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%I #50 Aug 14 2023 10:32:14
%S 2,5,9,8,0,7,6,2,1,1,3,5,3,3,1,5,9,4,0,2,9,1,1,6,9,5,1,2,2,5,8,8,0,8,
%T 5,5,0,4,1,4,2,0,7,8,8,0,7,1,5,5,7,0,9,4,2,0,8,3,7,1,0,4,6,9,1,7,7,8,
%U 9,9,5,2,5,3,6,3,2,0,0,0,5,5,6,2,1,7,1,9,2,8,0,1,3,5,8,7,2,8,6,3,5,1,3,4,3
%N Decimal expansion of the area of the regular hexagon with circumradius 1.
%C Equivalently, the area in the complex plane of the smallest convex set containing all order-6 roots of unity.
%C Subtracting 2.5 (i.e., dropping the first two digits) we obtain 0.09807.... which is a limiting mean cluster density for a bond percolation model at probability 1/2 [Finch]. - _R. J. Mathar_, Jul 26 2007
%C This constant is also the minimum radius of curvature of the exponential curve (occurring at x = -log(2)/2 = -0.34657359...). - _Jean-François Alcover_, Dec 19 2016
%C Luminet proves that this is the critical impact parameter of a bare black hole, in multiples of the Schwarzschild radius. That is, light from a distant source coming toward a black hole is captured by the black hole at smaller distances and deflected at larger distances. - _Charles R Greathouse IV_, May 21 2022
%C For any triangle ABC, sin(A) + sin(B) + sin(C) <= 3*sqrt(3)/2, equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - _Bernard Schott_, Sep 16 2022
%H G. C. Greubel, <a href="/A104956/b104956.txt">Table of n, a(n) for n = 1..10000</a>
%H S. R. Finch, <a href="http://dx.doi.org/10.1007/BF01608791">Several Constants Arising in Statistical Mechanics</a>, Annals Combinat. vol 3 (1999) issue (2-4) pp. 323-335.
%H Kiran S. Kedlaya, <a href="https://igor-kortchemski.perso.math.cnrs.fr/olympiades/Cours/ineqs-080299.pdf">A < B</a>, (1999), Problem 6.1, p. 6.
%H J.-P. Luminet, <a href="https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1979A&A....75..228L">Image of a spherical black hole with thin accretion disk</a>, Astronomy and Astrophysics, vol. 75, no. 1-2 (May 1979), pp. 228-235.
%H Michael Penn, <a href="https://www.youtube.com/watch?v=UNMYya7yYGg">not as bad as it seems...</a>, YouTube video, 2021.
%H Eric Weisstein et al., <a href="http://mathworld.wolfram.com/RootofUnity.html">Root of Unity</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/deMoivreNumber.html">de Moivre Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Twenty-VertexEntropyConstant.html">Twenty-Vertex Entropy Constant</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hexagon">Hexagon</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Regular_polygon">Regular polygon</a>
%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%F Equals (3*sqrt(3))/2, that is, 2*A104954.
%e 2.59807621135331594029116951225880855041420788071557094208371046917789952536320...
%t Floor[n/2]*Sin[(2*Pi)/n] - Sin[(4*Pi*Floor[n/2])/n]/2 /. n -> 6
%t RealDigits[(3*Sqrt[3])/2, 10, 50][[1]] (* _G. C. Greubel_, Jul 03 2017 *)
%o (PARI) 3*sqrt(3)/2 \\ _G. C. Greubel_, Jul 03 2017
%Y Cf. A002194, A104954, A104955, A104957.
%Y Cf. Areas of other regular polygons: A120011, A102771, A178817, A090488, A256853, A178816, A256854, A178809.
%K nonn,cons
%O 1,1
%A Joseph Biberstine (jrbibers(AT)indiana.edu), Mar 30 2005
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