Reuleaux triangle

A Reuleaux triangle is a unique geometric shape that resembles a triangle with curved sides.

While it looks like a triangle, it possesses a mathematical property called constant width. This means the distance between any two parallel lines touching its edges remains identical, regardless of the shape's orientation. This property makes it a member of a special group known as curves of constant width. Most people assume only circles have this capability, but the Reuleaux triangle is the simplest and most famous non-circular example. This characteristic allows it to function in specialized mechanical roles where a circle might not be ideal.

To understand its mechanism, we can look at how it is constructed.

One method uses a compass to draw three intersecting circles of equal radii. First, you mark two points and draw a circle centered at one that passes through the other. You then draw a second circle of the same size centered at the second point. Finally, a third circle is drawn using one of the intersection points as its center. The central area where all three circles overlap forms the Reuleaux triangle. Alternatively, you can start with a perfect equilateral triangle and draw circular arcs between its vertices. Each arc must be centered on one vertex and connect the remaining two.

There are several ways to view the mathematical properties of this shape. The most fundamental is the constant width, which is defined by parallel supporting lines.

When these lines touch the shape, one line will always contact a vertex. The opposite line will touch the arc directly across from that vertex. The distance between these lines always equals the radius of the arc used in construction. This constant distance is why the shape can rotate within a square while always touching all four sides. This specific ability has earned the shape the nickname of a "Reuleaux rotor."

The history of this shape involves many famous thinkers.

While named after the 19th-century German engineer Franz Reuleaux, the shape was known much earlier. Leonardo da Vinci utilized the shape for a map projection. The mathematician Leonhard Euler studied these shapes in the 18th century, specifically in his 1781 publication "De curvis triangularibus." Euler referred to these constant-width shapes as "orbiforms." Reuleaux himself was a pioneer in studying how machines translate different types of motion. He used these triangles in his engineering designs to solve specific mechanical problems.

In mathematics, the Reuleaux triangle is considered an extremal shape, meaning it sits at the edge of what is possible.

According to the Blaschke–Lebesgue theorem, it has the smallest possible area of any curve with a given constant width. It also features the sharpest possible corners for a constant-width shape, with angles of exactly 120 degrees. Despite its pointiness, it can rotate inside a square and cover approximately 98.8% of that square's area. However, it cannot reach the very corners of the square, leaving small uncovered gaps. Furthermore, it is noted for being highly asymmetrical compared to a circle.

Practical applications of the Reuleaux triangle are surprisingly common in daily life.

You can find this shape in the design of many guitar picks. It is also used for the nuts on fire hydrants and in certain types of pencils. Engineers use special Reuleaux-shaped drill bits to create filleted square holes. Because of its unique geometry, it can even be used as a manhole cover to prevent it from falling through its own opening. Beyond tools, its distinct look is often used in graphic design for corporate logos and various signs.

The concept can even be expanded into higher dimensions. One way to generalize it is through the Reuleaux tetrahedron, which is formed by the intersection of four balls.

Interestingly, a standard Reuleaux tetrahedron does not actually have constant width. To fix this, mathematicians create a Meissner tetrahedron by rounding its edges. Another method for three-dimensional expansion is to create a surface of revolution from the 2D triangle. These advanced shapes continue to help mathematicians understand the relationship between symmetry, area, and motion in complex systems.