In geometry, a Reuleaux polygon is a

curve of constant width In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width ...

made up of

circular arc Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fallacy * Circular ...

s of constant

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

. These shapes are named after their prototypical example, the Reuleaux triangle, which in turn, is named after 19th-century German engineer

Franz Reuleaux Franz Reuleaux (; ; 30 September 1829 – 20 August 1905), was a German mechanical engineer and a lecturer of the Berlin Royal Technical Academy, later appointed as the President of the Academy. He was often called the father of kinematics. He wa ...

. The Reuleaux triangle can be constructed from an

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...

by connecting each two vertices by a circular arc centered on the third vertex, and Reuleaux polygons can be formed by a similar construction from any

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...

with an odd number of sides, or from certain irregular polygons. Every curve of constant width can be accurately approximated by Reuleaux polygons. They have been applied in

coinage shapes Although the vast majority of coins are round, coins are made in a variety of other shapes, including squares, diamonds, hexagons, heptagons, octagons, decagons, and dodecagons. They have also been struck with scalloped (wavy) edges, and with hole ...

Construction

P

is a

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...

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

P

by an arc centered at its opposite vertex produces a Reuleaux polygon. As a special case, this construction is possible for every

with an odd number of sides. 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 In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

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.

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. Every curve of constant width can be approximated arbitrarily closely by a (possibly irregular) Reuleaux polygon of the same width. Reinhardt 15-gons

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 In geometry, a Reinhardt polygon is an equilateral polygon inscribed in a Reuleaux polygon. As in the regular polygons, each vertex of a Reinhardt polygon participates in at least one defining pair of the diameter of the polygon. Reinhardt polyg ...

. 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.

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 20-pence and 50-pence coins in the shape of a regular Reuleaux heptagon. The Canadian

loonie The loonie (french: huard), formally the Canadian one-dollar coin, is a gold-coloured Canadian coin that was introduced in 1987 and is produced by the Royal Canadian Mint at its facility in Winnipeg. The most prevalent versions of the coin sh ...

dollar coin uses another regular Reuleaux polygon with 11 sides. However, some coins with rounded-polygon sides, such as the 12-sided 2017

British pound Sterling (abbreviation: stg; Other spelling styles, such as STG and Stg, are also seen. ISO code: GBP) is the currency of the United Kingdom and nine of its associated territories. The pound ( sign: £) is the main unit of sterling, and t ...

coin, do not have constant width and are not Reuleaux polygons. Although Chinese inventor Guan Baihua has made a bicycle with Reuleaux polygon wheels, the invention has not caught on.

References

{{reflist, refs= {{citation, url=https://www.thetimes.co.uk/article/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 {{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 {{citation , last = Firey , first = W. J. , journal =

Pacific Journal of Mathematics The Pacific Journal of Mathematics is a mathematics research journal supported by several universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisati ...

, 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 {{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 {{citation , last1 = Hare , first1 = Kevin G. , last2 = Mossinghoff , first2 = Michael J. , doi = 10.1007/s10711-018-0326-5 , journal =

Geometriae Dedicata ''Geometriae Dedicata'' is a mathematical journal, founded in 1972, concentrating on geometry and its relationship to topology, group theory and the theory of dynamical systems. It was created on the initiative of Hans Freudenthal in Utrecht, the N ...

, mr = 3933447 , pages = 1–18 , title = Most Reinhardt polygons are sporadic , volume = 198 , year = 2019, arxiv = 1405.5233 , s2cid = 119629098 {{citation , last1 = Alsina , first1 = Claudi , last2 = Nelsen , first2 = Roger B. , at
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, 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 {{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 {{citation , last = Freiberger , first = Marianne , date = December 13, 2016 , magazine =

Plus Magazine ''Plus Magazine'' is an online popular mathematics magazine run under the Millennium Mathematics Project at the University of Cambridge. ''Plus'' contains: * feature articles on all aspects of mathematics; * reviews of popular maths books and ...

, title = New £1 coin gets even , url = https://plus.maths.org/content/new-1-coin-gets-even Piecewise-circular curves Constant width