Editing Reuleaux polygon

{{Short description|Constant-width curve of equal-radius arcs}}
[[File:ReuleauxTriangle.svg|thumb|upright=0.75|A [[Reuleaux triangle]] replaces the sides of an [[equilateral triangle]] by circular arcs]]
{{multiple image|total_width=360|
|image1=Reuleaux polygons.svg|caption1=Regular Reuleaux polygons
|image2=Reuleaux polygon construction.svg|caption2=Irregular Reuleaux [[heptagon]]
}}
[[File:Gambia 1 dalasi.JPG|thumb|upright|[[Gambian dalasi]] coin, a Reuleaux heptagon]]
In geometry, a '''Reuleaux polygon''' is a [[curve of constant width]] made up of [[circular arc]]s of constant [[radius]].{{r|mmo}} These shapes are named after their prototypical example, the [[Reuleaux triangle]], which in turn is named after 19th-century German engineer [[Franz Reuleaux]].{{r|icons}} The Reuleaux triangle can be constructed from an [[equilateral triangle]] by connecting each pair of adjacent vertices with a circular arc centered on the opposing vertex, and Reuleaux polygons can be formed by a similar construction from any [[regular polygon]] with an odd number of sides as well as certain irregular polygons. Every curve of constant width can be accurately approximated by Reuleaux polygons. They have been applied in [[coinage shapes]].

==Construction==
If <math>P</math> is a [[convex polygon]] with an odd number of sides, in which each vertex is equidistant to the two opposite vertices and closer to all other vertices, then replacing each side of <math>P</math> by an arc centered at its opposite vertex produces a Reuleaux polygon. As a special case, this construction is possible for every [[regular polygon]] with an odd number of sides.{{r|mmo}}

Every Reuleaux polygon must have an odd number of circular-arc sides, and can be constructed in this way from a polygon, the [[convex hull]] of its arc endpoints. However, it is possible for other curves of constant width to be made of an even number of arcs with varying radii.{{r|mmo}}

==Properties==
The Reuleaux polygons based on regular polygons are the only curves of constant width whose boundaries are formed by finitely many circular arcs of equal length.{{r|firey}}

Every curve of constant width can be approximated arbitrarily closely by a (possibly irregular) Reuleaux polygon of the same width.{{r|mmo}}

[[File:Reinhardt 15-gons.svg|thumb|Four 15-sided Reinhardt polygons, formed from four different Reuleaux polygons with 9, 3, 5, and 15 sides]]
A regular Reuleaux polygon has sides of equal length. More generally, when a Reuleaux polygon has sides that can be split into arcs of equal length, the convex hull of the arc endpoints is a [[Reinhardt polygon]]. These polygons are optimal in multiple ways: they have the largest possible perimeter for their diameter, the largest possible width for their diameter, and the largest possible width for their perimeter.{{r|hm}}

==Applications==
The constant width of these shapes allows their use as coins that can be used in coin-operated machines. For instance, the United Kingdom has made [[twenty pence (British coin)|20-pence]] and [[fifty pence (British coin)|50-pence]] coins in the shape of a regular Reuleaux heptagon.{{r|gardner}} The Canadian [[loonie]] dollar coin uses another regular Reuleaux polygon with 11 sides.{{r|chamberland}} However, some coins with rounded-polygon sides, such as the 12-sided 2017 [[British pound]] coin, do not have constant width and are not Reuleaux polygons.{{r|plus}}

Although Chinese inventor Guan Baihua has made a bicycle with Reuleaux polygon wheels, the invention has not caught on.{{r|bicycle}}

==References==
{{reflist|refs=

<ref name=bicycle>{{citation|url=https://www.thetimes.com/travel/destinations/asia-travel/china/a-new-bicycle-reinvents-the-wheel-with-a-pentagon-and-triangle-vb8jkdlf8qw|newspaper=[[The Times]]|first=Marcus|last=du Sautoy|authorlink=Marcus du Sautoy|title=A new bicycle reinvents the wheel, with a pentagon and triangle|date=May 27, 2009}}. See also {{citation|url=https://io9.gizmodo.com/inventor-creates-a-math-infused-bicycle-with-seriously-1640798248#!|title=Inventor creates seriously cool wheels|work=Gizmodo|first=Annalee|last=Newitz|date=September 30, 2014}}</ref>

<ref name=chamberland>{{citation
 | last = Chamberland | first = Marc
 | isbn = 9781400865697
 | pages = 104–105
 | publisher = Princeton University Press
 | title = Single Digits: In Praise of Small Numbers
 | url = https://books.google.com/books?id=n9iqBwAAQBAJ&pg=PA104
 | year = 2015}}</ref>

<ref name=firey>{{citation
 | last = Firey | first = W. J.
 | journal = [[Pacific Journal of Mathematics]]
 | doi = 10.2140/pjm.1960.10.823
 | doi-access = free
 | mr = 0113176
 | pages = 823–829
 | title = Isoperimetric ratios of Reuleaux polygons
 | volume = 10
 | issue = 3
 | year = 1960}}</ref>
 
<ref name=gardner>{{citation
 | last = Gardner | first = Martin | author-link = Martin Gardner
 | contribution = Chapter 18: Curves of Constant Width
 | isbn = 0-226-28256-2
 | pages = 212–221
 | publisher = University of Chicago Press
 | title = The Unexpected Hanging and Other Mathematical Diversions
 | year = 1991}}</ref>

<ref name=hm>{{citation
 | last1 = Hare | first1 = Kevin G.
 | last2 = Mossinghoff | first2 = Michael J.
 | doi = 10.1007/s10711-018-0326-5
 | journal = [[Geometriae Dedicata]]
 | mr = 3933447
 | pages = 1–18
 | title = Most Reinhardt polygons are sporadic
 | volume = 198
 | year = 2019| arxiv = 1405.5233
 | s2cid = 119629098
 }}</ref>

<ref name=icons>{{citation
 | last1 = Alsina | first1 = Claudi
 | last2 = Nelsen | first2 = Roger B.
 | at = [https://books.google.com/books?id=4DavMl7-aFgC&pg=PA155 p. 155]
 | isbn = 978-0-88385-352-8
 | publisher = Mathematical Association of America
 | series = Dolciani Mathematical Expositions
 | title = Icons of Mathematics: An Exploration of Twenty Key Images
 | title-link = Icons of Mathematics
 | volume = 45
 | year = 2011}}</ref>

<ref name=mmo>{{citation
 | last1 = Martini | first1 = Horst
 | last2 = Montejano | first2 = Luis
 | last3 = Oliveros | first3 = Déborah | author3-link = Déborah Oliveros
 | contribution = Section 8.1: Reuleaux Polygons
 | doi = 10.1007/978-3-030-03868-7
 | isbn = 978-3-030-03866-3
 | mr = 3930585
 | pages = 167–169
 | publisher = Birkhäuser
 | title = Bodies of Constant Width: An Introduction to Convex Geometry with Applications
 | year = 2019| s2cid = 127264210
 }}</ref>

<ref name=plus>{{citation
 | last = Freiberger | first = Marianne
 | date = December 13, 2016
 | magazine = [[Plus Magazine]]
 | title = New £1 coin gets even
 | url = https://plus.maths.org/content/new-1-coin-gets-even}}</ref>

}}

[[Category:Piecewise-circular curves]]
[[Category:Constant width]]