Reuleaux triangle

Because its width is constant, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?

Among constant-width shapes with a given width, the Reuleaux triangle has the minimum area and the sharpest (smallest) possible angle (120°) at its corners.

[5] In a paper that he presented in 1771 and published in 1781 entitled De curvis triangularibus, Euler studied curvilinear triangles as well as the curves of constant width, which he called orbiforms.

For the Reuleaux triangle, the two centrally symmetric shapes that determine the measures of asymmetry are both hexagonal, although the inner one has curved sides.

That is, the maximum ratio of areas on either side of a diameter, another measure of asymmetry, is bigger for the Reuleaux triangle than for other curves of constant width.

[21] Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite that can be inscribed into a Reuleaux triangle.

[22] By Barbier's theorem all curves of the same constant width including the Reuleaux triangle have equal perimeters.

[26] It has also been conjectured, but not proven, that the Reuleaux triangles have the highest packing density of any curve of constant width.

[9] As it rotates, its axis does not stay fixed at a single point, but instead follows a curve formed by the pieces of four ellipses.

[30] The existence of Reuleaux triangles and other curves of constant width shows that diameter measurements alone cannot verify that an object has a circular cross-section.

[32] Several types of machinery take the shape of the Reuleaux triangle, based on its property of being able to rotate within a square.

The Watts Brothers Tool Works square drill bit has the shape of a Reuleaux triangle, modified with concavities to form cutting surfaces.

When mounted in a special chuck which allows for the bit not having a fixed centre of rotation, it can drill a hole that is nearly square.

[9][33] Panasonic's RULO robotic vacuum cleaner has its shape based on the Reuleaux triangle in order to ease cleaning up dust in the corners of rooms.

[9][37] This concept was used in a science fiction short story by Poul Anderson titled "The Three-Cornered Wheel".

[39] Another class of applications of the Reuleaux triangle involves using it as a part of a mechanical linkage that can convert rotation around a fixed axis into reciprocating motion.

In its use in Gothic church architecture, the three-cornered shape of the Reuleaux triangle may be seen both as a symbol of the Trinity,[51] and as "an act of opposition to the form of the circle".

[42] Modern buildings that have been claimed to use a Reuleaux triangle shaped floorplan include the MIT Kresge Auditorium, the Kölntriangle, the Donauturm, the Torre de Collserola, and the Mercedes-Benz Museum.

[56] Many guitar picks employ the Reuleaux triangle, as its shape combines a sharp point to provide strong articulation, with a wide tip to produce a warm timbre.

Because all three points of the shape are usable, it is easier to orient and wears less quickly compared to a pick with a single tip.

[58] Following a suggestion of Keto (1997),[59] the antennae of the Submillimeter Array, a radio-wave astronomical observatory on Mauna Kea in Hawaii, are arranged on four nested Reuleaux triangles.

[60][61] Placing the antennae on a curve of constant width causes the observatory to have the same spatial resolution in all directions, and provides a circular observation beam.

As the most asymmetric curve of constant width, the Reuleaux triangle leads to the most uniform coverage of the plane for the Fourier transform of the signal from the array.

[62][63] Another corporate logo framed in the Reuleaux triangle, the south-pointing compass of Bavaria Brewery, was part of a makeover by design company Total Identity that won the SAN 2010 Advertiser of the Year award.

Triangular curves of constant width with smooth rather than sharp corners may be obtained as the locus of points at a fixed distance from the Reuleaux triangle.

The resulting shape consists of circular arcs (at most as many as sides of the polygon), can be constructed algorithmically in linear time, and can be drawn with compass and straightedge.

[74] In the classical presentation of a three-set Venn diagram as three overlapping circles, the central region (representing elements belonging to all three sets) takes the shape of a Reuleaux triangle.

[77] Relatives of the Reuleaux triangle arise in the problem of finding the minimum perimeter shape that encloses a fixed amount of area and includes three specified points in the plane.

For a wide range of choices of the area parameter, the optimal solution to this problem will be a curved triangle whose three sides are circular arcs with equal radii.

[79] Other planar shapes with three curved sides include the arbelos, which is formed from three semicircles with collinear endpoints,[80] and the Bézier triangle.