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%I #114 Nov 03 2023 11:16:36
%S 6,2,8,3,1,8,5,3,0,7,1,7,9,5,8,6,4,7,6,9,2,5,2,8,6,7,6,6,5,5,9,0,0,5,
%T 7,6,8,3,9,4,3,3,8,7,9,8,7,5,0,2,1,1,6,4,1,9,4,9,8,8,9,1,8,4,6,1,5,6,
%U 3,2,8,1,2,5,7,2,4,1,7,9,9,7,2,5,6,0,6,9,6,5,0,6,8,4,2,3,4,1,3
%N Decimal expansion of 2*Pi.
%C Pi/5 or 2*Pi/10 is the expected surface area containing completely a Brownian curve (trajectory) on a plane. - _Lekraj Beedassy_, Jul 28 2005
%C Bob Palais considers this a more fundamental constant than Pi, see the Palais reference and link. - _Jonathan Vos Post_, Sep 10 2010
%C The Persian mathematician Jamshid al-Kashi seems to have been the first to use the circumference divided by the radius as the circle constant. In Treatise on the Circumference published 1424 he calculated the circumference of a unit circle to 9 sexagesimal places. - Peter Harremoës, _John W. Nicholson_, Aug 02 2012
%C "Proponents of a new mathematical constant tau (τ), equal to two times π, have argued that a constant based on the ratio of a circle's circumference to its radius rather than to its diameter would be more natural and would simplify many formulas" (from Wikipedia). - _Jonathan Sondow_, Aug 15 2012
%C The constant 2*Pi appears in the formula for the period T of a simple gravity pendulum. For small angles this period is given by Christiaan Huygens’s law, i.e., T = 2*Pi*sqrt(L/g), see for more information A223067. - _Johannes W. Meijer_, Mar 14 2013
%C There are seven consecutive nines at positions 762 to 768. - _Roland Kneer_, Jul 05 2013
%C Volume of a cylinder in which a sphere of radius 1 can be inscribed. - _Omar E. Pol_, Sep 25 2013
%C 2*Pi is also the surface area of a sphere whose diameter equals the square root of 2. More generally, x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - _Omar E. Pol_, Dec 18 2013
%C From _Bernard Schott_, Jan 31 2020: (Start)
%C Also, (2*Pi)*a^2 is the area of the deltoid (an hypocycloid with three cusps) whose Cartesian parametrization is:
%C x = a * ((2*cos(t) + cos(2*t)),
%C y = a * ((2*sin(t) - sin(2*t)).
%C The length of this deltoid is 16*a. See the curve at the Mathcurve link. (End)
%C Pi/5 = 0.1 * 2*Pi is the mean area of the plane triangles formed by 3 points independently and uniformly chosen at random on the surface of a unit-radius sphere. - _Amiram Eldar_, Aug 06 2020
%H Harry J. Smith, <a href="/A019692/b019692.txt">Table of n, a(n) for n = 1..20000</a>
%H Robert Ferréol, <a href="https://www.mathcurve.com/courbes2d.gb/deltoid/deltoid.shtml">Deltoid</a>, Mathcurve.
%H Christophe Garban and José A. Trujillo Ferreras, <a href="https://doi.org/10.1007/s00220-006-1555-2">The expected area of the filled planar Brownian loop is pi/5</a>, Communications in mathematical physics, Vol. 264, No. 3 (2006), pp. 797-810, <a href="https://arxiv.org/abs/math/0504496">preprint</a>, arXiv:math/0504496 [math.PR], 2005.
%H Peter Harremoës, <a href="http://www.harremoes.dk/Peter/Undervis/Turnpage/Turnpage1.html">Al-Kashi’s constant</a>
%H Michael Hartl, <a href="http://tauday.com">The Tau Manifesto</a>
%H Melissa Larson, <a href="https://www.d.umn.edu/~jgreene/masters_reports/BBP%20Paper%20final.pdf">Verifying and discovering BBP-type formulas</a>, 2008.
%H Bob Palais, <a href="http://www.math.utah.edu/~palais/pi.html">Web page about "Pi is wrong!"</a>
%H Bob Palais, <a href="http://dx.doi.org/10.1007/BF03026846">Pi is wrong!</a>, The Mathematical Intelligencer Volume 23, Number 3, 2001, pp. 7-8.
%H Grant Sanderson, <a href="https://www.youtube.com/watch?v=bcPTiiiYDs8">How pi was almost 6.283185...</a>, 3Blue1Brown video (2018).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Turn_(geometry)#Tau_proposals">Tau proposals</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula">Bailey-Borwein-Plouffe formula</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F e^(Zeta'(0)/Zeta(0)) = 2*Pi. - _Peter Luschny_, Jun 17 2018
%F From _Peter Bala_, Oct 30 2019: (Start)
%F 2*Pi = Sum_{n >= 0} (-1)^n*( 1/(n + 1/6) + 1/(n + 5/6) ).
%F 2*Pi = Sum_{n >= 0} (-1)^n*( 1/(n + 1/10) - 1/(n + 3/10) - 1/(n + 7/10) + 1/(n + 9/10) ). Cf. A091925 and A244979. (End)
%F From _Amiram Eldar_, Aug 06 2020: (Start)
%F Equals Gamma(1/6)*Gamma(5/6).
%F Equals Integral_{x=0..oo} log(1 + 1/x^6) dx.
%F Equals Integral_{x=0..oo} log(1 + 4/x^2) dx.
%F Equals Integral_{x=-oo..oo} exp(x/6)/(exp(x) + 1) dx.
%F Equals Sum_{k>=0} 1/((k + 1/4)*(k + 3/4)). (End)
%e 6.283185307179586476925286766559005768394338798750211641949889184615632...
%t RealDigits[N[2 Pi, 6! ]] (* _Vladimir Joseph Stephan Orlovsky_, Dec 02 2009 *)
%o (PARI) default(realprecision, 20080); x=2*Pi; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019692.txt", n, " ", d)); \\ _Harry J. Smith_, May 31 2009
%o (Magma) R:= RealField(100); 2*Pi(R); // _G. C. Greubel_, Mar 08 2018
%o (Julia)
%o using Nemo
%o RR = RealField(334)
%o tau = const_pi(RR) + const_pi(RR)
%o tau |> println # _Peter Luschny_, Mar 14 2018
%o (Python) # Use some guard digits when computing.
%o # BBP formula P(1, 16, 8, (0, 8, 4, 4, 0, 0, -1, 0)).
%o from decimal import Decimal as dec, getcontext
%o def BBPtau(n: int) -> dec:
%o getcontext().prec = n
%o s = dec(0); f = dec(1); g = dec(16)
%o for k in range(n):
%o ek = dec(8 * k)
%o s += f * ( dec(8) / (ek + 2) + dec(4) / (ek + 3)
%o + dec(4) / (ek + 4) - dec(1) / (ek + 7))
%o f /= g
%o return s
%o print(BBPtau(200)) # _Peter Luschny_, Nov 03 2023
%Y Cf. A058291 (continued fraction).
%Y Cf. A000796, A019693, A019699, A091925, A244979.
%Y Cf. A093828 (astroid), A180434 (loop of strophoid), A197723 (cardioid).
%K nonn,cons
%O 1,1
%A _N. J. A. Sloane_
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