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| [[File:ReuleauxTriangle.svg|frame|right|The Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the opposite vertex.]]
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| A '''Reuleaux triangle''' is the simplest and best known Reuleaux polygon. It is a [[curve of constant width]], meaning that the separation of two parallel lines tangent to the curve is independent of their orientation. Because all diameters are the same, 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?" The term derives from [[Franz Reuleaux]], a 19th-century German engineer who did pioneering work on ways that machines translate one type of motion into another, although the concept was known before his time.
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| == Construction ==
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| [[File:Construction triangle Reuleaux.svg|thumb|left|150px|To construct a Reuleaux triangle]]
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| With a [[Compass (drafting)|compass]], sweep an arc sufficient to enclose the desired figure. With radius unchanged, sweep a sufficient arc centred at a point on the first arc to intersect that arc. With the same radius and the centre at that intersection sweep a third arc to intersect the other arcs. The result is a curve of constant width.
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| Equivalently, given an [[equilateral triangle]] ''T'' of side length ''s'', take the boundary of the [[intersection (set theory)|intersection]] of the disks with [[radius]] ''s'' centered at the vertices of ''T''.
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| By the [[Blaschke–Lebesgue theorem]], the Reuleaux triangle has the least area of any curve of given constant width. This area is <math>{1\over2}(\pi - \sqrt3)s^2</math>, where ''s'' is the constant width. The existence of Reuleaux polygons shows that diameter measurements alone cannot verify that an object has a circular cross-section.
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| The area of Reuleaux triangle is smaller than that of the [[Disk (mathematics)|disk]] of the same width (i.e. diameter); the area of such a disk is <math>\pi s^2 \over 4</math>.
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| == Reuleaux polygons ==
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| [[File:Reuleaux polygons.svg|thumb|Reuleaux polygons]]
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| The Reuleaux triangle can be generalized to [[regular polygon]]s with an odd number of sides, yielding a '''Reuleaux polygon'''. The most commonly used of these is the Reuleaux heptagon, which is the shape of several coins:
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| * [[Botswana pula]] coins in the denominations of 2 pula, 1 pula, 25 thebe and 5 thebe.
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| * [[Coins of the Cypriot pound#Decimal - cents|Cypriot]] 50-cent coin, from 1991 until Cyprus joined the [[Euro]] in 2008.
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| * [[Jordan]]ian quarter-[[Jordanian dinar|dinar]] and half-dinar coins.
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| * [[Mauritius|Mauritian]] 10-[[Mauritian rupee|rupee]] coin.
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| * [[United Kingdom|British]] [[twenty pence (British coin)|20-pence]] and [[fifty pence (British coin)|50-pence]] coins.
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| * [[Canada|Canadian]] [[Loonie]] dollar coin (eleven sides).
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| The constant width of such coins allows their use in coin-operated machines.
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| == Other uses ==
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| [[File:Rouleaux triangle Animation.gif|frame|right|The Reuleaux triangle rotating inside a square]]
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| * The rotor of the [[Wankel engine]] is easily mistaken for a Reuleaux triangle but its curved sides are somewhat flatter than those of a Reuleaux triangle and so it does not have constant width.<ref>[http://www.der-wankelmotor.de/Techniklexikon/techniklexikon.html Ein Wankel-Rotor ist kein Reuleux-Dreieck!] German [http://translate.google.de/translate?u=http%3A%2F%2Fwww.der-wankelmotor.de%2FTechniklexikon%2Ftechniklexikon.html&sl=de&tl=en&hl=de&ie=UTF-8 Translation A Wankel-Rotor is not a Reuleux-Triangle!]</ref>
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| * The [[Watts Brothers Tool Works]] square [[drill bit]] has the shape of a Reuleaux triangle and can, if mounted in a special chuck which allows for the bit not having a fixed centre of rotation, drill a hole that is nearly square;<ref>{{cite book
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| |author=Watts Brothers Tool Works
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| |title=How to drill square hexagon octagon pentagon holes
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| |publisher=Wilmerding, Pa. : The Company,
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| |location = New York
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| |year=1950–1951
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| |contribution=27 p. : ill. ; 23 x 15 cm. + price list & brochure.
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| }}</ref> the corners of the square are slightly rounded, as can be seen by tracing any vertex in this figure, and the drill bit covers 0.9877... of the area of the square.<ref>[[Clifford Pickover|Pickover, Clifford A.]], ''The Math Book'', Sterling, 2009: p. 266.</ref> The Harry Watt square is often used in [[Mortise and tenon|mortising]]<ref>[http://upper.us.edu/faculty/smith/reuleaux.htm Drilling Square Holes]</ref><ref>[http://mathworld.wolfram.com/ReuleauxTriangle.html Reuleaux Triangle – from Wolfram MathWorld]</ref> Other Reuleaux polygons are used to drill pentagonal, hexagonal, and octagonal holes.
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| * A Reuleaux triangle (along with all other [[curve of constant width|curves of constant width]]) can roll but makes a poor wheel because it does not roll about a fixed center of rotation. An object on top of rollers with cross-sections that were Reuleaux triangles would roll smoothly and flatly, but an axle attached to Reuleaux triangle wheels 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."<ref>[http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf613 "Three-Cornered Wheel"]</ref>
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| * Several pencils are manufactured in this shape, rather than the more traditional round or hexagonal barrels.<ref>http://www.pencilrevolution.com/2006/04/review-of-staedtler-noris-ergosoft-hb/</ref> They are usually promoted as being more comfortable or encouraging proper grip, as well as being less likely to roll off tables (since the center of gravity moves up and down more than a rolling hexagon).
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| * The shape is used for signage for the [[National Trails System]] administered by the United States National Park Service,<ref name=NPS>{{cite web|publisher=[[National Park Service]]|title=National Trails System – Visit The Trails|url=http://www.fs.fed.us/r3/prescott/recreation/trails/index.shtml|accessdate=2009-01-18}}</ref> as well as the logo of [[Colorado School of Mines]] and the Connecticut Collegiate Mathematics Competition.
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| * The corporate logo of [[Petrofina]] (Fina), a Belgian oil company with major operations in Europe, North America and Africa, utilized a Reuleaux triangle with the Fina name from 1950 until Petrofina's merger with [[Total S.A.]] in 2000.<ref>{{cite web|title=Fina Logo History: from Petrofina to Fina|url=http://www.total.com/en/about-total/group-presentation/group-history/fina-logo-history-922651.html|work=Total: Group Presentation|publisher=Total S.A.|accessdate=24 June 2013}}</ref> A rotated version of Fina's Reuleaux triangle is utilized by [[Alon USA]], which acquired the American Petrofina operations spun off by Total in 2006.<ref>{{cite web|title=Retail/Branded Marketing|url=http://www.alonusa.com/content/retailbranded-marketing|publisher=Alon USA|accessdate=24 June 2013}}</ref>
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| * Valve covers used in the Mission Bay Project of [[San Francisco]] to differentiate reclaimed water from potable water are in the shape of a Reuleaux triangle.<ref>[http://www.maa.org/FoundMath/08week21.html A picture of Reuleaux triangle water valve cover] in MMA's ''Found Math'' gallery</ref>
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| [[File:Reuleaux kite.svg|thumb|An [[equidiagonal quadrilateral|equidiagonal]] [[kite (geometry)|kite]] that maximizes the ratio of perimeter to diameter, inscribed in a Reuleaux triangle]]
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| * Among all [[quadrilateral]]s, the shape that has the greatest ratio of its [[perimeter]] to its [[diameter]] is an [[equidiagonal quadrilateral|equidiagonal]] [[kite (geometry)|kite]] that can be inscribed into a Reuleaux triangle.<ref>{{Cite journal |first=D.G. |last=Ball |title=A generalisation of π |journal=Mathematical Gazette |volume=57 |issue=402 |year=1973 |pages=298–303 |doi=10.2307/3616052 |ref=harv}}; {{Cite journal |first1=David |last1= Griffiths |first2=David |last2=Culpin |title=Pi-optimal polygons |journal=Mathematical Gazette |volume=59 |issue=409 |year=1975 |pages=165–175 |doi=10.2307/3617699 |ref=harv |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.</ref>
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| * Many [[guitar picks]] employ the Reuleaux triangle, as its unique shape combines a sharp point to provide strong articulation, with a wide tip to produce a warm timbre. Many players find the shape ergonomic, since it naturally tends to point in the proper direction. Its three equal tips also prevent wear and extend lifespan, as compared to the single tip of a pick shaped like an [[isosceles triangle]]. <ref>Hoover, Will (November 1995). Picks!: The Colorful Saga of Vintage Celluloid Guitar Plectrums. (pages 32-33) Backbeat Books. ISBN 978-0-87930-377-8.</ref>
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| * Many windows in buildings designed by Berlage in Amsterdam's Plan Zuid extension are the shape of the Reuleaux triangle.
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| == Three-dimensional version == <!-- This section is linked from [[Sphere]] -->
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| The intersection of four spheres 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]].<ref name=weber>{{cite web
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| | author = Weber, Christof
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| | year = 2009
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| | url = http://www.swisseduc.ch/mathematik/geometrie/gleichdick/docs/meissner_en.pdf
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| | title = What does this solid have to do with a ball?}} There are also [http://www.swisseduc.ch/mathematik/material/gleichdick/index.html films of both types of Meissner body rotating].</ref> It can, however, be made into a surface of constant width, called [[Reuleaux tetrahedron#Meissner bodies|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 known surfaces of revolution of given constant width ({{harvtxt|Campi|Colesanti|Gronchi|1996}}).
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| == See also ==
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| * [[Deltoid curve]]
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| * [[Superellipse]]
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| * [[Vesica piscis]]
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| ==Notes==
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| {{Reflist}}
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| * [[Heinrich Guggenheimer]] (1977) ''Applicable Geometry'', page 58, Krieger, Huntington ISBN 0-88275-368-1 .
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| ==External links==
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| {{Commons category|Reuleaux triangles}}
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| *[http://www.howround.com How Round is Your Circle?] – book about various geometric properties, including curves and solids of constant width
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| *[http://www.cut-the-knot.org/do_you_know/cwidth.shtml Shapes of constant width] at [[cut-the-knot]]
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| *{{cite web|title=Shapes and Solids of Constant Width|url=http://www.numberphile.com/videos/shapes_constant.html|work=Numberphile|publisher=[[Brady Haran]]|author=Mould, Steve}}
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| {{Expand Russian|date=September 2011|Треугольник Рёло}}
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| {{DEFAULTSORT:Reuleaux Triangle}}
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| [[Category:Polygons]]
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| [[Category:Curves]]
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| [[Category:Triangles]]
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| {{Link GA|uk}}
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| {{Link FA|ru}}
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| {{Link FA|uk}}
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