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The Reuleaux triangle is the simplest nontrivial example of a curve of constant width - that is, a curve in which all diameters are the same length. It is 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.
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 vertexes. Do the same with the compass centered at each of the other vertexes. 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 vertexes of T.
By the Blaschke-Lebesgue theorem, the Reuleaux triangle has the least area of any curve of given constant width.
The Reuleaux triangle can be generalized to regular polygons with 2n + 1 sides. See Reuleaux polygon; also British coin Twenty Pence, British coin Fifty Pence.
Trivia
- Because all of its diameters are the same length, the Reuleaux triangle - actually, all Reuleaux polygons - is the non-obvious answer to the Mensa-like question "What shape can you make a manhole cover so that it cannot fall down through the hole?" The obvious answer is a circle.
- Altough a Reuleaux triangle rolls easily, it does not make a good wheel because it doesn't have a fixed center of rotation. Car axels rolling on circular wheels stay the same height above the ground, but an axle attached to Reuleaux triangles would wobble up and down.
Three-dimensional version
The intersection of the balls of radius s centered at the vertexes 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 in two ways.
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