Shapes of Constant Width: Beyond Circles

Imagine a triangle that can roll as smoothly as a wheel. Sounds impossible? Welcome to the curious world of shapes of constant width—figures whose width measured across any parallel pair of lines is always the same. These aren’t just math curiosities; they blur the line between circles and other polygons and show how geometry can surprise you.

A brief history

The first systematic study of constant-width curves dates back to 1875, when German engineer Franz Reuleaux explored them in his book on machine design. A century earlier, 17th-century French mathematician Charles de Bovelles hinted at similar ideas. Enthusiasts like Sam Lloyd later challenged puzzle fans to find non-circular shapes of constant width—which led to the famous Reuleaux triangle.

Where you’ll see this in real life

1. Coin design: British 20p and 50p coins use constant-width curves so vending machines can measure their diameter reliably. 2. Drill bits: Special Reuleaux-triangle bits can carve perfect square holes in wood or metal. 3. Engineering cams: Machine parts often use constant-width cams for smooth motion, turning rotary into linear movement. 4. Architecture: Some modern buildings employ constant-width arches or windows to create uniform lighting and unique aesthetics.

A common misconception

Many students assume that only circles have constant width. In fact, any convex shape whose boundary is formed by arcs of circles (like the Reuleaux triangle) or other curves can share that property. The key is how the curves are arranged, not just their individual radii.