# Square Wheels and Catenary Roads

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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.

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:

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