In geometry, a Reuleaux polygon is a curve of constant width made up of

circular arc A circular arc is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than radians (180 ...

s of constant

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

. 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 at ''Technische Hochschule Berlin'' (today Technische Universität Berlin), later appointed as the president of the academy. He was often c ...

. The Reuleaux triangle can be constructed from an

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

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 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 ...

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

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 ...

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, 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. 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 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 (Currency symbol, symbol: Pound sign, £; ISO 4217, currency code: GBP) is the currency of the United Kingdom and nine of its associated territories. The pound is the main unit of account, unit of sterling, and the word ''Pound (cu ...

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

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The Times ''The Times'' is a British Newspaper#Daily, daily Newspaper#National, national newspaper based in London. It began in 1785 under the title ''The Daily Universal Register'', adopting its modern name on 1 January 1788. ''The Times'' and its si ...

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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 organisa ...

, 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 =

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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 an ...

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