A010527 - OEIS

%I #154 Jan 27 2025 16:46:29

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

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

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

%N Decimal expansion of sqrt(3)/2.

%C This is the ratio of the height of an equilateral triangle to its base.

%C Essentially the same sequence arises from decimal expansion of square root of 75, which is 8.6602540378443864676372317...

%C Also the real part of i^(1/3), the cubic root of i. - _Stanislav Sykora_, Apr 25 2012

%C Gilbert & Pollak conjectured that this is the Steiner ratio rho_2, the least upper bound of the ratio of the length of the Steiner minimal tree to the length of the minimal tree in dimension 2. (See Ivanov & Tuzhilin for the status of this conjecture as of 2012.) - _Charles R Greathouse IV_, Dec 11 2012

%C Surface area of a regular icosahedron with unit edge is 5*sqrt(3), i.e., 10 times this constant. - _Stanislav Sykora_, Nov 29 2013

%C Circumscribed sphere radius for a cube with unit edges. - _Stanislav Sykora_, Feb 10 2014

%C Also the ratio between the height and the pitch, used in the Unified Thread Standard (UTS). - _Enrique Pérez Herrero_, Nov 13 2014

%C Area of a 30-60-90 triangle with shortest side equal to 1. - _Wesley Ivan Hurt_, Apr 09 2016

%C If a, b, c are the sides of a triangle ABC and h_a, h_b, h_c the corresponding altitudes, then (h_a+h_b+h_c) / (a+b+c) <= sqrt(3)/2; equality is obtained only when the triangle is equilateral (see Mitrinovic reference). - _Bernard Schott_, Sep 26 2022

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.2, 8.3 and 8.6, pp. 484, 489, and 504.

%D Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), pp. 450-451.

%D D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, and J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.8, page 114.

%H Harry J. Smith, <a href="/A010527/b010527.txt">Table of n, a(n) for n = 0..20000</a>

%H E. N. Gilbert and H. O. Pollak, <a href="http://dx.doi.org/10.1137/0116001">Steiner minimal trees</a>, SIAM J. Appl. Math. 16, (1968), pp. 1-29.

%H A. O. Ivanov and A. A. Tuzhilin, <a href="http://dx.doi.org/10.1007/s00453-011-9508-3">The Steiner ratio Gilbert-Pollak conjecture is still open</a>, Algorithmica 62:1-2 (2012), pp. 630-632.

%H Matt Parker, <a href="https://www.youtube.com/watch?v=ErBbyLu-M94">The mystery of 0.866025403784438646763723170752936183471402626905190314027903489</a>, Stand-up Maths, YouTube video, Feb 14 2024.

%H Simon Plouffe, Plouffe's Inverter, <a href="https://wayback.cecm.sfu.ca/projects/ISC/dataB/isc/C/sqrt32.txt">sqrt(3)/2 to 10000 digits</a>.

%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap84.html">Sqrt(3)/2 to 5000 digits</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LebesgueMinimalProblem.html">Lebesgue Minimal Problem</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Icosahedron">Icosahedron</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Platonic solid">Platonic solid</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Unified_Thread_Standard">Unified Thread Standard</a>.

%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.

%F Equals cos(30 degrees). - _Kausthub Gudipati_, Aug 15 2011

%F Equals A002194/2. - _Stanislav Sykora_, Nov 30 2013

%F From _Amiram Eldar_, Jun 29 2020: (Start)

%F Equals sin(Pi/3) = cos(Pi/6).

%F Equals Integral_{x=0..Pi/3} cos(x) dx. (End)

%F Equals 1/(10*A020832). - _Bernard Schott_, Sep 29 2022

%F Equals x^(x^(x^...)) where x = (3/4)^(1/sqrt(3)) (infinite power tower). - _Michal Paulovic_, Jun 25 2023

%e 0.86602540378443864676372317...

%p Digits:=100: evalf(sqrt(3)/2); # _Wesley Ivan Hurt_, Apr 09 2016

%t RealDigits[Sqrt[3]/2, 10, 200][[1]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 21 2011 *)

%o (PARI)  default(realprecision, 20080); x=10*(sqrt(3)/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b010527.txt", n, " ", d));  \\ _Harry J. Smith_, Jun 02 2009

%o (PARI) sqrt(3)/2 \\ _Michel Marcus_, Apr 10 2016

%o (Magma) SetDefaultRealField(RealField(100)); Sqrt(3)/2; // _G. C. Greubel_, Nov 02 2018

%Y Cf. A010153.

%Y Cf. Platonic solids surfaces: A002194 (tetrahedron), A010469 (octahedron), A131595 (dodecahedron).

%Y Cf. Platonic solids circumradii: A010503 (octahedron), A019881 (icosahedron), A179296 (dodecahedron), A187110 (tetrahedron).

%Y Cf. A126664 (continued fraction), A144535/A144536 (convergents).

%Y Cf. A002194, A010502, A020821, A104956, A152623 (other geometric inequalities).

%K nonn,cons,easy,changed

%O 0,1

%A _N. J. A. Sloane_

%E Last term corrected and more terms added by _Harry J. Smith_, Jun 02 2009