50p coins make great wheels

50p coins make great wheels

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:

50p coins make great wheels

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.

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  • Lou Davis

    liking the blog steve!

    • http://www.stevemould.com admin

      Fank you!

  • Lou Davis

    liking the blog steve!

    • http://www.stevemould.com admin

      Fank you!

  • http://www.wheels.net wheels

    ye s
    i like it
    50p coins make great wheels

  • Hamzah

    Great talk today at UCL!

    • http://stevemould.com Steve Mould

      Thanks Hamzah!

  • KirstyB

    Went to the Maths Inspiration Lectures on Tues in Liverpool – a brilliant afternoon!  My year 11 top set particularly loved your lecture and one of them brought in some shapes of constant width that he had made the next day!!!  I have been told that I will score MAJOR cool points if I can obtain some of the 3D ones (fairly unlikely I think, from what you were saying) – I love the fact that I teach kids who equate that with cool :-)

    • http://stevemould.com Steve Mould

       That’s amazing! So glad they liked my talk. You can in fact get solids of constant width. We’ll be selling some at mathsgear.co.uk very soon (a cool version based on a rounded tetrahedron like this http://www.shapeways.com/model/166136/meissner-tetrahedra.html as opposed to the solids of rotation I had at the talk) but in the mean time you can get them here http://www.grand-illusions.com/acatalog/Solids_of_Constant_Width.html – these ones are solids of rotation too.

      • KirstyB

         Fantastic, thanks so much!  Was sat front row at the talk, wondering if I could sneak on stage and get one in my bag without anyone noticing, but think this is a much better idea!!  Have just spent half an hour investigating the distinction between the solids of constant width that are or aren’t solids of rotation, so now I want some of each.