OFFSET
0,1
COMMENTS
One of the five constructible squarable lunes. It was discovered by Hippocrates of Chios in the 5th century BC.
Called the "isosceles trapezoid lune" by Shelburne (2005).
A lune is a crescent-shaped plane figure bounded by two circular arcs of unequal radii.
A squarable lune is one that can be squared, i.e., a square with an identical area can be constructed using only a compass and a straightedge.
Chebotarev (1935) and Dorodnov (1947) proved that there are precisely five squarable lunes that are also constructible, i.e., can be constructed with a compass and a straightedge.
For a squarable lune the central angles of its two arcs must be commensurable, i.e., their ratio must be a rational number, or equivalently, they are integral multiples of a common angle.
Following the notation in Postnikov (2000), let the central angles of the two arcs be 2*alpha and 2*beta. There exist positive integers k and m (k > m) such that alpha = k * theta and beta = m * theta.
A lune is squarable if it satisfies the condition sin(k*theta)^2/k = sin(m*theta)^2/m, and it is constructible if the angle theta is constructible.
By inscribing a polygonal chain of k equal segments in the outer arc and m equal segments in the inner arc, a resulting (k+m)-sided polygon is formed, and its area is equal to the area of the lune.
Assuming the common chord of the arcs is of unit length, the radii of the two arcs are 1/(2*sin(alpha)) and 1/(2*sin(beta)). The area of the lune is given by alpha/sin(alpha)^2 - beta/sin(beta)^2 + cot(betha)/4 - cot(alpha)/4.
The five lunes are often characterized in literature by the ratio of their central angles or by the shape of the polygon with the same area. The first lune, corresponding to the 2:1 ratio, is known as the Lune of Hippocrates.
k | m | theta | area | polygon
--+------------------------+------------------- +--------------------------
REFERENCES
Claudi Alsina and Roger Nelsen, Charming Proofs: A Journey into Elegant Mathematics, MAA, 2010, section 9.1 Squarable lunes, pp. 137-144.
Anatolii Vasilievich Dorodnov, O krugovykh lunochkakh, kvadriruemykh pri pomoshchi tsirkulya i lineıki (in Russian; On circular lunes quadrable with the use of ruler and compass), Dokl. Akad. Nauk SSSR., Vol. 58 (1947), pp. 965-968.
William Dunham, Journey Through Genius, Wiley, 1990, Chapter 1, pp. 1-26.
Julian Havil, Curves for the Mathematically Curious, Princeton University Press, 2019, Section 5.4, The Quadratrix and Circle Squaring, pp. 78-84.
Stacy G. Langton, The Quadrature of Lunes, from Hippocrates to Euler, pp. 53-62 in: Robert E. Bradley, Lawrence A. D'Antonio, Charles Edward Sandifer (eds.), Euler at 300: An Appreciation, MAA, 2007.
David S. Richeson, Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity, Princeton University Press, 2021, pp. 99-104.
Abe Shenitzer and John Stillwell, eds., Mathematical Evolutions, MAA, 2001. Includes Postnikov's paper (2000), pp. 181-187.
LINKS
Author?, The Lunes of Hippocrates, 2021.
Alexander Bogomolny, Hippocrates' Squaring of Lunes, Cut The Knot.
Kevin Brown, The Five Squarable Lunes, MathPages.
Thomas Clausen,Vier neue mondförmige Flächen, deren Inhalt quadrirbar ist, Journal für die reine und angewandte Mathematik, Vol. 21 (1840), pp. 375-376; alternative link.
Amiram Eldar, Illustration.
Kurt Girstmair, Hippocrates' lunes and transcendence, Expositiones Mathematicae, Vol. 21, No. 2 (2003), pp. 179-183.
Brady Haran and Barry Mazur, Spearheading a Lune, YouTube Numberphile video, 2014.
Mikhail Mikhailovich Postnikov, The Problem of Squarable Lunes, The American Mathematical Monthly, Vol. 107, No. 7 (2000), pp. 645-651. Translated from Russian by Abe Shenitzer.
Brian J. Shelburne, The Five Quadrable (Squarable) Lunes, Wittenberg University Springfield, 2005.
Peter Stevenhagen and Hendrik W. Lenstra, Jr., Chebotarëv and his density theorem, The Mathematical Intelligencer, Vol. 18 (1996), pp. 26-37.
Nikolaj Tschebotaröw (Nikolay Grigorievich Chebotarev), Über quadrierbare Kreisbogenzweiecke. I, Mathematische Zeitschrift, Vol. 39 (1935), pp. 161-175.
Martin Johan Wallenius, Dissertatio gradualis, lunulas quasdam circulares quadrabiles, 1766; Graduate Dissertation: Showing Certain Squarable Circular Lunes, translated and annotated by Ian Bruce.
Eric Weisstein's World of Mathematics, Lune.
Wikipedia, Lune (geometry).
Wikipedia, Lune of Hippocrates.
FORMULA
Equals sqrt(3 + 2*sqrt(3))/6.
Equals f(3*theta) - f(theta), where theta = A395469, and f(x) = x/sin(x)^2 - cot(x)/4.
Minimal polynomial: 432*x^4 - 72*x^2 - 1.
Equals A388924/2. - Hugo Pfoertner, Apr 24 2026
EXAMPLE
0.423743292806235413045204330340354992242804776075592...
MATHEMATICA
RealDigits[Sqrt[3 + 2*Sqrt[3]]/6, 10, 120][[1]]
PROG
(PARI) sqrt(3 + 2*sqrt(3))/6
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Apr 24 2026
STATUS
approved