Reuleaux’s Polygons

A plank slides on wheels, always parallel and at the same distance from the base. However, the wheels are not round! What is happening?

The engineer Franz Reuleaux (19th century, German, but pronounced “roy-luh” with an “r” rolled and the first “o” pronounced like “aw”) developed various mechanisms for transforming motion.

To construct a Reuleaux triangle, start with an equilateral triangle with sides of length L. With the center at each vertex, open the compass and describe the (smaller) arc of a circle passing through the other two vertices. The resulting closed curve has a “constant width” L, which is the distance between the two parallel tangents. The construction can be generalized to other polygons and (after some interventions) also to polyhedra.

Note that, as it rolls, the axis of the Reuleaux “wheel” goes up and down, causing a car with such wheels to vibrate on a flat road. On the other hand, in the case of the plank supported on cylinders with this cross-section, the rolling has two points of static equilibrium, one stable (with the lower axis and of practical interest) and the other unstable (with the higher axis).

Any curve of constant width can be inscribed in a square and rotate within it (with sliding), although, again, its center does not remain fixed. Some drills for creating polygonal holes are manufactured according to this principle.

Construction of the Polygon

Rotating Polygon