Editing Reuleaux triangle (section)

{{Short description|Curved triangle with constant width}}
{{good article}}
[[File:ReuleauxTriangle.svg|thumb|The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the opposite vertex.]]
A '''Reuleaux triangle''' {{IPA|fr|ʁœlo|}} is a [[circular triangle|curved triangle]] with [[curve of constant width|constant width]], the simplest and best known curve of constant width other than the circle.<ref>{{harvtxt|Gardner|2014}} calls it the simplest, while {{harvtxt|Gruber|1983|page=59}} calls it "the most notorious".</ref> It is formed from the intersection of three [[circle|circular disks]], each having its center on the boundary of the other two. Constant width means that the separation of every two parallel [[supporting line]]s is the same, independent of their orientation. 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?"<ref>{{citation
 | last = Klee | first = Victor | author-link = Victor Klee
 | doi = 10.2307/3026963
 | issue = 2
 | journal = The Two-Year College Mathematics Journal
 | pages = 14–27
 | title = Shapes of the future
 | volume = 2
 | year = 1971| jstor = 3026963 }}.</ref>

They are named after [[Franz Reuleaux]],<ref name="icons">{{citation|title=Icons of Mathematics: An Exploration of Twenty Key Images|volume=45|series=Dolciani Mathematical Expositions|first1=Claudi|last1=Alsina|first2=Roger B.|last2=Nelsen|publisher=Mathematical Association of America|year=2011|isbn=978-0-88385-352-8|at=[https://books.google.com/books?id=4DavMl7-aFgC&pg=PA155 p. 155]|title-link= Icons of Mathematics}}.</ref> a 19th-century German engineer who pioneered the study of machines for translating one type of motion into another, and who used Reuleaux triangles in his designs.<ref>{{citation|title=The Machines of Leonardo Da Vinci and Franz Reuleaux: Kinematics of Machines from the Renaissance to the 20th Century|volume=2|series=History of Mechanism and Machine Science|first=F. C.|last=Moon|publisher=Springer|year=2007|isbn=978-1-4020-5598-0}}.</ref> However, these shapes were known before his time, for instance by the designers of [[Gothic architecture|Gothic]] church windows, by [[Leonardo da Vinci]], who used it for a [[Octant projection|map projection]], and by [[Leonhard Euler]] in his study of constant-width shapes. Other applications of the Reuleaux triangle include giving the shape to [[guitar pick]]s, [[fire hydrant]] nuts, [[pencil]]s, and [[drill bit]]s for drilling [[Fillet (mechanics)|filleted]] square holes, as well as in graphic design in the shapes of some signs and corporate logos.

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. By several numerical measures it is the farthest from being [[central symmetry|centrally symmetric]]. It provides the largest constant-width shape avoiding the points of an [[integer lattice]], and is closely related to the shape of the quadrilateral maximizing the ratio of perimeter to diameter. It can perform a complete rotation within a square while at all times touching all four sides of the square, and has the smallest possible area of shapes with this property. However, although it covers most of the square in this rotation process, it fails to cover a small fraction of the square's area, near its corners. Because of this property of rotating within a square, the Reuleaux triangle is also sometimes known as the '''Reuleaux rotor'''.<ref name="howround" />

The Reuleaux triangle is the first of a sequence of [[Reuleaux polygon]]s whose boundaries are curves of constant width formed from [[regular polygon]]s with an odd number of sides. Some of these curves have been used as the [[Coinage shapes|shapes of coins]]. The Reuleaux triangle can also be generalized into three dimensions in multiple ways: the [[Reuleaux tetrahedron]] (the intersection of four [[ball (mathematics)|balls]] whose centers lie on a regular [[tetrahedron]]) does not have constant width, but can be modified by rounding its edges to form the [[Meissner body|Meissner tetrahedron]], which does. Alternatively, the [[surface of revolution]] of the Reuleaux triangle also has constant width.