Cut It Down the Middle

What’s going on

Cut a plain paper ring along its middle line, all the way round, and it falls into two rings, each half as wide. No surprise there.

Now do the same with a Möbius strip. Two Möbius strips? Make a guess with the Guess buttons above, then press Cut.

One piece, twice as long

The strip does not fall apart. You get one band, twice as long, and it is no longer a Möbius strip: it has two sides and two edges, and it is twisted by two full turns, 720°. Follow the bead: it runs round twice before it gets back.

Why one piece? Remember the pen from the previous gem: the middle line of a Möbius strip comes back on the other face after one lap. The scissors follow that line, so they cut the strip into “the half above the line” and “the half below it”. But on a Möbius strip, going round once takes “above” to “below”. The two halves are joined end to end into one long loop.

A third from the edge

Take a fresh strip and cut one third of the way in from the edge, keeping the same distance all the way. Choose A third from the edge and cut.

Watch the scissors: they go round twice before the cut closes. The line a third from one edge, followed once round, becomes the line a third from the other edge. It is the same line.

The result is two pieces, hooked together: the middle third, a thin Möbius strip as long as the original, and the outer two thirds joined into one long band with two full twists, the same band the middle cut gave.

More twists

The scene lets you choose the number of half-turns. The pattern, which you can check with paper:

• Even number of half-turns: cutting down the middle gives two rings of the same length. With 0 they are separate; with 2 (a full turn) they are linked once; with 4, twice.
• Odd number: always one band, twice as long. With 3 half-turns that band is tied in a trefoil knot, the simplest knot there is. Turn the model to find it.

Try it yourself

1. Cut it again

Cut a Möbius strip down the middle, then cut the long band you got down the middle once more. What do you get?

Answer

Two bands, each as long as the long band, linked together twice. The long band has two full twists, an even number of half-turns (four), so it behaves like an even ring: it falls into two rings linked 4 ÷ 2 = 2 times.

2. Guess the laps

You cut a ring with three half-turns a third of the way from the edge. How many times do the scissors go round before the cut closes?

Answer

Twice. Any odd number of half-turns turns the strip upside down after one lap, so the line a third from one edge continues as the line a third from the other edge.

3. A full turn, cut in thirds

A ring with one full turn (two half-turns) is cut a third from each edge. How many pieces?

Answer

Three rings of the same length, each with one full twist, and every one is linked with the other two. With an even number of half-turns the two cut lines are separate, so the strip falls into three bands. Set 2 half-turns and A third from the edge in the scene to see them.

Go deeper

• Magicians call the paper-band version of this the Afghan bands: three loops that look alike, cut in half in front of the audience, give two rings, two linked rings and one big ring. The secret is 0, 2 and 1 half-turns.
• A band with three half-turns cut down the middle gives a trefoil knot. Mathematicians who study knots use exactly this kind of model: the knot is the band’s middle line, the twists are how the band winds around it.