Displaying 1-10 of 29 results found.
Decimal expansion of sqrt(3)/2.
+10
86
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
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
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.
FORMULA
Equals sin(Pi/3) = cos(Pi/6).
Equals Integral_{x=0..Pi/3} cos(x) dx. (End)
Equals x^(x^(x^...)) where x = (3/4)^(1/sqrt(3)) (infinite power tower). - Michal Paulovic, Jun 25 2023
EXAMPLE
0.86602540378443864676372317...
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 (Magma) SetDefaultRealField(RealField(100)); Sqrt(3)/2; // G. C. Greubel, Nov 02 2018
EXTENSIONS
Last term corrected and more terms added by Harry J. Smith, Jun 02 2009
Decimal expansion of 1/sqrt(2).
+10
76
7, 0, 7, 1, 0, 6, 7, 8, 1, 1, 8, 6, 5, 4, 7, 5, 2, 4, 4, 0, 0, 8, 4, 4, 3, 6, 2, 1, 0, 4, 8, 4, 9, 0, 3, 9, 2, 8, 4, 8, 3, 5, 9, 3, 7, 6, 8, 8, 4, 7, 4, 0, 3, 6, 5, 8, 8, 3, 3, 9, 8, 6, 8, 9, 9, 5, 3, 6, 6, 2, 3, 9, 2, 3, 1, 0, 5, 3, 5, 1, 9, 4, 2, 5, 1, 9, 3, 7, 6, 7, 1, 6, 3, 8, 2, 0, 7, 8, 6, 3, 6, 7, 5, 0, 6
COMMENTS
The decimal expansion of sqrt(50) = 5*sqrt(2) = 7.0710678118654752440... gives essentially the same sequence.
Also real and imaginary part of the square root of the imaginary unit. - Alonso del Arte, Jan 07 2011 1/sqrt(2) = (1/2)^(1/2) = (1/4)^(1/4) (see the comments in A072364). If a triangle has sides whose lengths form a harmonic progression in the ratio 1 : 1/(1 + d) : 1/(1 + 2d) then the triangle inequality condition requires that d be in the range -1 + 1/sqrt(2) < d < 1/sqrt(2). - Frank M Jackson, Oct 11 2011 Let s_2(n) be the sum of the base-2 digits of n and epsilon(n) = (-1)^s_2(n), the Thue-Morse sequence A010060, then Product_{n >= 0} ((2*n + 1)/(2*n + 2))^epsilon(n) = 1/sqrt(2). - Jonathan Vos Post, Jun 03 2012 The square root of 1/2 and thus it follows from the Pythagorean theorem that it is the sine of 45 degrees (and the cosine of 45 degrees). - Alonso del Arte, Sep 24 2012 Circumscribed sphere radius for a regular octahedron with unit edges. In electrical engineering, ratio of effective amplitude to peak amplitude of an alternating current/voltage. - Stanislav Sykora, Feb 10 2014 Radius of midsphere (tangent to edges) in a cube with unit edges. - Stanislav Sykora, Mar 27 2014
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Sections 1.1, 7.5.2, and 8.2, pp. 1-3, 468, 484, 487.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.
LINKS
Ovidiu Furdui, Problem 1, Problem Corner, Research Group in Mathematical Inequalities and Applications, 2010. Donald R. Woods, Problem E 2692, Elementary Problems, The American Mathematical Monthly, Vol. 85, No. 1 (1978), p. 48; A Transcendental Function Satisfy a Duplication Formula, by David Robbins, ibid., Vol. 86, No. 5 (1979), pp. 394-395.
FORMULA
Equals sin(Pi/4) = cos(Pi/4).
Equals Integral_{x=0..Pi/4} cos(x) dx. (End)
Equals hypergeom([-1/2, -3/4], [5/4], -1). - Peter Bala, Mar 02 2022 Limit_{n->oo} (sqrt(T(n+1)) - sqrt(T(n))) = 1/sqrt(2), where T(n) = n(n+1)/2 = A000217(n) is the triangular numbers. - Jules Beauchamp, Sep 18 2022 Equals Product_{k>=0} ((2*k+1)/(2*k+2))^((-1)^ A000120(k)) (Woods, 1978). - Amiram Eldar, Feb 04 2024 Equals 1 + Sum_{k>=1} (-1)^k*binomial(2*k,k)/2^(2*k) [Newton].
Equal Product_{k>=1} 1 - 1/(4*(2*k - 1)^2). (End)
Equals Product_{k>=0} (1 - (-1)^k/(6*k+3)). - Amiram Eldar, Nov 22 2024
PROG
(PARI) default(realprecision, 20080); x=10*(1/sqrt(2)); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b010503.txt", n, " ", d)); \\ Harry J. Smith, Jun 02 2009
CROSSREFS
Cf. A073084 (infinite tetration limit).
Decimal expansion of length of edge of a regular icosahedron with radius of circumscribed sphere = 1.
+10
31
1, 0, 5, 1, 4, 6, 2, 2, 2, 4, 2, 3, 8, 2, 6, 7, 2, 1, 2, 0, 5, 1, 3, 3, 8, 1, 6, 9, 6, 9, 5, 7, 5, 3, 2, 1, 4, 5, 7, 0, 9, 9, 5, 8, 6, 4, 4, 8, 6, 6, 8, 3, 5, 6, 3, 0, 5, 7, 8, 7, 1, 0, 4, 6, 4, 8, 2, 4, 2, 2, 2, 9, 2, 8, 0, 6, 4, 2, 8, 0, 3, 6, 7, 4, 3, 2, 6, 5, 2, 5, 7, 6, 6, 3, 1, 0, 5, 1, 4, 1, 9, 1, 3, 3, 9
COMMENTS
Regular icosahedron: A three-dimensional figure with 20 congruent equilateral triangle faces, 12 vertices, and 30 edges.
Shorter diagonal of golden rhombus with unit edge length. - Eric W. Weisstein, Dec 11 2018 The length of the shorter side of a golden rectangle inscribed in a unit circle. - Michal Paulovic, Sep 01 2022 The side length of a square inscribed within a golden ellipse with a unit semi-major axis. - Amiram Eldar, Oct 02 2022
FORMULA
Equals sqrt(50-10*sqrt(5))/5.
Equals 1/Im(e^(3*i*Pi/5)) = 1/Im(e^(3*i*Pi/5) - 1) = sqrt(2 - 2/sqrt(5)). - Karl V. Keller, Jr., Jun 11 2020 Equals Product_{k >= 1} ((10*k - 1)*(10*k + 1))/((10*k - 2)*(10*k + 2)).
Equals Product_{k >= 1} 1/(1 - 1/(25*(2*k - 1)^2)). (End)
EXAMPLE
1.051462224238267212051338169695753214570995864486683563057871046482422...
PROG
(Python)
from decimal import *
getcontext().prec = 110
c = Decimal.sqrt(2 - 2 / Decimal.sqrt(Decimal(5)))
print([int(i) for i in str(c) if i != '.'])
CROSSREFS
Cf. A179290 (longer golden rhombus diagonal).
Decimal expansion of 2*sin(Pi/5).
+10
21
1, 1, 7, 5, 5, 7, 0, 5, 0, 4, 5, 8, 4, 9, 4, 6, 2, 5, 8, 3, 3, 7, 4, 1, 1, 9, 0, 9, 2, 7, 8, 1, 4, 5, 5, 3, 7, 1, 9, 5, 3, 0, 4, 8, 7, 5, 2, 8, 6, 2, 9, 1, 9, 8, 2, 1, 4, 4, 5, 4, 4, 9, 6, 1, 5, 1, 4, 5, 5, 6, 9, 4, 8, 3, 2, 4, 7, 0, 3, 9, 1, 5, 0, 1, 7, 0, 0
COMMENTS
The golden ratio phi is the real part of 2*exp(i*Pi/5), while this constant c is the corresponding imaginary part. It is handy, for example, in simplifying metric expressions for Platonic solids (particularly for regular icosahedron and dodecahedron).
Edge length of a regular pentagon with unit circumradius. - Stanislav Sykora, May 07 2014 This is a constructible number (see A003401 for more details). Moreover, since phi is also constructible, (2^k)*exp(i*Pi/5), for any integer k, is a constructible complex number. - Stanislav Sykora, May 02 2016 rms(c, phi) := sqrt((c^2+phi^2)/2) = sqrt(2) = A002193.
LINKS
Eric Weisstein's World of Mathematics, Pentagon.
EXAMPLE
1.1755705045849462583374119...
MATHEMATICA
RealDigits[2*Sin[Pi/5], 10, 120][[1]] (* Harvey P. Dale, Sep 29 2012 *)
PROG
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); 2*Sin(Pi(R)/5); // G. C. Greubel, Nov 02 2018
Decimal expansion of the volume of square cupola with edge length 1.
+10
20
1, 9, 4, 2, 8, 0, 9, 0, 4, 1, 5, 8, 2, 0, 6, 3, 3, 6, 5, 8, 6, 7, 7, 9, 2, 4, 8, 2, 8, 0, 6, 4, 6, 5, 3, 8, 5, 7, 1, 3, 1, 1, 4, 5, 8, 3, 5, 8, 4, 6, 3, 2, 0, 4, 8, 7, 8, 4, 4, 5, 3, 1, 5, 8, 6, 6, 0, 4, 8, 8, 3, 1, 8, 9, 7, 4, 7, 3, 8, 0, 2, 5, 9, 0, 0, 2, 5, 8, 3, 5, 6, 2, 1, 8, 4, 2, 7, 7, 1, 5, 1, 5, 6, 6, 7
COMMENTS
Square cupola: 12 vertices, 20 edges, and 10 faces.
Also, decimal expansion of 1 + Product_{n>0} (1-1/(4*n+2)^2). - Bruno Berselli, Apr 02 2013 Decimal expansion of 1 + (least possible ratio of the side length of one inscribed square to the side length of another inscribed square in the same non-obtuse triangle). - L. Edson Jeffery, Nov 12 2014 2*sqrt(2)/3 is the radius of the base of the maximum-volume right cone inscribed in a unit-radius sphere. - Amiram Eldar, Sep 25 2022
FORMULA
Equals (3 + 2*sqrt(2))/3.
EXAMPLE
1.942809041582063365867792482806465385713114583584632048784453158660...
MATHEMATICA
RealDigits[N[1+(2*Sqrt[2])/3, 200]]
(* From the second comment: *) RealDigits[N[1 + Product[1 - 1/(4 n + 2)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)
CROSSREFS
Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A224268. Cf. A131594 (decimal expansion of sqrt(2)/3).
Decimal expansion of sine of 36 degrees.
+10
19
5, 8, 7, 7, 8, 5, 2, 5, 2, 2, 9, 2, 4, 7, 3, 1, 2, 9, 1, 6, 8, 7, 0, 5, 9, 5, 4, 6, 3, 9, 0, 7, 2, 7, 6, 8, 5, 9, 7, 6, 5, 2, 4, 3, 7, 6, 4, 3, 1, 4, 5, 9, 9, 1, 0, 7, 2, 2, 7, 2, 4, 8, 0, 7, 5, 7, 2, 7, 8, 4, 7, 4, 1, 6, 2, 3, 5, 1, 9, 5, 7, 5, 0, 8, 5, 0, 4, 0, 4, 9, 8, 6, 2, 7, 4, 1, 3, 3, 5
COMMENTS
This sequence is also decimal expansion of cosine of 54 degrees. - Mohammad K. Azarian, Jun 29 2013 Perimeter length of a regular pentagon with circumscribed unit circle. - R. J. Mathar, Aug 24 2023
FORMULA
sin 36 degrees = sin Pi/5 radians = sqrt((1/8)(5 - sqrt(5))).
EXAMPLE
sin 36 degrees = 0.587785252292473129168705954639...
MATHEMATICA
RealDigits[Sin[36 Degree], 10, 120][[1]] (* Harvey P. Dale, Aug 14 2018 *)
Decimal expansion of tangent of 72 degrees.
+10
12
3, 0, 7, 7, 6, 8, 3, 5, 3, 7, 1, 7, 5, 2, 5, 3, 4, 0, 2, 5, 7, 0, 2, 9, 0, 5, 7, 6, 0, 3, 6, 9, 0, 9, 8, 2, 4, 0, 0, 6, 7, 0, 2, 1, 4, 3, 5, 3, 7, 7, 9, 2, 4, 2, 7, 0, 3, 9, 1, 5, 6, 2, 5, 0, 3, 7, 4, 8, 6, 3, 2, 8, 8, 4, 9, 5, 0, 9, 0, 9, 1, 8, 4, 5, 4, 5, 9, 3, 7, 2, 1, 6, 6, 7, 1, 0, 5, 4, 3
COMMENTS
Length of the second longest diagonal in a regular 10-gon with unit side. - Mohammed Yaseen, Nov 12 2020
FORMULA
Equals tan(66 degrees) + tan(36 degrees) + tan(6 degrees). - Amiram Eldar, Apr 07 2022
EXAMPLE
3.077683537175253402570290576036909824006702143537792427...
MATHEMATICA
RealDigits[Tan[72 Degree], 10, 120][[1]] (* Harvey P. Dale, Apr 30 2012 *) RealDigitis[Sqrt[5 + 2*Sqrt[5]], 10, 100][[1]] (* G. C. Greubel, Nov 21 2018 *)
PROG
(Magma) SetDefaultRealField(RealField(100)); Sqrt(5+2*Sqrt(5)); // G. C. Greubel, Nov 21 2018 (Sage) numerical_approx(tan(2*pi/5), digits=100) # G. C. Greubel, Nov 21 2018
Decimal expansion of sine of 75 degrees.
+10
11
9, 6, 5, 9, 2, 5, 8, 2, 6, 2, 8, 9, 0, 6, 8, 2, 8, 6, 7, 4, 9, 7, 4, 3, 1, 9, 9, 7, 2, 8, 8, 9, 7, 3, 6, 7, 6, 3, 3, 9, 0, 4, 8, 3, 9, 0, 0, 8, 4, 0, 4, 5, 5, 0, 4, 0, 2, 3, 4, 3, 0, 7, 6, 3, 1, 0, 4, 2, 3, 2, 1, 3, 9, 7, 9, 8, 5, 5, 5, 1, 6, 3, 4, 7, 5, 6, 1, 7, 4, 1, 8, 5, 8, 0, 7, 0, 4, 5, 1
COMMENTS
Length of one side of the new Type 15 Convex Pentagon. - Michel Marcus, Aug 04 2015
EXAMPLE
0.96592582628906828674974319972889736763390483900840455040234307631042...
Decimal expansion of circumradius of a regular dodecahedron with edge length 1.
+10
10
1, 4, 0, 1, 2, 5, 8, 5, 3, 8, 4, 4, 4, 0, 7, 3, 5, 4, 4, 6, 7, 6, 6, 7, 7, 9, 3, 5, 3, 2, 2, 0, 6, 7, 9, 9, 4, 4, 4, 3, 9, 3, 1, 7, 3, 9, 7, 7, 5, 4, 9, 2, 8, 6, 3, 6, 6, 0, 8, 4, 5, 1, 8, 6, 3, 9, 1, 3, 5, 4, 0, 2, 7, 2, 1, 1, 4, 4, 4, 7, 6, 7, 6, 5, 0, 1, 0, 8, 3, 9, 0, 9, 0, 3, 9, 8, 0, 5, 2, 3, 3, 9, 7, 9, 8
COMMENTS
Dodecahedron: A three-dimensional figure with 12 faces, 20 vertices, and 30 edges.
Appears as a coordinate in a degree-7 quadrature formula on 12 points over the unit circle [Stroud & Secrest]. - R. J. Mathar, Oct 12 2011
REFERENCES
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.
FORMULA
Equals (sqrt(3) + sqrt(15))/4 = sqrt((9 + 3*sqrt(5))/8).
The minimal polynomial is 16*x^4 - 36*x^2 + 9. - Joerg Arndt, Feb 05 2014
EXAMPLE
1.40125853844407354467667793532206799444393173977549286366084518639135...
MATHEMATICA
RealDigits[(Sqrt[3]+Sqrt[15])/4, 10, 175][[1]]
Decimal expansion of (diagonal)/(shortest side) of a golden rectangle.
+10
10
1, 9, 0, 2, 1, 1, 3, 0, 3, 2, 5, 9, 0, 3, 0, 7, 1, 4, 4, 2, 3, 2, 8, 7, 8, 6, 6, 6, 7, 5, 8, 7, 6, 4, 2, 8, 6, 8, 1, 1, 3, 9, 7, 2, 6, 8, 2, 5, 1, 5, 0, 0, 4, 4, 4, 8, 9, 4, 6, 1, 1, 2, 8, 8, 8, 6, 0, 3, 0, 6, 3, 4, 0, 1, 7, 0, 3, 8, 7, 0, 0, 3, 4, 3, 7, 5, 8, 5, 6, 2, 1, 9, 4, 1, 6, 2, 2, 7, 6, 3, 3, 5, 1, 7, 9, 9, 4, 3, 5, 1, 0, 2, 8, 0, 6, 0, 0, 8, 4, 1, 7, 9, 7, 4, 1, 3, 2, 3, 8, 7
COMMENTS
A rectangle of length L and width W is a golden rectangle if L/W = r = (1+sqrt(5))/2. The diagonal has length D = sqrt(L^2+W^2), so D/W = sqrt(r^2+1) = sqrt(r+2).
This is the case n=10 of (Gamma(1/n)/Gamma(2/n))*(Gamma((n-1)/n)/Gamma((n-2)/n)) = 2*cos(Pi/n). - Bruno Berselli, Dec 13 2012 Edge length of a pentagram (regular star pentagon) with unit circumradius. - Stanislav Sykora, May 07 2014 The ratio diagonal/side of the shortest diagonal in a regular 10-gon. - Mohammed Yaseen, Nov 04 2020
LINKS
Eric Weisstein's World of Mathematics, Pentagram.
EXAMPLE
1.902113032590307144232878666758764286811397268251...
MATHEMATICA
r = (1 + 5^(1/2))/2; RealDigits[(2 + r)^(1/2), 10, 130]][[1]]
RealDigits[Sqrt[GoldenRatio+2], 10, 130][[1]] (* Harvey P. Dale, Oct 27 2023 *)
PROG
(PARI) sqrt((5+sqrt(5))/2)
(Magma) SetDefaultRealField(RealField(100)); Sqrt((5+Sqrt(5))/2); // G. C. Greubel, Nov 02 2018
CROSSREFS
Cf. A188594 (D/W for the silver rectangle, r=1+sqrt(2)).
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