keep going. After going round twice, you meet the start of your line. Now look: there is a line on every part of the strip, on both "sides". There was only ever one side.

Test 2: how many edges?

An edge is where the paper stops. The ordinary ring has two edges: the top rim and the bottom rim. Run your finger along the top rim: you go round once and come back, and you never touch the bottom rim.

Run your finger along the edge of the Möbius strip. Go round once ... and your finger is on the "other" edge! Keep going, and after going round twice, you are back where you started. The Möbius strip has only one edge.

Sides Edges
Ordinary ring 2 2
Möbius strip 1 1

Why one side? The two-lane road

Here is a way to understand what the half twist does. Pretend the strip is a road with two lanes: an orange lane along the top half and a blue lane along the bottom half, on the front of the paper.

On the ordinary ring, the orange lane joins up with itself, and the blue lane joins up with itself. Two separate lanes.

On the Möbius strip, the half twist turns one end over. So at the tape, the orange lane at one end meets the blue lane at the other end. Drive along the orange lane all the way round, and when you cross the tape you are in the blue lane. Drive along the blue lane all the way round, and you are back in the orange lane.

trip 1 round the striptrip 2 round the strip the twist sends the top lane into the bottom lane after two trips you are back at the start

So the two lanes are really one long lane, twice as long as the strip. That is why the crayon line needed two trips round, and why it ended up on "both sides": there is only one road, and it visits every part of the paper.

Try it: cut down the middle

Now the big surprise. Take the scissors, poke a hole in the middle of the strip, and cut along your crayon line, all the way round.

  • Ordinary ring: you get two thin rings, the same length as before. No surprise.
  • Möbius strip: you get one loop, twice as long and half as wide! Try it, and see.

Why? Use the two lanes. Cutting down the middle splits the orange lane from the blue lane. On the ordinary ring each lane is its own loop, so you get two loops. But on the Möbius strip, the orange lane and the blue lane are joined end to end into one long lane. Cutting them apart from each other does not cut that long lane anywhere. So you get one long loop, twice as long. (Look closely: the new loop has more twist in it, and if you test it with a crayon, it has two sides. Cutting changed it.)

Try it: cut one third of the way in

Make a new Möbius strip. This time, start cutting one third of the way in from the edge, not in the middle. Keep the scissors the same distance from the edge, and keep cutting. You will go round twice before you get back to the start.

What do you get? Two loops, linked together like a chain: a thin Möbius strip, the same length as the one you started with, and a long loop, twice as long, hooked through it.

Why? Now the road has three lanes: top, middle and bottom. The half twist swaps top and bottom, but the middle lane stays in the middle. So:

  • the middle lane joins up with itself after one trip: that is a thin Möbius strip, as long as the original;
  • the top lane joins the bottom lane, and the bottom lane joins the top lane: together they make one loop, twice as long.

The two loops are linked because the long loop runs right alongside the middle one, on both of its "sides".

Worked examples

1. One trip or two? An ant walks along the middle of a Möbius strip that is 30 centimetres long. How far does it walk before it is back where it started, standing the right way up?

One trip round is 30 centimetres, but after one trip the ant is on the "other side" of the paper, upside down under its starting spot. It needs a second trip. So it walks \(30 + 30 = 60\) centimetres.

2. The edge. The strip is 30 centimetres long. How long is the one edge of the Möbius strip?

Your finger went round twice along the edge before it came back. Each trip is 30 centimetres, so the edge is \(30 + 30 = 60\) centimetres long. (An ordinary ring of the same strip has two edges, each 30 centimetres long.)

3. More twists. Make a loop with two half twists (one full twist). How many sides does it have?

Use the lanes. The first half twist swaps the lanes, and the second half twist swaps them back. So the orange lane meets the orange lane again: two separate lanes, two sides. It is a twisted ordinary ring. If you cut it down the middle you get two loops, but they are linked together.

4. Three half twists. Swap, swap back, swap again: the lanes end up swapped. So a loop with three half twists has one side, like a Möbius strip.

The rule: an odd number of half twists gives one side; an even number gives two sides. Each half twist swaps the lanes, and you just need to know whether they end up swapped or not.

Try it

  1. How many sides and how many edges does an ordinary ring have? A Möbius strip?
  2. A Möbius strip is made from a strip 20 centimetres long. How far does a crayon line along the middle go before it meets its start?
  3. You cut an ordinary ring down the middle. What do you get?
  4. You cut a Möbius strip down the middle. What do you get? How long is it, if the strip was 20 centimetres long?
  5. Will a loop with 4 half twists have one side or two? What about 5 half twists?
  6. Make a Möbius strip and colour the "outside" red. Where does the red stop?
  7. Cut a Möbius strip one third of the way in. Describe what you get and explain it with three lanes.
Answers
  1. Ordinary ring: 2 sides and 2 edges. Möbius strip: 1 side and 1 edge.
  2. It goes round twice: \(20 + 20 = 40\) centimetres.
  3. Two thin rings, each as long as the first ring.
  4. One loop, twice as long: \(20 + 20 = 40\) centimetres long (and half as wide).
  5. 4 half twists: swap, back, swap, back. The lanes end up not swapped, so two sides. 5 half twists: they end up swapped, so one side. (Even number: two sides. Odd number: one side.)
  6. The red never stops at an edge: if you keep colouring without crossing an edge, the whole strip turns red. There is no "inside" to keep clean.
  7. You get two linked loops: a thin Möbius strip, as long as the original, and a loop twice as long. The middle lane stays in the middle at the twist, so it joins itself after one trip: the short Möbius strip. The top and bottom lanes swap at the twist, so they join each other into one long loop.

Common mistake

Guessing instead of trying. Almost everyone guesses "two loops" for the cut down the middle, and "two loops, not linked" for the cut at one third. Make the strips and cut them: the paper is the best judge. Then use the lanes to see why.

Stopping the crayon too early. After one trip round, the line seems to have "missed". Keep going for a second trip, and the line meets itself.

Where it leads