Reuleaux polygon

AReuleaux trianglereplaces the sides of anequilateral triangleby circular arcs

Regular Reuleaux polygons

Irregular Reuleauxheptagon

In geometry, aReuleaux polygonis acurve of constant widthmade up ofcircular arcsof constantradius.^[1]These shapes are named after their prototypical example, theReuleaux triangle,which in turn is named after 19th-century German engineerFranz Reuleaux.^[2]The Reuleaux triangle can be constructed from anequilateral triangleby 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 anyregular polygonwith 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 incoinage shapes.

Construction[edit]

If $P$ is aconvex polygonwith 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 everyregular polygonwith an odd number of sides.^[1]

Every Reuleaux polygon must have an odd number of circular-arc sides, and can be constructed in this way from a polygon, theconvex hullof 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.^[1]

Properties[edit]

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.^[3]

Every curve of constant width can be approximated arbitrarily closely by a (possibly irregular) Reuleaux polygon of the same width.^[1]

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 aReinhardt 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.^[4]

Applications[edit]

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 made20-penceand50-pencecoins in the shape of a regular Reuleaux heptagon.^[5]The Canadianlooniedollar coin uses another regular Reuleaux polygon with 11 sides.^[6]However, some coins with rounded-polygon sides, such as the 12-sided 2017British poundcoin, do not have constant width and are not Reuleaux polygons.^[7]

Although Chinese inventor Guan Baihua has made a bicycle with Reuleaux polygon wheels, the invention has not caught on.^[8]

References[edit]

^^a ^b ^c ^dMartini, Horst; Montejano, Luis;Oliveros, Déborah(2019), "Section 8.1: Reuleaux Polygons",Bodies of Constant Width: An Introduction to Convex Geometry with Applications,Birkhäuser, pp. 167–169,doi:10.1007/978-3-030-03868-7,ISBN 978-3-030-03866-3,MR 3930585,S2CID 127264210
^Alsina, Claudi; Nelsen, Roger B. (2011),Icons of Mathematics: An Exploration of Twenty Key Images,Dolciani Mathematical Expositions, vol. 45, Mathematical Association of America,p. 155,ISBN 978-0-88385-352-8
^Firey, W. J. (1960), "Isoperimetric ratios of Reuleaux polygons",Pacific Journal of Mathematics,10(3): 823–829,doi:10.2140/pjm.1960.10.823,MR 0113176
^Hare, Kevin G.; Mossinghoff, Michael J. (2019), "Most Reinhardt polygons are sporadic",Geometriae Dedicata,198:1–18,arXiv:1405.5233,doi:10.1007/s10711-018-0326-5,MR 3933447,S2CID 119629098
^Gardner, Martin(1991), "Chapter 18: Curves of Constant Width",The Unexpected Hanging and Other Mathematical Diversions,University of Chicago Press, pp. 212–221,ISBN 0-226-28256-2
^Chamberland, Marc (2015),Single Digits: In Praise of Small Numbers,Princeton University Press, pp. 104–105,ISBN 9781400865697
^Freiberger, Marianne (December 13, 2016),"New £1 coin gets even",Plus Magazine
^du Sautoy, Marcus(May 27, 2009),"A new bicycle reinvents the wheel, with a pentagon and triangle",The Times.See alsoNewitz, Annalee (September 30, 2014),"Inventor creates seriously cool wheels",Gizmodo

[mmo-1] Martini, Horst; Montejano, Luis;Oliveros, Déborah(2019), "Section 8.1: Reuleaux Polygons",Bodies of Constant Width: An Introduction to Convex Geometry with Applications,Birkhäuser, pp. 167–169,doi:10.1007/978-3-030-03868-7,ISBN 978-3-030-03866-3,MR 3930585,S2CID 127264210

[icons-2] Alsina, Claudi; Nelsen, Roger B. (2011),Icons of Mathematics: An Exploration of Twenty Key Images,Dolciani Mathematical Expositions, vol. 45, Mathematical Association of America,p. 155,ISBN 978-0-88385-352-8

[firey-3] Firey, W. J. (1960), "Isoperimetric ratios of Reuleaux polygons",Pacific Journal of Mathematics,10(3): 823–829,doi:10.2140/pjm.1960.10.823,MR 0113176

[hm-4] Hare, Kevin G.; Mossinghoff, Michael J. (2019), "Most Reinhardt polygons are sporadic",Geometriae Dedicata,198:1–18,arXiv:1405.5233,doi:10.1007/s10711-018-0326-5,MR 3933447,S2CID 119629098

[gardner-5] Gardner, Martin(1991), "Chapter 18: Curves of Constant Width",The Unexpected Hanging and Other Mathematical Diversions,University of Chicago Press, pp. 212–221,ISBN 0-226-28256-2

[chamberland-6] Chamberland, Marc (2015),Single Digits: In Praise of Small Numbers,Princeton University Press, pp. 104–105,ISBN 9781400865697

[plus-7] Freiberger, Marianne (December 13, 2016),"New £1 coin gets even",Plus Magazine

[bicycle-8] u Sautoy, Marcus(May 27, 2009),"A new bicycle reinvents the wheel, with a pentagon and triangle",The Times.See alsoNewitz, Annalee (September 30, 2014),"Inventor creates seriously cool wheels",Gizmodo

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