Editing Reuleaux triangle (section)

== Mathematical properties ==
[[File:Reuleaux supporting lines.svg|thumb|Parallel [[supporting line]]s of a Reuleaux triangle]]
The most basic property of the Reuleaux triangle is that it has constant width, meaning that for every pair of parallel [[supporting line]]s (two lines of the same slope that both touch the shape without crossing through it) the two lines have the same [[Euclidean distance]] from each other, regardless of the orientation of these lines.<ref name="gardner" /> In any pair of parallel supporting lines, one of the two lines will necessarily touch the triangle at one of its vertices. The other supporting line may touch the triangle at any point on the opposite arc, and their distance (the width of the Reuleaux triangle) equals the radius of this arc.<ref name="mj14">{{citation|title=Beautiful Geometry|first1=Eli|last1=Maor|first2=Eugen|last2=Jost|publisher=Princeton University Press|year=2014|isbn=978-1-4008-4833-1|contribution=46 The Reuleaux Triangle|pages=154–156|url=https://books.google.com/books?id=0fOKAQAAQBAJ&pg=PA154}}.</ref>

The first mathematician to discover the existence of curves of constant width, and to observe that the Reuleaux triangle has constant width, may have been [[Leonhard Euler]].<ref name="howround">{{citation|title=How Round Is Your Circle?: Where Engineering and Mathematics Meet|title-link=How Round Is Your Circle|first1=John|last1=Bryant|first2=Chris|last2=Sangwin|publisher=Princeton University Press|year=2011|isbn=978-0-691-14992-9|at=[https://books.google.com/books?id=2lXgO3xKcxAC&pg=PA190 p. 190]}}.</ref> In a paper that he presented in 1771 and published in 1781 entitled ''De curvis triangularibus'', Euler studied [[Curve|curvilinear]] triangles as well as the curves of constant width, which he called orbiforms.<ref name="reich">{{citation|contribution=Euler's contribution to differential geometry and its reception|first=Karin|last=Reich|author-link= Karin Reich |title=Leonhard Euler: Life, Work and Legacy|series=Studies in the History and Philosophy of Mathematics|volume=5|year=2007|pages=479–502|doi=10.1016/S0928-2017(07)80026-0|editor1-first=Robert E.|editor1-last=Bradley|editor2-first=Ed|editor2-last=Sandifer|publisher=Elsevier|isbn=9780444527288}}. See in particular section 1.4, "Orbiforms, 1781", [https://books.google.com/books?id=75vJL_Y-PvsC&pg=PA484 pp. 484–485].</ref><ref>{{citation|first=Leonhard|last=Euler|author-link=Leonhard Euler|title=De curvis triangularibus|url=http://eulerarchive.maa.org/pages/E513.html|journal=Acta Academiae Scientiarum Imperialis Petropolitanae|volume=1778|year=1781|pages=3–30|language=la}}. See in particular p.&nbsp;7 for the definition of orbiforms.</ref>

=== Extremal measures ===
By many different measures, the Reuleaux triangle is one of the most extreme curves of constant width.

By the [[Blaschke–Lebesgue theorem]], the Reuleaux triangle has the smallest possible area of any curve of given constant width. This area is
:<math>\frac{1}{2}(\pi - \sqrt3)s^2 \approx 0.705s^2,</math>
where ''s'' is the constant width. One method for deriving this area formula is to partition the Reuleaux triangle into an inner equilateral triangle and three curvilinear regions between this inner triangle and the arcs forming the Reuleaux triangle, and then add the areas of these four sets. At the other extreme, the curve of constant width that has the maximum possible area is a [[Disk (mathematics)|circular disk]], which has area <math>\pi s^2 / 4\approx 0.785s^2</math>.<ref name="gruber">{{citation|title=Convexity and its Applications|first=Peter M.|last=Gruber|publisher=Birkhäuser|year=1983|isbn=978-3-7643-1384-5|page=[https://archive.org/details/convexityitsappl0000unse/page/67 67]|url=https://archive.org/details/convexityitsappl0000unse/page/67}}</ref>

The angles made by each pair of arcs at the corners of a Reuleaux triangle are all equal to 120°. This is the sharpest possible angle at any [[vertex (geometry)|vertex]] of any curve of constant width.<ref name="gardner" /> Additionally, among the curves of constant width, the Reuleaux triangle is the one with both the largest and the smallest inscribed equilateral triangles.<ref>{{harvtxt|Gruber|1983|page=76}}</ref> The largest equilateral triangle inscribed in a Reuleaux triangle is the one connecting its three corners, and the smallest one is the one connecting the three [[midpoint]]s of its sides. The subset of the Reuleaux triangle consisting of points belonging to three or more diameters is the interior of the larger of these two triangles; it has a larger area than the set of three-diameter points of any other curve of constant width.<ref>{{citation
 | last = Makeev | first = V. V.
 | doi = 10.1023/A:1021287302603
 | issue = Geom. i Topol. 5
 | journal = Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)
 | mr = 1809823
 | pages = 152–155, 329
 | title = An extremal property of the Reuleaux triangle
 | volume = 267
 | year = 2000| s2cid = 116027099
 | doi-access = free
 }}.</ref>

[[File:Symmetry measure of Reuleaux triangle.svg|thumb|Centrally symmetric shapes inside and outside a Reuleaux triangle, used to measure its asymmetry]]
Although the Reuleaux triangle has sixfold [[dihedral symmetry]], the same as an [[equilateral triangle]], it does not have [[central symmetry]].
The Reuleaux triangle is the least symmetric curve of constant width according to two different measures of central asymmetry, the [[Kovner–Besicovitch measure]] (ratio of area to the largest [[central symmetry|centrally symmetric]] shape enclosed by the curve) and the [[Estermann measure]] (ratio of area to the smallest centrally symmetric shape enclosing the curve). For the Reuleaux triangle, the two centrally symmetric shapes that determine the measures of asymmetry are both [[hexagon]]al, although the inner one has curved sides.<ref name="finch">{{citation|contribution-url=http://www.people.fas.harvard.edu/~sfinch/csolve/rx.pdf|contribution=8.10 Reuleaux Triangle Constants|first=Steven R.|last=Finch|pages=[https://archive.org/details/mathematicalcons0000finc/page/513 513–514]|title=Mathematical Constants|series=Encyclopedia of Mathematics and its Applications|publisher=Cambridge University Press|year=2003|isbn=978-0-521-81805-6|url=https://archive.org/details/mathematicalcons0000finc/page/513}}.</ref> The Reuleaux triangle has diameters that split its area more unevenly than any other curve of constant width. That is, the maximum ratio of areas on either side of a diameter, another measure of asymmetry, is bigger for the Reuleaux triangle than for other curves of constant width.<ref>{{citation
 | last1 = Groemer | first1 = H.
 | last2 = Wallen | first2 = L. J.
 | issue = 2
 | journal = Beiträge zur Algebra und Geometrie
 | mr = 1865537
 | pages = 517–521
 | title = A measure of asymmetry for domains of constant width
 | volume = 42
 | year = 2001}}.</ref>

Among all shapes of constant width that avoid all points of an [[integer lattice]], the one with the largest width is a Reuleaux triangle. It has one of its axes of symmetry parallel to the coordinate axes on a half-integer line. Its width, approximately 1.54, is the root of a degree-6 polynomial with integer coefficients.<ref name="finch" /><ref>{{harvtxt|Gruber|1983|page=78}}</ref><ref>{{citation
 | last = Sallee | first = G. T.
 | journal = [[Pacific Journal of Mathematics]]
 | mr = 0240724
 | pages = 669–674
 | title = The maximal set of constant width in a lattice
 | url = http://projecteuclid.org/euclid.pjm/1102983320
 | volume = 28
 | issue = 3
 | year = 1969
 | doi=10.2140/pjm.1969.28.669| doi-access = free
 }}.</ref>

Just as it is possible for a circle to be surrounded by six congruent circles that touch it, it is also possible to arrange seven congruent Reuleaux triangles so that they all make contact with a central Reuleaux triangle of the same size. This is the maximum number possible for any curve of constant width.<ref>{{citation
 | last = Fejes Tóth | first = L. | author-link = László Fejes Tóth
 | journal = Studia Scientiarum Mathematicarum Hungarica
 | mr = 0221388
 | pages = 363–367
 | title = On the number of equal discs that can touch another of the same kind
 | volume = 2
 | year = 1967}}; {{citation
 | last = Schopp | first = J.
 | journal = Studia Scientiarum Mathematicarum Hungarica
 | language = de
 | mr = 0285983
 | pages = 475–478
 | title = Über die Newtonsche Zahl einer Scheibe konstanter Breite
 | volume = 5
 | year = 1970}}.</ref>

[[File:Reuleaux kite.svg|thumb|An [[equidiagonal quadrilateral|equidiagonal]] [[kite (geometry)|kite]] that maximizes the ratio of perimeter to diameter, inscribed in a Reuleaux triangle]]
Among all [[quadrilateral]]s, the shape that has the greatest ratio of its [[perimeter]] to its [[diameter]] is an [[equidiagonal quadrilateral|equidiagonal]] [[kite (geometry)|kite]] that can be inscribed into a Reuleaux triangle.<ref name="kite">{{citation |first=D.G. |last=Ball |title=A generalisation of π |journal=[[The Mathematical Gazette]] |volume=57 |issue=402 |year=1973 |pages=298–303 |doi=10.2307/3616052|jstor=3616052 |s2cid=125396664 }}; {{citation |first1=David |last1= Griffiths |first2=David |last2=Culpin |title=Pi-optimal polygons |journal=[[The Mathematical Gazette]] |volume=59 |issue=409 |year=1975 |pages=165–175 |doi=10.2307/3617699|jstor= 3617699 |s2cid= 126325288 }}.</ref>

=== Other measures ===
By [[Barbier's theorem]] all curves of the same constant width including the Reuleaux triangle have equal [[perimeter]]s. In particular this perimeter equals the perimeter of the circle with the same width, which is <math>\pi s</math>.<ref>{{citation
 | last = Lay | first = Steven R.
 | at = Theorem 11.11, pp. 81–82
 | isbn = 978-0-486-45803-8
 | publisher = Dover
 | title = Convex Sets and Their Applications
 | url = https://books.google.com/books?id=U9eOPjmaH90C&pg=PA81
 | year = 2007}}.</ref><ref>{{citation
 | last = Barbier | first = E.
 | journal = [[Journal de Mathématiques Pures et Appliquées]] | series = 2<sup>e</sup> série
 | language = fr
 | pages = 273–286
 | title = Note sur le problème de l'aiguille et le jeu du joint couvert
 | url = http://sites.mathdoc.fr/JMPA/PDF/JMPA_1860_2_5_A18_0.pdf
 | volume = 5
 | year = 1860}}. See in particular pp. 283–285.</ref><ref name="gardner">{{citation|title=Knots and Borromean Rings, Rep-Tiles, and Eight Queens|volume=4|series=The New Martin Gardner Mathematical Library|first=Martin|last=Gardner|author-link=Martin Gardner|publisher=Cambridge University Press|year=2014|isbn=978-0-521-75613-6|contribution=Chapter 18: Curves of Constant Width|pages=223–245}}.</ref>

The radii of the largest [[inscribed circle]] of a Reuleaux triangle with width ''s'', and of the [[circumscribed circle]] of the same triangle, are
:<math>\displaystyle\left(1-\frac{1}{\sqrt 3}\right)s\approx 0.423s \quad \text{and} \quad \displaystyle\frac{s}{\sqrt 3}\approx 0.577s</math>
respectively; the sum of these radii equals the width of the Reuleaux triangle. More generally, for every curve of constant width, the largest inscribed circle and the smallest circumscribed circle are concentric, and their radii sum to the constant width of the curve.<ref>{{harvtxt|Lay|2007}}, Theorem 11.8, [https://books.google.com/books?id=U9eOPjmaH90C&pg=PA80 pp.&nbsp;80–81].</ref>

{{unsolved|mathematics|How densely can Reuleaux triangles be packed in the plane?}}
The optimal [[packing density]] of the Reuleaux triangle in the plane remains unproven, but is conjectured to be
:<math>\frac{2(\pi-\sqrt 3)}{\sqrt{15}+\sqrt{7}-\sqrt{12}} \approx 0.923, </math>
which is the density of one possible [[double lattice]] packing for these shapes. The best proven upper bound on the packing density is approximately 0.947.<ref>{{citation
 | last1 = Blind | first1 = G.
 | last2 = Blind | first2 = R. | author2-link = Roswitha Blind
 | issue = 2–4
 | journal = Studia Scientiarum Mathematicarum Hungarica
 | language = de
 | mr = 787951
 | pages = 465–469
 | title = Eine Abschätzung für die Dichte der dichtesten Packung mit Reuleaux-Dreiecken
 | volume = 18
 | year = 1983}}. See also {{citation
 | last1 = Blind | first1 = G.
 | last2 = Blind | first2 = R.
 | doi = 10.1007/BF03323256
 | issue = 1–2
 | journal = [[Results in Mathematics]]
 | language = de
 | mr = 880190
 | pages = 1–7
 | title = Reguläre Packungen mit Reuleaux-Dreiecken
 | volume = 11
 | year = 1987| s2cid = 121633860
 }}.</ref> It has also been conjectured, but not proven, that the Reuleaux triangles have the highest packing density of any curve of constant width.<ref>{{citation|arxiv=1504.06733|title=On Curves and Surfaces of Constant Width|year=2015|author-link1=Howard L. Resnikoff|first=Howard L.|last=Resnikoff|bibcode=2015arXiv150406733R}}.</ref>

=== Rotation within a square ===
[[File:Rotation of Reuleaux triangle.gif|thumb|Rotation of a Reuleaux triangle within a square, showing also the curve traced by the center of the triangle]]
Any curve of constant width can form a rotor within a [[square]], a shape that can perform a complete rotation while staying within the square and at all times touching all four sides of the square. However, the Reuleaux triangle is the rotor with the minimum possible area.<ref name="gardner" /> As it rotates, its axis does not stay fixed at a single point, but instead follows a curve formed by the pieces of four [[ellipse]]s.<ref>{{citation
 | last1 = Gleiftner | first1 = Winfried
 | last2 = Zeitler | first2 = Herbert
 | date = May 2000
 | doi = 10.1007/bf03322004
 | issue = 3–4
 | journal = [[Results in Mathematics]]
 | pages = 335–344
 | title = The Reuleaux triangle and its center of mass
 | volume = 37| s2cid = 119600507
 }}.</ref> Because of its 120° angles, the rotating Reuleaux triangle cannot reach some points near the sharper angles at the square's vertices, but rather covers a shape with slightly rounded corners, also formed by elliptical arcs.<ref name="gardner" />

{{multiple image|align=center|image1=Reuleaux triangle rotation center.svg|alt1=Reuleaux triangle in a square, with ellipse governing the path of motion of the triangle center|caption1=One of the four ellipses followed by the center of a rotating Reuleaux triangle in a square|image2=Reuleaux triangle rotation corners.svg|alt2=Reuleaux triangle in a square, with ellipse bounding the region swept by the triangle|caption2=Ellipse separating one of the corners (lower left) of a square from the region swept by a rotating Reuleaux triangle}}

At any point during this rotation, two of the corners of the Reuleaux triangle touch two adjacent sides of the square, while the third corner of the triangle traces out a curve near the opposite vertex of the square. The shape traced out by the rotating Reuleaux triangle covers approximately 98.8% of the area of the square.<ref>{{citation|author-link=Clifford A. Pickover|last=Pickover|first=Clifford A.|title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics|publisher=Sterling Publishing Company|year=2009|isbn= 978-1-4027-5796-9|pages=266|contribution=Reuleaux Triangle|url=https://books.google.com/books?id=JrslMKTgSZwC&pg=PA266}}.</ref>

=== As a counterexample ===
Reuleaux's original motivation for studying the Reuleaux triangle was as a counterexample, showing that three single-point contacts may not be enough to fix a planar object into a single position.<ref>{{harvtxt|Moon|2007}}, [https://books.google.com/books?id=lw7lN8JibZsC&pg=PA239 p.&nbsp;239].</ref> The existence of Reuleaux triangles and other curves of constant width shows that diameter measurements alone cannot verify that an object has a circular cross-section.<ref>{{citation
 | last1 = Granovsky | first1 = V. A.
 | last2 = Siraya | first2 = T. N.
 | editor1-last = Pavese | editor1-first = F.
 | editor2-last = Bär | editor2-first = M.
 | editor3-last = Filtz | editor3-first = J.-R.
 | editor4-last = Forbes | editor4-first = A. B.
 | editor5-last = Pendrill | editor5-first = L.
 | editor6-last = Shirono | editor6-first = K.
 | contribution = Metrological traceability and quality of industrial tests measurements
 | pages = 194–201
 | publisher = World Scientific
 | title = Advanced Mathematical and Computational Tools in Metrology and Testing IX}}. See in particular [https://books.google.com/books?id=2dwn4M7IEWUC&pg=PA200 p.&nbsp;200].</ref>

In connection with the [[inscribed square problem]], {{harvtxt|Eggleston|1958}} observed that the Reuleaux triangle provides an example of a constant-width shape in which no regular polygon with more than four sides can be inscribed, except the regular hexagon, and he described a small modification to this shape that preserves its constant width but also prevents regular hexagons from being inscribed in it. He generalized this result to three dimensions using a cylinder with the same shape as its [[Cross section (geometry)|cross section]].<ref>{{citation
 | last = Eggleston | first = H. G.
 | doi = 10.2307/2308878
 | journal = [[American Mathematical Monthly]]
 | mr = 0097768
 | pages = 76–80
 | title = Figures inscribed in convex sets
 | volume = 65
 | issue = 2
 | year = 1958| jstor = 2308878
 }}.</ref>