An interesting little thing about the catenary function is that it lets you ride square-wheeled bicycles, or any other wheeled vehicle really, just as easily as ordinary ones. If you want to ride a square-wheeled bicycle smoothly, you can do so by riding along a surface composed of inverted catenaries placed end-to end, with the length along each catenary's surface being equal to the length along one side of the bicycle's wheels. So the ground will look like a washboard.

When the lower-most side of the wheel is horizontal, the centre of the wheel will be directly above the centre of a catenary, and when a corner of the wheel is pointing straight down it will sit exactly in the join between two adjacent catenaries. This keeps the distance from the centre of the wheel to an arbitrary horizontal plane constant; or in ordinary terms, it'll be just as smooth as a riding a regular ole' bike down a regular ole' flat road, only with much less structural integrity in your wheels. Also you won't be able to turn without losing synch with the road and not looking nearly as cool as before.

In fact, this works for any regular polygon with more than four sides (equilateral triangles bump into the the next catenary as they move into the little valley in between), but the function will have to be adjusted to give each catenary the proper convexity, by way of varying *a* (see vireo's wu above) in proportion to the number of sides. So when the number of sides becomes larger, so will *a*, and the convexity will decrease. This makes sense if you think about it: as the number of sides increases, the ground surface needn't be as bumpy, and since a circular wheel is a regular polygon with infinitely many sides, the ground's function will be (&infin/2)(e^{0} + e^{0}), which is a flat plane (the infinity doesn't sit all too well with me, but the derivative is zero, so just fugghedaboudit). So as the number of sides approaches infinity, the ground approaches complete flatness.

Here is a photo of, and an article about, a mathematician who researched and built a square-wheeled-tricycle and catenary floor, which are both open to the public at Macalester College for maths-related fun.