Shapes of constant width are more of a curiosity, and quite difficult to explain without demonstrating it physically, but I’ll take my best shot!
So, let’s first talk about what a shape of constant width is (if it weren’t already semi-apparent from its name). A shape of constant width is a convex planar shape whose width is the same regardless of the orientation of the curve. Here, we define the width as the perpendicular distance between two distinct parallel lines. Each of these lines must have at least one point in common with the shape’s boundary. However, they cannot have any point in common with the shape’s interior. It’s a bit of a mouthful, but think of it this way: put this shape between two parallel rulers and no matter what way you orient the shape, the distance between these two rulers will always be the same.
One of the best known of these shapes is a Reuleaux Triangle. You can see at the top left of the image above. Here’s a drawing of what it looks like:
In order to generate a Reuleaux Triangle, start off with a regular equilateral triangle. Then, “curve” out the flat edges by adding a circular arc between two points (centered around the other remaining point). You can extend this to other polygons with an odd number of vertices. What you get are Reuleaux Polygons. You can see some of these below:
As far as I know, they have no particularly useful practical application, but that doesn’t mean they’re not fun to play with! As a matter of fact, some countries’ coins are shaped as various Reuleaux Polygons. For example, the Canadian 1-dollar coin (the “Loonie”) and the British 20p and 50p coins. Why do coins not come as triangles or pentagons? Because they need to be able to be used in coin-operated machines such as parking meters and vending machines.
So, these objects have constant width and roll freely, just like wheels. So, why do you not use these shapes as wheels for cars and bicycles? The reason is that a circle rotates about a fixed axis. These shapes do not. For example, you mounted a car on wheels that were Reuleaux Triangles, the car would bounce up and down three times for every revolution of the wheels.
These shapes also have 3-D equivalents (pictured below):
Similar in construction, they’re have smooth, curving edges. If you were to place these shapes underneath a book, the book would roll freely as if it were resting on spheres, rather than pyramid-shaped objects.
These shapes are really fun to play with, because at first glance, you wouldn’t expect them to roll as smoothly as they do. It’s like a little magic trick, all in the power of mathematics!