Reuleaux triangle

Constant-width curves and Franz Reuleaux

A curve of constant width is a convex shape that has the same width regardless of the direction in which it is measured — place it between two parallel plates, and the gap is always the same no matter how you orient the shape. The circle is the obvious example, but infinitely many other such curves exist. The simplest non-circular one is the Reuleaux triangle, named after the German mechanical engineer Franz Reuleaux (1829–1905), who studied it extensively in his 1876 treatise on kinematics. It is constructed by drawing three circular arcs, each centered on one vertex of an equilateral triangle and passing through the other two vertices. The resulting shape has sharp 120° corners but a perfectly constant width equal to the side length of the underlying triangle. Higher-order Reuleaux polygons (pentagonal, heptagonal, etc.) can be constructed similarly from any regular polygon with an odd number of sides.

Barbier’s theorem and square holes

One of the most elegant results about constant-width curves is Barbier’s theorem (1860): every curve of constant width w has the same perimeter, namely πw, regardless of its shape. The circle, the Reuleaux triangle, and every other constant-width curve of the same width all share the same perimeter. Among all such curves, the Reuleaux triangle encloses the least area (a result proved by Blaschke and Lebesgue), while the circle encloses the most. Perhaps the most surprising application is drilling square holes: when a Reuleaux triangle is rotated inside a square, its boundary traces out a path that covers nearly all of the square — about 98.77% of its area. In 1914, Harry James Watts patented a drill bit based on this principle, and Watts Brothers Tool Works produced them commercially for decades. The drill produces a square with slightly rounded corners, but for engineering purposes the result is remarkably close to a true square.

Manhole covers and the Meissner tetrahedron

The reason manhole covers are round is often explained by the fact that a circle cannot fall through its own hole — but any constant-width curve shares this property. A Reuleaux triangle manhole cover would work just as well, and some cities have actually used them. In three dimensions, the analog is the Meissner tetrahedron, a body of constant width derived from the Reuleaux tetrahedron (the intersection of four spheres centered at the vertices of a regular tetrahedron). The Reuleaux tetrahedron itself is not of constant width — its edges are too sharp — but by rounding three of its six edges with a specific spindle-shaped surface, Meissner and Schiller (1912) produced a body that is. Whether the Meissner tetrahedron is the minimum-volume body of constant width in three dimensions remains an open conjecture.