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 curve of constant width - that is, a curve in which all diameters are the same length. In Geometry, a curve of constant width is a convex planar Shape whose width measured by the distance between two opposite parallellines touching its boundary 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. Franz Reuleaux ( September 30, 1829 &ndash August 20, 1905) was a mechanical engineer and a lecturer of the Berlin Royal Technical

The Reuleaux triangle rotating inside a constant sized square

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. In Geometry, a curve of constant width is a convex planar Shape whose width measured by the distance between two opposite parallellines touching its boundary (The trivial example would be a circle. )

To construct the Reuleaux triangle, start with an equilateral triangle. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line Center a compass at one vertex and sweep out the (minor) arc between the other two vertices. A compass or pair of compasses is a Technical drawing instrument that can be used for inscribing Circles or arcs They can also be used as In Geometry, a vertex (plural "vertices" is a special kind of point. 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. In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers

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

The Reuleaux triangle can be generalized to regular polygons with an odd number of sides. General properties These properties apply to both convex and star regular polygons See also the British Twenty Pence and Fifty Pence coins. The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom, the UK or Britain,is a Sovereign state located

Other uses

• Because all of its diameters are the same length, the Reuleaux triangle, along 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?"

• The rotor of the Wankel engine is similar to a Reuleaux triangle. A manhole cover is a removable plate forming the lid over the opening of a Manhole, to prevent anyone from falling in and to keep unauthorized persons out The Wankel engine is a type of Internal combustion engine which uses a rotary design to convert pressure into a rotating motion instead of using reciprocating

• A drill bit in the shape of a Reuleaux triangle can drill a hole that is very nearly a perfect square. For the ficitonal character see Drill Bit (Transformers. Drill bits are cutting tools used to create cylindrical holes ^[1]

• A Reuleaux triangle rolls smoothly and easily, but does not make a good wheel because it does not have a fixed center of rotation. A wheel is a circular device that is capable of rotating on its axis facilitating movement or transportation whilst supporting a load ( Mass) or performing labour in machines While an object on top of rollers with cross-sections that were 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. Poul William Anderson ( November 25, 1926 – July 31, 2001) was an American Science fiction author who wrote during a Golden "

• 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.

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. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. The Reuleaux tetrahedron is the intersection of four spheres of Radius s centered at the vertices of a regular Tetrahedron with side length In Geometry, a surface of constant width is a convex Form whose width measured by the distance between two opposite parallelplanes touching its boundary 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. A surface of revolution is a Surface created by rotating a Curve lying on some plane (the Generatrix) around a Straight line (the Axis

References

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