50p coins are too pointy to use as wheels. Or so I thought.

Intuitively, you'd think that wheels need to be round. I'm talking about rollers here really as opposed to wheels on an axis. This sort of thing:

Anyway, surly they need to be circles (have a circular cross section) because they need to be the same width all the way round for a smooth ride.

Well they do need to be the same width all the way round but that doesn't necessarily mean circular: I give you... the 50p coin.

To prove it I made a video:

So there you go. They're called curves of constant width. One way to draw a curve of constant width is to take an odd numbered regular polygon like a triangle or a pentagon and round off the edges in a special way: use the opposite corner as a guide. Using a compass, place the pin on a corner and draw an arc joining the opposite pair of corners. Like this:

There's a reason 50p and 20p coins are shaped this way. Vending machines detect what coins are coming in based on their width. So coins need to be the same width all the way round!

What about ball bearings? Do they need to be spherical? It turns out they don't. To prove it I made another video:

These *solids* of constant width in the video were given to me by Chris Sangwin who wrote a book about them and other things. He's awesome. He also has much better videos than me! Thanks Chris!

I couldn't find a good animation of this particular sold of constant width so I made my own. I used Google sketchUp (incredibly easy to pick up) and screenr to animate it:

Tried to get them on Blue Peter but they didn't go for it. I'll get maths on the show one day!

Know any funny shapes. Tell me about them in the comments.

Here are some more coins from around the world that are shapes of constant width.

UPDATE 27/01/2014

Here's the Sketchup file for of the solid of rotation as requested by a commenter.

Tags: 50p, coins, explainer, solids of constant width, video