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A010527 Decimal expansion of sqrt(3)/2. 75
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
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
REFERENCES
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
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
Eric Weisstein's World of Mathematics, Lebesgue Minimal Problem
Wikipedia, Icosahedron
Wikipedia, Platonic solid
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
EXAMPLE
0.86602540378443864676372317...
MAPLE
Digits:=100: evalf(sqrt(3)/2); # Wesley Ivan Hurt, Apr 09 2016
MATHEMATICA
RealDigits[N[Sqrt[3]/2, 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)
PROG
(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
(Magma) SetDefaultRealField(RealField(100)); Sqrt(3)/2; // G. C. Greubel, Nov 02 2018
CROSSREFS
Cf. A010153.
Cf. Platonic solids surfaces: A002194 (tetrahedron), A010469 (octahedron), A131595 (dodecahedron). - Stanislav Sykora, Nov 30 2013
Cf. Platonic solids circumradii: A010503 (octahedron), A019881 (icosahedron), A179296 (dodecahedron), A187110 (tetrahedron). - Stanislav Sykora, Feb 10 2014
Cf. A126664 (continued fraction), A144535/A144536 (convergents).
Cf. A002194, A010502, A020821, A104956, A152623 (other geometric inequalities).
Sequence in context: A360167 A046266 A165104 * A270137 A269846 A316136
KEYWORD
nonn,cons,easy
AUTHOR
EXTENSIONS
Last term corrected and more terms added by Harry J. Smith, Jun 02 2009
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)