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. Image File history File links A Reuleaux Triangle has equal widths. ... Look up polygon in Wiktionary, the free dictionary. ... For closed convex planar bodies whose boundary is a smooth curve, one notes that there are exactly two parallel tangent lines to the boundary curve in any given direction. ... Franz Reuleaux (September 30, 1829 - August 20, 1905), was a mechanical engineer and a lecturer of Berlin Royal Technical Academy, later appointed as the President of the Academy. ...

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.) For closed convex planar bodies whose boundary is a smooth curve, one notes that there are exactly two parallel tangent lines to the boundary curve in any given direction. ...

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. A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ... a compass In drafting, a compass (or pair of compasses) is an instrument]] used by mathematicians and craftsmen in for drawing or inscribing a circle or arc. ... Look up vertex in Wiktionary, the free dictionary. ... 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, all elements of B that also belong to A), but no other elements. ... Circle illustration In classical geometry, a radius (plural: radii) of a circle or sphere is any line segment from its center to its boundary. ...

By the Blaschke-Lebesgue theorem, the Reuleaux triangle has the least area of any curve of given constant width. This area is , 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. A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... The British decimal Twenty Pence (20p) coin was issued in June 1982 to fill in the obvious gap between the Ten Pence and Fifty Pence coins; it rapidly gained acceptance and very large numbers now circulate [1]. The coin is minted from an alloy of 84% copper and 16% nickel... The British decimal fifty pence (50p) coin – often pronounced fifty pee – was issued on October 14, 1969 in the run-up to decimalisation to replace the ten shilling note. ...

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 engine has been used by Japanese car company Mazda.

• 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.

It has been suggested that Densa be merged into this article or section. ... Princeton University manhole cover, Princeton, NJ, USA Pick holes in manhole cover, Palo Alto, CA, USA Kraków manhole cover (note integral hinge) Painted manhole cover in Matsumoto, Japan. ... Wankel Engine in Deutsches Museum Munich, Germany The Wankel rotary engine is a type of internal combustion engine, invented by German engineer Felix Wankel, which uses a rotor instead of reciprocating pistons. ... Drill bits are the cutters of drill tools. ... The force bearing on the axle has an eccentricity e with the point of contact to the rolling surface and exerts a moment about the contact point A wheel is a circular device capable of rotating on its axis, facilitating movement or transportation or performing labour in machines. ... Poul Anderson portrayed on the cover of a special edition of The Magazine of Fantasy and Science Fiction; painting by Kelly Freas. ...

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. A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Animation of a Reuleaux tetrahedron, showing also the tetrahedron from which it is formed. ... The parabola y=x2 rotated about the z-axis A surface of revolution is a surface created by rotating a curve lying on some plane (the generatrix) around a straight line (the axis of rotation) that lies on the same plane. ...

External link

• Shapes of constant width at cut-the-knot
• Film about the Reuleaux triangle, also showing the Wankel engine and the mechanics of a famous sovietic film projector (self-explaing, with no sound and only some russian words)
• Explanations (in german) and pictures with some three-dimensional generalizations of a Reuleaux triangle, including films of both types of Meissner bodies rotating