Editing Reuleaux triangle (section)

=== 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>