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204:. 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.
200:
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
111:
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
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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
186:
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.
165:
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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
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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
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322:
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dollar coin uses another regular
Reuleaux polygon with 11 sides. However, some coins with rounded-polygon sides, such as the 12-sided 2017
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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.
540:
467:
441:
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Every curve of constant width can be approximated arbitrarily closely by a (possibly irregular) Reuleaux polygon of the same width.
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by an arc centered at its opposite vertex produces a
Reuleaux polygon. As a special case, this construction is possible for every
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Every
Reuleaux polygon must have an odd number of circular-arc sides, and can be constructed in this way from a polygon, the
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Although
Chinese inventor Guan Baihua has made a bicycle with Reuleaux polygon wheels, the invention has not caught on.
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Four 15-sided
Reinhardt polygons, formed from four different Reuleaux polygons with 9, 3, 5, and 15 sides
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Hare, Kevin G.; Mossinghoff, Michael J. (2019), "Most Reinhardt polygons are sporadic",
313:, Dolciani Mathematical Expositions, vol. 45, Mathematical Association of America,
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Bodies of Constant Width: An Introduction to Convex Geometry with Applications
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Firey, W. J. (1960), "Isoperimetric ratios of Reuleaux polygons",
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67:
18:
507:"A new bicycle reinvents the wheel, with a pentagon and triangle"
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coin, do not have constant width and are not Reuleaux polygons.
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coins in the shape of a regular Reuleaux heptagon. The Canadian
95:. These shapes are named after their prototypical example, the
99:, which in turn is named after 19th-century German engineer
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Icons of Mathematics: An Exploration of Twenty Key Images
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The Unexpected Hanging and Other Mathematical Diversions
153:
129:
103:. The Reuleaux triangle can be constructed from an
159:
135:
436:, University of Chicago Press, pp. 212–221,
462:, Princeton University Press, pp. 104–105,
432:(1991), "Chapter 18: Curves of Constant Width",
8:
482:Freiberger, Marianne (December 13, 2016),
259:(2019), "Section 8.1: Reuleaux Polygons",
459:Single Digits: In Praise of Small Numbers
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352:
307:Alsina, Claudi; Nelsen, Roger B. (2011),
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16:Constant-width curve of equal-radius arcs
518:"Inventor creates seriously cool wheels"
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516:Newitz, Annalee (September 30, 2014),
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248:
246:
244:
7:
14:
255:Martini, Horst; Montejano, Luis;
263:, Birkhäuser, pp. 167–169,
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39:
340:Pacific Journal of Mathematics
1:
171:with an odd number of sides.
562:
456:Chamberland, Marc (2015),
541:Piecewise-circular curves
401:10.1007/s10711-018-0326-5
269:10.1007/978-3-030-03868-7
75:coin, a Reuleaux heptagon
46:Regular Reuleaux polygons
27:replaces the sides of an
484:"New ÂŁ1 coin gets even"
354:10.2140/pjm.1960.10.823
85:curve of constant width
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105:equilateral triangle
29:equilateral triangle
378:Geometriae Dedicata
58:Irregular Reuleaux
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157:
133:
77:
33:
503:du Sautoy, Marcus
324:978-0-88385-352-8
278:978-3-030-03866-3
257:Oliveros, DĂ©borah
202:Reinhardt polygon
160:{\displaystyle P}
136:{\displaystyle P}
97:Reuleaux triangle
25:Reuleaux triangle
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505:(May 27, 2009),
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31:by circular arcs
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79:In geometry, a
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347:(3): 823–829,
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145:convex polygon
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113:coinage shapes
101:Franz Reuleaux
73:Gambian dalasi
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489:Plus Magazine
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208:Applications
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119:Construction
91:of constant
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514:. See also
176:convex hull
87:made up of
535:Categories
235:References
182:Properties
511:The Times
417:119629098
392:1405.5233
295:127264210
385:: 1–18,
218:50-pence
214:20-pence
60:heptagon
522:Gizmodo
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363:0113176
287:3930585
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315:p. 155
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222:loonie
93:radius
413:S2CID
387:arXiv
291:S2CID
143:is a
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464:ISBN
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