Reuleaux triangle

The Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the opposite vertex.

A Reuleaux triangle is, apart from the trivial case of the circle, the simplest and best known Reuleaux polygon, a curve of constant width. The separation of two parallel lines tangent to the curve is independent of their orientation. The term derives from the name of Franz Reuleaux, a 19th-century German engineer who did pioneering work on ways that machines translate one type of motion into another, although the concept was known before his time.

Construction

To construct a Reuleaux triangle

With a compass, sweep an arc sufficient to enclose the desired figure. With radius unchanged, sweep a sufficient arc centred at a point on the first arc to intersect that arc. With the same radius and the centre at that intersection sweep a third arc to intersect the other arcs. The result is a curve of constant width. Equivalently, given an equilateral triangle T of side length s, take the boundary of the intersection of the disks with radius s centered at the vertices of T.

By the Blaschke-Lebesgue theorem, the Reuleaux triangle has the least area of any curve of given constant width. This area is , where s is the constant width.

The Reuleaux triangle can be generalized to regular polygons with an odd number of sides. The British twenty pence and fifty pence coins are approximately Reuleaux heptagons with rounded apexes

Other uses

The Reuleaux triangle rotating inside a constant sized square

• Because all diameters are the same length, the Reuleaux triangle, with all other Reuleaux polygons, is an answer to the question "Other than a circle, what shape can you make a manhole cover so that it cannot fall down through the hole?" However, in practice manhole covers are not built in these shapes, due to difficulties in machining and lack of compelling reason.

• The rotor of the Wankel engine is easily mistaken for a Reuleaux triangle but its curved sides are somewhat flatter than those of a Reuleaux triangle and so it does not have constant width.^[1]

• A drill bit in the shape of a Reuleaux triangle can, if mounted in a special chuck which allows for the bit not having a fixed centre of rotation, drill a hole that is very nearly a perfect square.^[2] Other Reuleaux polygons are used for drill bits for pentagonal, hexagonal, and octagonal holes

• A Reuleaux triangle rolls smoothly and easily, but makes a poor wheel because it does not roll about a fixed center of rotation. An object on top of rollers with cross-sections that were Reuleaux triangles would roll smoothly and flatly, but an axle attached to Reuleaux triangle wheels would bounce up and down three times per revolution. This concept was used in a science fiction short story by Poul Anderson titled "The Three-Cornered Wheel."

• The existence of Reuleaux polygons shows that diameter measurements alone cannot verify that an object has a circular cross-section.

• Several pencils are manufactured in this shape, rather than the more traditional round or hexagonal barrels. They are usually promoted as being more comfortable or encouraging proper grip (if marketed for children), as well as having the advantage of not rolling off tables.

• The shape is used for signing for the National Trails System administered by the United States National Park Service.^[3]

Three-dimensional version

The intersection of the balls of radius s centered at the vertices of a regular tetrahedron with side length s is called the Reuleaux tetrahedron, but is not a surface of constant width. It can, however, be made into a surface of constant width, called Meissner's tetrahedron, by replacing its edge arcs by curved surface patches; alternatively, the surface of revolution of a Reuleaux triangle through one of its symmetry axes forms a surface of constant width, with minimum volume among all surfaces of revolution of given constant width.

See also

• Deltoid curve
• Superellipse
• Vesica piscis
• Watts Brothers Tool Works

Notes

External links

• How Round is Your Circle? - book about various geometric properties, including curves and solids of constant width
• Shapes of constant width at cut-the-knot
• Russian-language site with downloadable films about the Reuleaux triangle, also showing the Wankel engine and the mechanics of a famous Soviet film projector