Reuleaux triangle

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The Reuleaux triangle is a constant width curve based on an equilateral triangle. The distances from any point on a side to the opposite vertex are all equal.

A Reuleaux polygon is a polygon that is a curve of constant width - that is, a curve in which all diameters are the same length. The best-known version is the Reuleaux triangle. Both are named after Franz Reuleaux, a 19th-century German engineer who did pioneering work on ways that machines translate one type of motion into another, although it was known before his time.

The Reuleaux triangle is the simplest nontrivial example of a curve of constant width - a curve in which the distance between two opposite parallel tangent lines to its boundary is the same, regardless of the direction of those two parallel lines. (The trivial example would be a circle.)

To construct the Reuleaux triangle, start with an equilateral triangle. Center a compass at one vertex and sweep out the (minor) arc between the other two vertices. Do the same with the compass centered at each of the other vertices. Delete the original triangle. 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 ${1\over2}(\pi - \sqrt3)d^2$ , where d is the constant diameter.

The Reuleaux triangle can be generalized to regular polygons with an odd number of sides. See also the British Twenty Pence and Fifty Pence coins.

[edit] Trivia

Because all of its diameters are the same length, the Reuleaux triangle (along with all other Reuleaux polygons) is an answer to the Mensa-like question "Other than as a circle, what shape can you make a manhole cover so that it cannot fall down through the hole?"

The rotor of the Wankel engine is similar to a Reuleaux triangle.

This is the mathematical basis for drill bits which can drill a square hole.

Although a Reuleaux triangle rolls smoothly and easily, it does not make a good wheel because it does not have a fixed center of rotation. While an object on top of rollers with cross-sections that were Reuleaux triangles (like using logs as rollers, but shaped like Reuleaux triangles) would roll smoothly and flatly, an axle attached to wheels shaped like Reuleaux triangles 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 is a good demonstration of why you cannot use diameter measurements alone to verify that an object has a circular cross-section.

[edit] 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 given constant width.