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[[File:Reuleaux polygons.svg|thumb|These [[Reuleaux polygon]]s have constant width, and all have the same width; therefore by Barbier's theorem they also have equal perimeters.]]
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In [[geometry]], '''Barbier's theorem''' states that every [[curve of constant width]]  has perimeter [[Pi|π]] times its width, regardless of its precise shape.<ref>{{citation
| last = Lay | first = Steven R.
  | at = Theorem 11.11, pp. 81–82
| isbn = 9780486458038
| publisher = Dover
| title = Convex Sets and Their Applications
| url = http://books.google.com/books?id=U9eOPjmaH90C&pg=PA81
| year = 2007}}.</ref> This theorem was first published by [[Joseph-Émile Barbier]] in 1860.<ref>{{citation
| last = Barbier | first = E.
| journal = Journal de mathématiques pures et appliquées | series = 2<sup>e</sup> série
| language = French
| pages = 273–286
| title = Note sur le problème de l’aiguille et le jeu du joint couvert
| url = http://portail.mathdoc.fr/JMPA/PDF/JMPA_1860_2_5_A18_0.pdf
| volume = 5
| year = 1860}}. See in particular pp. 283–285.</ref>
 
==Examples==
The most familiar examples of curves of constant width are the [[circle]] and the [[Reuleaux triangle]].  For a circle, the width is the same as the [[diameter]]; a circle of width ''w'' has [[perimeter]] π''w''.  A Reuleaux triangle of width ''w'' consists of three [[Arc (geometry)|arc]]s of circles of [[radius]] ''w''.  Each of these arcs has [[central angle]] π/3, so the perimeter of the Reuleaux triangle of width ''w'' is equal to half the perimeter of a circle of radius ''w'' and therefore is equal to π''w''.  A similar analysis of other simple examples such as [[Reuleaux polygon]]s gives the same answer.
 
==Proofs==
One proof of the theorem uses the properties of [[Minkowski sum]]s. If ''K'' is a body of constant width ''w'', then the Minkowski sum of ''K'' and its 180° rotation is a disk with radius ''w'' and perimeter 2π''w''. However, the Minkowski sum acts linearly on the perimeters of convex bodies, so the perimeter of ''K'' must be half the perimeter of this disk, which is π''w'' as the theorem states.<ref>[http://www.cut-the-knot.org/ctk/Barbier.shtml The Theorem of Barbier (Java)] at [[cut-the-knot]].</ref>
 
More generally, the theorem follows immediately from the [[Crofton formula]] in [[integral geometry]] according to which the length of any curve equals the measure of the set of lines that cross the curve, multiplied by their numbers of crossings. Any two curves that have the same constant width are crossed by sets of lines with the same measure, and therefore they have the same length. Historically, Crofton derived his formula later than, and independently of, Barbier's theorem.<ref>{{citation
| last = Sylvester | first = J. J. | author-link = James Joseph Sylvester
| doi = 10.1007/BF02413320
| issue = 1
| journal = Acta Mathematica
| pages = 185–205
| title = On a funicular solution of Buffon's “problem of the needle” in its most general form
| volume = 14
| year = 1890}}.</ref>
 
An elementary probabilistic proof of the theorem can be found at [[Buffon's noodle]].
 
==Higher dimensions==
The analogue of Barbier's theorem for [[surface of constant width|surfaces of constant width]] is false. In particular, the [[unit sphere]] has surface area <math>4\pi\approx 12.566</math>, while the [[surface of revolution]] of a [[Reuleaux triangle]] with the same constant width has surface area <math>8\pi-\tfrac{4}{3}\pi^2\approx 11.973</math>.<ref>{{citation
| last1 = Bayen | first1 = Térence
| last2 = Henrion | first2 = Didier
| doi = 10.1080/10556788.2010.547580
| issue = 6
| journal = Optimization Methods and Software
| pages = 1073–1099
| title = Semidefinite programming for optimizing convex bodies under width constraints
| url = http://hal.archives-ouvertes.fr/hal-00495031
| volume = 27
| year = 2012}}.</ref>
 
==References==
{{reflist}}
 
[[Category:Theorems in geometry]]
[[Category:Pi]]
[[Category:Length]]

Latest revision as of 07:29, 15 December 2014

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