Mathematical striptease

Here's a challenge: can you turn your shirt inside out if your hands are tied together?

Discuss this video, and watch other thought-provoking films on the OU YouTube channel.

The explanation

Well it's obvious how George the first turned his shirt inside out, but you might be wondering why it worked when George the second did it with his hands tied together. And should the pole really have caused George the third so much trouble? Like so many things, it can all be explained by good old mathematics.

Production team George’s task of turning the shirt inside out with his hands tied together may be simpler than it seems. His shirt comes off his head onto his arms and down the rope.

While it is on the rope, it’s now the right way around…but it’s not yet on George.

However, while it is on the rope it can fairly easily be turned outside in again, by pushing the shirt down one of its own sleeves.

Once back on the rope, it is now inside out – but not yet on George.

He can then reverse his first move and put it back on again, which in the process turns it right way out.

Maths isn’t just about numbers. There’s a major area of maths called topology. This is the study of the properties of space.

It seems that topologists don’t care much about size, texture, shape, distance – they’re interested in very abstract notions of space.

It’s perhaps easier to imagine a topologist's view of the world as made of infinitely stretchable clay. So a cube can be made into a sphere, or a pyramid – so they are the same. A ring can be made into a loop, a doughnut shape, even a doorway – so they are all the same. However, a cube CAN’T be made into a ring without tearing a hole in the middle – so they are not the same.

Basically, it seems that in topology any object is equivalent to any other object if, when it is stretched/deformed, it can take the same shape without tearing apart or having distinct parts stuck together.

Okay – now we think that a topologist might be able to solve George the third’s problem. Like George the second, he’s got his shirt on inside out, and his hands are tied together. However, he’s also got his hands around a pole.

Let’s start with how a topologist might describe George the second. To a topologist:

• George the second (with his rope) is just part of a ring – formed by his arms and the rope.
• His head and lower torso are just odd, unfortunate lumps on the ring.
• Around this ring is another ring – his shirt – which could be stretched and rolled.
• His ‘shirt’ ring could easily twist around – turning outside in and inside out.

So if his head and body are shrunk down then his shirt can slide all along the ring, made of rope and arms (with a tiny head and tiny body), without any fuss, like his head or body getting in the way.

The closest George the second gets to this in the real world is when he’s removed the shirt over his head and the shirt is sitting on the rope in front of him – where is can easily twist around and be turned outside in – and then place back on again.

Production team To a topologist, George the third is just the same as George the second – except the ring his arms and rope make now also circles around a pole (think of it as another infinite ring).

Now, a topologists could repeat what happened with George the second. He shrinks his head and torso – allows the shirt to sit on the ring and twist outside in again. The pole seems irrelevant to completing the task in the world of topology.

Of course, George is not an ideal topological object – but subject to the constraints of the real world (he can’t be stretched and shrunk).

The fact that George the third can’t morph the shape of his body into a ring may mean he'll be stuck around the pole for some time…

Though you might want to think about this: if the shirt had been very stretchy would a real world George have managed to turn it outside in and put it back on again (if for example, he could pass his body down the sleeve?)

What could you do next?

• Can you get out of the George the third challenge – post your clips on the 'Mathematical Striptease' You Tube page
• Post clips of other topological challenges
• Check out the mathematics courses at the Open University