Most of us are generally familiar with the concept that wheels are usually round. But do they have to be this way? What are the properties that make a round wheel useful? Yes, you might think that eight years of math podcasting has finally driven me insane, to question something so obvious. But math geeks are famous for requiring proofs of the obvious-- and this is a case where common instincts might lead us astray. Now of course, for a wide variety of reasons, circular shapes do tend to make the best wheels. In certain cases, though, there is a more general class of figures that can be substituted for the circular shape, with some important real-life applications. These are known as curves of constant width.

To simplify the discussion and avoid the complications of axles, let's discuss simple rollers. Suppose you want to smoothly roll a large plank across the top of a bunch of logs. If the logs have a circular cross section, it's pretty obvious that the plank can roll smoothly along, without wobbling up and down. But what is the property that enables this? The reason for the plank's smooth rolling is that the circle is a curve of constant width. This means that if you put parallel lines above and below a circle and touching it, the distance will be a constant, the diameter of a circle. However, a surprising fact discovered by Euler in the 18th century is that there are many other curves of constant width that could be used instead and still allow smooth rolling.

The most famous non-circular curve in this class is known as the Reuleaux Triangle, a kind of equilateral triangle with rounded edges. To create one, start with an ordinary equilateral triangle. Then, for each vertex, replace the opposite side with the arc of a circle whose center is that vertex, and whose radius matches the side of the triangle. If you think about it for a minute, you should see that this curve will be of constant diameter: if a plank is rolling over the top, at any given moment either the plank or the ground will be touching a vertex, and the opposite surface will be touching a curved edge. Since the circle used to form that curved edge is defined as the set of points equidistant from its center, the opposite vertex in this case, the distance between the plank and the ground will be a constant value equal to the circle's radius. Thus, logs with a Reuleaux cross section will be rolled over just as smoothly as circular ones.

As you can probably see from how we constructed it, the Reuleaux Triangle is just one representative of a large class of curves of constant width. Take any regular polygon with an odd number of sides, and replace side opposite each vertex with an arc of a circle centered at that vertex. There are also many other curves in this class, with more complicated construction methods; you can read up on these in the show notes if you're curious.

The surprising discovery of this large class of shapes has led to some useful real-life applications. Reuleaux, the 19th-century engineer for whom the triangle is named (despite Euler's earlier knowledge of it), became famous for investigating a variety of uses based on converting circular into other types of motion. Later this led to applications in mechanisms as diverse as film projectors and automotive engines. Since a rotating Reuleaux triangle traces a shape that is nearly square, it has also been used to construct a special drill bit that enables woodworkers to drill square holes. By basing the drill on other curves of constant width, a similar method can be used to drill pentagon, hexagon, or octagon-shaped holes as well. This shape has also been used in the design of pencils, with the claim that the constant diameter but non-circular shape provide a comfortable grip while reducing the chance of spontaneously rolling off a table. And in several countries, non-circular curves of constant width have been used as the shape of coins, with their constant diameter providing advantages in the design of vending machines. You can see nice pictures of these and other applications at the links in the show notes.

But one of the most useful aspects of the Reuleaux Triangle and related shapes is as a non-circular counterexample, forcing us to question basic assumptions about simple geometric properties. According to some sources, engineers working on the doomed space shuttle Challenger tried to verify the cylindrical shape of some components by measuring their width at various sampling points, not being aware of the existence of non-circular curves of constant diameter. Too bad they didn't have math podcasts back then, though techncially the engineers could have read Martin Gardner's classic essay on the topic. Anyway, if the shapes were not circular, this would mean that various types of stress would affect the parts unevenly. This may have contributed to the shuttle's eventual destruction.

So, be sure to think about the existence of these non-circular curves of constant width, next time you are assembling a mechanical device, minting your nation's currency, or designing a space shuttle.

And this has been your math mutation for today.

References:

- https://en.wikipedia.org/wiki/Reuleaux_triangle
- http://web.mat.bham.ac.uk/C.J.Sangwin/howroundcom/roundness/applications.html
- https://en.wikipedia.org/wiki/Curve_of_constant_width
- http://www.amazon.com/Knots-Borromean-Rings-Rep-Tiles-Queens/dp/0521758718