Square Wheels and Catenary Roads

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Academic, Play

The other day Paul told me that square wheels (and, indeed, any regular polygon except a triangle) can ride on a road constructed of inverted catenaries. A catenary is the curve made by a telegraph wire when stretched between two poles — a cable hanging under its own weight.

Inverted catenary with height to the vertex labelled h

A little bit of thinking about the road makes you realise that the length of the catenary s must be the same as the length of the side of the square, and the height h must be

\[h = \frac{s}{2} (\sqrt{2} - 1)\]

This is because the centre of the square, the axle, must remain at a constant height from the ground for a smooth ride.

Anyway, armed with this information, I spent this afternoon making a real-life version of this. I used some string to mark out a catenary of the right size, and used it as a template on two bits of plywood. A fair bit of jigsawing and filing later, I had this:

Model of a catenary road

I was really surprised how well it worked! Check the video to see how smoothly the “wheel” runs along the “road”!


  1. Ivor Biggun

    erm..how is your masters work going? have much spare time? How have I gone so long without knowing what a catenary was, Until till now I thought it was the love child of Sylvester and Tweety Pie. thank you for testing that we can recognise blurred numbers

  2. Chris Buckley

    Why are you on AOL?

  3. Chris Buckley

    I’m really glad that OliOli UAE were able to create a more recent full-sized version after chatting about this process! See it on Instagram

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