A089491 - OEIS

%I #53 Apr 22 2022 05:41:13

%S 9,5,4,9,2,9,6,5,8,5,5,1,3,7,2,0,1,4,6,1,3,3,0,2,5,8,0,2,3,5,0,8,6,1,

%T 7,2,2,0,6,7,5,7,8,7,4,4,4,2,7,3,8,6,9,2,4,8,6,0,0,4,0,6,4,3,5,3,3,8,

%U 0,7,8,5,8,0,5,3,5,9,2,1,0,5,4,0,6,8,2,8,1,6,5,9,7,5,1,8,5,1,5,7,3,6,4,3,7

%N Decimal expansion of Buffon's constant 3/Pi.

%C Whereas 2/Pi (A060294) is the probability that a needle will land on one of many parallel lines, this is the probability that a needle will land on one of many lines making up a grid.

%C The probability that the boundary of an equilateral triangle will intersect one of the parallel lines if the triangle edge length l (almost) equals the distance d between each pair of lines. This follows directly from the Weisstein/MathWorld Buffon's Needle Problem link's statement P=p/(Pi*d), where P is the probability of intersection with any convex polygon's boundary if the generalized diameter of that polygon is less than d and p is the perimeter of the polygon. (Take d=l, then p=3d.) - _Rick L. Shepherd_, Jan 11 2006

%C Related grid problems are discussed in the Weisstein/MathWorld Buffon-Laplace Needle Problem link. - _Rick L. Shepherd_, Jan 11 2006

%C The area of a regular dodecagon circumscribed in a unit-area circle. - _Amiram Eldar_, Nov 05 2020

%C From _Bernard Schott_, Apr 19 2022: (Start)

%C For any non-obtuse triangle ABC (see Mitrinović and Oppenheim links):

%C (a/A + b/B + c/C)/(a+b+c) >= 3/Pi,

%C (a^2/A + b^2/B + c^2/C)/(a^2+b^2+c^2) <= 3/Pi,

%C where (A,B,C) are the angles (measured in radians) and (a,b,c) the side lengths of this triangle.

%C Equality stands iff triangle ABC is equilateral. (End)

%D Joe Portney, Portney's Ponderables, Litton Systems, Inc., Appendix 2, 'Buffon's Needle' by Lawrence R. Weill, 200, pp. 135-138.

%H Harry Khamis, <a href="http://www.wright.edu/~harry.khamis/buffons_needle_problem/">Buffon's Needle Problem</a>.

%H D. S. Mitrinović, J. E. Pečarić, and V. Volenec, <a href="https://doi.org/10.1007/978-94-015-7842-4_9">Recent Advances In Geometric Inequalities</a>, Kluwer Academic Publishers, 1989, Inequalities 4.11, p. 170.

%H A. Oppenheim, <a href="https://www.jstor.org/stable/2318876?seq=1">Problem E 2649</a>, American Mathematical Monthly, 84 (1977), p. 294.

%H Kevin Peterson, <a href="https://faculty.lynchburg.edu/peterson_km/buffon.pdf">A Problem in Geometric Probability: Buffon's Needle Problem</a>.

%H George Reese, <a href="http://mste.illinois.edu/activity/buffon/">Buffon's Needle, An Analysis and Simulation</a>.

%H Shodor Education Foundation, Inc., <a href="http://www.shodor.org/interactivate/activities/buffon/#">Buffon's needle</a>.

%H Washington and Lee University, <a href="http://home.wlu.edu/~mcraea/GeometricProbabilityFolder/ApplicationsConvexSets/Problem18/Problem18.html">Problem 18: Buffon's Needle Again</a>. [Broken link]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BuffonsNeedleProblem.html">Buffon's needle problem</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Buffon-LaplaceNeedleProblem.html">Buffon-Laplace needle problem</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GeneralizedDiameter.html">Generalized Diameter</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals sinc(Pi/6). - _Peter Luschny_, Oct 04 2019

%F From _Amiram Eldar_, Aug 20 2020: (Start)

%F Equals Product{k>=1} cos(Pi/(6*2^k)).

%F Equals Product{k>=1} (1 - 1/(6*k)^2). (End)

%e 3/Pi = 0.95492965855137201461330258023508617220675787444273869248600...

%t RealDigits[ N[ 3/Pi, 111]][[1]]

%o (PARI) 3/Pi \\ _Michel Marcus_, Nov 05 2020

%Y Cf. A000796 (Pi), A060294 (2/Pi).

%K cons,nonn

%O 0,1

%A _Robert G. Wilson v_, Nov 04 2003