There is nothing very exciting about a single loop of paper, is there? But give the paper a single twist before gluing the ends together, and it is a different story. The loop with the twist is called a Möbius strip, named after its discoverer.

Would you believe that the Möbius strip has only one surface, and not two? Try colouring it in. It only has one edge too!

Now suppose we take a pair of scissors and cut the Möbius strip lengthwise along its centre line. How many pieces will we get? Starting again, cut a Möbius strip lengthwise keeping one third the width from one of the edges. How many pieces this time? Preoccupation with Möbius strips can have dire consequences, as the following rhyme shows.
 

A burleycue dancer, a pip,
Named Virginia could peel in a zip.
But she read science fiction And died of constriction
Attempting a Möbius strip. – C. Kornbluth

Hint 1

See if you can mentally visualize the answers before actually trying the experiments.

Solution

The Möbius strip is both an object of serious mathematical interest, and an item in recreational mathematics. It belongs to the part of mathematics called topology.

It is easy to check that it has just one surface and just one edge.

What happens when we cut it along its centre line? One way of expressing the solution is:

A mathematician confided
That a Mobius strip is one sided.
And you get quite a laugh
When you cut it in half,
For it stays in one piece when divided.

When cut along a line one third the way from one of the edges, we finish up with two linked twisted loops.
Most people would find these outcomes difficult to visualize!

Extensions

1. What happens if you take a loop which is given a double twist before gluing, and cut it along the centre line? along a line one third the width from one of the edges?

2. Can we generalize the Möbius strip to 3 dimensions? Use your library to find our about the Klein bottle.

Hint 1

Solution

Extensions