A010527 Decimal expansion of sqrt(3)/2.

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, 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, 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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy
• changed

Comments

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

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

Also the real part of i^(1/3), the cubic root of i. - Stanislav Sykora, Apr 25 2012

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

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

Circumscribed sphere radius for a cube with unit edges. - Stanislav Sykora, Feb 10 2014

Also the ratio between the height and the pitch, used in the Unified Thread Standard (UTS). - Enrique Pérez Herrero, Nov 13 2014

Area of a 30-60-90 triangle with shortest side equal to 1. - Wesley Ivan Hurt, Apr 09 2016

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

Examples

			0.86602540378443864676372317...
		

References

• 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.
• 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. 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.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM J. Appl. Math. 16, (1968), pp. 1-29.
• A. O. Ivanov and A. A. Tuzhilin, The Steiner ratio Gilbert-Pollak conjecture is still open, Algorithmica 62:1-2 (2012), pp. 630-632.
• Matt Parker, The mystery of 0.866025403784438646763723170752936183471402626905190314027903489, Stand-up Maths, YouTube video, Feb 14 2024.
• Simon Plouffe, Plouffe's Inverter, sqrt(3)/2 to 10000 digits.
• Simon Plouffe, Sqrt(3)/2 to 5000 digits.
• Eric Weisstein's World of Mathematics, Lebesgue Minimal Problem.
• Wikipedia, Icosahedron.
• Wikipedia, Platonic solid.
• Wikipedia, Unified Thread Standard.
• Index entries for algebraic numbers, degree 2.

Crossrefs

Cf. A010153.

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

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

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

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

Programs

Magma

SetDefaultRealField(RealField(100)); Sqrt(3)/2; // G. C. Greubel, Nov 02 2018

Maple

Digits:=100: evalf(sqrt(3)/2); # Wesley Ivan Hurt, Apr 09 2016

Mathematica

RealDigits[Sqrt[3]/2, 10, 200][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

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

PARI

sqrt(3)/2 \\ Michel Marcus, Apr 10 2016

Formula

Equals cos(30 degrees). - Kausthub Gudipati, Aug 15 2011

Equals A002194/2. - Stanislav Sykora, Nov 30 2013

From Amiram Eldar, Jun 29 2020: (Start)

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

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

Equals 1/(10*A020832). - Bernard Schott, Sep 29 2022

Equals x^(x^(x^...)) where x = (3/4)^(1/sqrt(3)) (infinite power tower). - Michal Paulovic, Jun 25 2023

Equals 2F1(-1/4,1/4 ; 1/2 ; 3/4) . - R. J. Mathar, Aug 31 2025

Extensions

Last term corrected and more terms added by Harry J. Smith, Jun 02 2009