Surprisingly shapes of constant width have a number of useful applications.
The United Kingdom fifty pence and twenty pence coins have constant width. This helps reduce the problems associated with them becoming stuck in slot machines. However, some other countries have coins with an even number of undulations. Did they simply just not get the geometry?
Cams, sometimes called box beater cams, were designed using shapes of constant width. The important feature is that the drive is positive at all stages, and no weight or spring is required at any part of the stroke.
By removing sections of a Reuleaux rotor, Harry James Watts, created a drill which cuts a square hole. You will notice that it does not remove a perfect square, and that small arcs remain in the corners. There is also the problem of how to guide the drill itself.
The problem of rounded corners is overcome by using a curve of constant width based upon the right angled isosceles triangle. If a cam of this shape is housed in a square hole then one point must follow a square path as the shape is rotated.
Notice the floating chuck which applies a torsional force to the cam yet allows it unrestricted movement in both the horizontal and vertical directions.
The final application of the Reuleaux rotor is in the design of the rotary engine car. In a traditional engine the piston reciprocates but in a rotary engine, as the name implies, direct circular motion is generated. This is done by taking a Reuleaux rotor and from the centre drilling a large hole to create an internal gear. This then rotates on a fixed gear.
By symmetry it is clear that the three vertices of the rotor all move along the same path, and the gaps left allow the fuel to be injected, compressed, ignited and for the combustion products to expand and finally be expelled.