A019824 -id:A019824 - cp's OEIS frontend

A010503 Decimal expansion of 1/sqrt(2).

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

Offset: 0

N. J. A. Sloane

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
Positive zero of the Hermite polynomial of degree 2. - A.H.M. Smeets, Jun 02 2025

			0.7071067811865475...

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.

Harry J. Smith, Table of n, a(n) for n = 0..20000
P. C. Fishburn and J. A. Reeds, Bell inequalities, Grothendieck's constant and root two, SIAM J. Discrete Math., Vol. 7, No. 1, Feb. 1994, pp. 48-56.
Ovidiu Furdui, Problem 1, Problem Corner, Research Group in Mathematical Inequalities and Applications, 2010.
Michael Penn, A surprisingly convergent limit, YouTube video, 2022.
Michael Penn, The infinite fraction of your dreams (nightmare?), YouTube video, 2022.
Jonathan Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; arXiv:1108.6096 [math.NT], 2011, see p. 3 in the link.
Eric Weisstein's World of Mathematics, Digit Product.
Wikipedia, Platonic solid.
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.
Index entries for algebraic numbers, degree 2.

Cf. A000120, A040042, A072364, A268682.
Cf. A073084 (infinite tetration limit).
Platonic solids circumradii: A010527 (cube), A019881 (icosahedron), A179296 (dodecahedron), A187110 (tetrahedron).
Platonic solids midradii: A020765 (tetrahedron), A020761 (octahedron), A019863 (icosahedron), A239798 (dodecahedron).
Cf. A000217, A010060, A019812, A019824, A019851, A019857, A019884, A019896.

1/Sqrt(2); // Vincenzo Librandi, Feb 21 2016

Digits:=100; evalf(1/sqrt(2)); Wesley Ivan Hurt, Mar 27 2014

N[ 1/Sqrt[2], 200]
RealDigits[1/Sqrt[2],10,120][[1]] (* Harvey P. Dale, Mar 25 2019 *)

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

1/sqrt(2) = cos(Pi/4) = sqrt(2)/2. - Eric Desbiaux, Nov 05 2008
a(n) = 9 - A268682(n). As constants, this sequence is 1 - A268682. - Philippe Deléham, Feb 21 2016
From Amiram Eldar, Jun 29 2020: (Start)
Equals sin(Pi/4) = cos(Pi/4).
Equals Integral_{x=0..Pi/4} cos(x) dx. (End)
Equals (1/2)*A019884 + A019824 * A010527 = A019851 * A019896 + A019812 * A019857. - R. J. Mathar, Jan 27 2021
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
From Stefano Spezia, Oct 15 2024: (Start)
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

More terms from Harry J. Smith, Jun 02 2009

A020761 Decimal expansion of 1/2.

Original entry on oeis.org

5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Real part of all nontrivial zeros of the Riemann zeta function (assuming the Riemann hypothesis to be true). - Alonso del Arte, Jul 02 2011
Radius of a sphere with surface area Pi. - Omar E. Pol, Aug 09 2012
Radius of the midsphere (tangent to the edges) in a regular octahedron with unit edges. Also radius of the inscribed sphere (tangent to faces) in a cube with unit edges. - Stanislav Sykora, Mar 27 2014
Construct a rectangle of maximal area inside an arbitrary triangle. The ratio of the rectangle's area to the triangle's area is 1/2. - Rick L. Shepherd, Jul 30 2014

			1/2 = 0.50000000000000...

Michael Penn, A creative approach to a scary looking integral., YouTube video, 2020.
Michael Penn, I really like this sum!, YouTube video, 2021.
Wikipedia, Riemann zeta function
Wikipedia, Platonic solid
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. In platonic solids:
midsphere radii:
A020765 (tetrahedron),
A010503 (cube),
A019863 (icosahedron),
A239798 (dodecahedron);
insphere radii:
A020781 (tetrahedron),
A020763 (octahedron),
A179294 (icosahedron),
A237603 (dodecahedron).

Digits:=100; evalf(1/2); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[1/2, 10, 128][[1]] (* Alonso del Arte, Dec 13 2013 *)
LinearRecurrence[{1},{5,0},99] (* Ray Chandler, Jul 15 2015 *)

{ default(realprecision); x=1/2*10; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Felix Fröhlich, Jul 24 2014

a(n) = 5*(n==0); \\ Michel Marcus, Jul 25 2014

Equals Sum_{k>=1} (1/3^k). Hence 1/2 = 0.1111111111111... in base 3.
Cosine of 60 degrees, i.e., cos(Pi/3).
-zeta(0), zeta being the Riemann function. - Stanislav Sykora, Mar 27 2014
a(0) = 5; a(n) = 0, n > 0. - Wesley Ivan Hurt, Mar 27 2014
a(n) = 5 * floor(1/(n + 1)). - Wesley Ivan Hurt, Mar 27 2014
Equals 2*A019824*A019884. - R. J. Mathar, Jan 17 2021

A019973 Decimal expansion of tangent of 75 degrees.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

An equivalent definition of this sequence: decimal expansion of x > 1 satisfying x^2 - 4*x + 1 = 0. - Arkadiusz Wesolowski, Nov 28 2011
An algebraic integer of degree 2 with minimal polynomial x^2 - 4*x + 1. - Charles R Greathouse IV, Oct 17 2016
Length of the second longest diagonal in a regular 12-gon with unit side. - Mohammed Yaseen, Dec 13 2020

			3.732050807568877293527446341505872366942805253810380628...

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Ivan Panchenko)
Wikipedia, Exact trigonometric constants.
Index entries for algebraic numbers, degree 2.

Cf. A002194, A019824, A019884, A019913.

Cf. A001075, A001353, A001835, A049310.

SetDefaultRealField(RealField(100)); 2 + Sqrt(3); // G. C. Greubel, Nov 20 2018

RealDigits[Tan[75 Degree],10,120][[1]] (* Harvey P. Dale, Nov 08 2011 *)
RealDigits[2+Sqrt[3], 10, 100][[1]] (* G. C. Greubel, Nov 20 2018 *)

sqrt(3)+2 \\ Charles R Greathouse IV, Oct 17 2016

numerical_approx(2+sqrt(3), digits=100) # G. C. Greubel, Nov 20 2018

Equals 2 + sqrt(3) = 2+A002194 = cotangent of 15 degrees. - Rick L. Shepherd, Jul 04 2004
Equals exp(arccosh(2)). - Amiram Eldar, Aug 07 2023
c^n = A001835(n) + (1 + sqrt(3)) * A001353(n) = A001075(n) + sqrt(3) * A001353(n); where c = 2 + sqrt(3). - Gary W. Adamson, Oct 14 2023
Equals lim_{n->oo} S(n, 4)/ S(n-1, 4), with the S-Chebyshev polynomial (see A049310) S(n, 4) = A001353(n+1). See the A001353 formula from Oct 06 2002 by Gregory V. Richardson. - Wolfdieter Lang, Nov 15 2023
Equals A019884 / A019824. - R. J. Mathar, Jan 12 2024
Equals 1/A019913. - Hugo Pfoertner, Mar 24 2024

Checked by Neven Juric (neven.juric(AT)apis-it.hr), Feb 04 2008

A101263 Decimal expansion of sqrt(2 - sqrt(3)), edge length of a regular dodecagon with circumradius 1.

Original entry on oeis.org

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

Offset: 0

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jan 25 2005

sqrt(2 - sqrt(3)) is the shape of the lesser sqrt(6)-contraction rectangle, as defined at A188739. - Clark Kimberling, Apr 16 2011
This is a constructible number, since 12-gon is a constructible polygon. See A003401 for more details. - Stanislav Sykora, May 02 2016
It is also smaller positive coordinate of (symmetrical) intersection points of x^2 + y^2 = 4 circle and y = 1/x hyperbola. The bigger coordinate is A188887. - Leszek Lezniak, Sep 18 2018
The greatest possible minimum distance between 8 points in a unit square (Schaer and Meir, 1965; Schaer, 1965; Croft et al., 1991). - Amiram Eldar, Feb 24 2025

			0.517638090205041524697797675248096656698137802639861027628006414630113....

Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved Problems in Geometry, Springer, 1991, Section D1, p. 108.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

G. C. Greubel, Table of n, a(n) for n = 0..5000
J. Schaer and A. Meir, On a geometric extremum problem, Canadian Mathematical Bulletin, Vol. 8, No. 1 (1965), pp. 21-27.
J. Schaer, The densest packing of 9 circles in a square, Canadian Mathematical Bulletin, Vol. 8, No. 3 (1965), pp. 273-277.
Eric Weisstein's World of Mathematics, Dodecagon.
Index entries for algebraic numbers, degree 4.

Cf. A003401, A019824, A019913, A091999, A120683, A188739, A188887, A329247.

r = 6^(1/2); t = (r - (-4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]  (*A101263*)
RealDigits[Sqrt[2-Sqrt[3]],10,120][[1]] (* Harvey P. Dale, Apr 24 2018 *)

2*sin(Pi/12) \\ Stanislav Sykora, May 02 2016

Equals sqrt(A019913). - R. J. Mathar, Apr 20 2009
Equals 2*sin(Pi/12) = 2*cos(Pi*5/12). - Stanislav Sykora, May 02 2016
Equals i^(5/6) + i^(-5/6). - Gary W. Adamson, Jul 07 2022
From Amiram Eldar, Nov 24 2024: (Start)
Equals A120683 / 2 = 2 * A019824 = 1 / A188887 = exp(-A329247).
Equals (sqrt(3)-1)/sqrt(2).
Equals Product_{k>=1} (1 + (-1)^k/A091999(k)). (End)

A019913 Decimal expansion of tangent of 15 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also, 2 - sqrt(3) = cotangent of 75 degrees. An equivalent definition of this sequence: decimal expansion of x < 1 satisfying x^2 - 4*x + 1 = 0. - Arkadiusz Wesolowski, Nov 29 2011
Multiplied by -1 (that is, -2 + sqrt(3)), this is one of three real solutions to x^3 = 15x + 4. The other two are 4 and -2 - sqrt(3), all of which can be found with Viete's formula. - Alonso del Arte, Dec 15 2012
Wentworth (1903) shows how to compute the tangent of 15 degrees to five decimal places by the laborious process of adding up the first few terms of Pi/12 + Pi^3/5184 + 2Pi^5/3732480 + 17Pi^7/11287019520 + ... - Alonso del Arte, Mar 13 2015
A quadratic integer. - Charles R Greathouse IV, Aug 27 2017
This is the radius of the largest sphere that can be placed in the space between a sphere of radius 1 and the corners of its circumscribing cube. - Amiram Eldar, Jul 11 2020

			0.2679491924311227064725536...

Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1). Princeton, New Jersey: Princeton University Press (1988): 22 - 23.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Willis F. Kern and James R. Bland, Solid Mensuration: With Proofs, 2nd ed., J. Wiley & Sons, Inc., New York, 1938. See pp. 91-92.
George Albert Wentworth, New Plane and Spherical Trigonometry, Surveying, and Navigation, Boston: The Atheneum Press (1903), p. 240.
Wikipedia, Exact trigonometric constants.
Index entries for algebraic numbers, degree 2

Cf. A002194 (sqrt(3)).

RealDigits[N[Tan[15 Degree], 200]][[1]] (* Arkadiusz Wesolowski, Nov 29 2011 *)

2-sqrt(3) \\ Charles R Greathouse IV, Aug 27 2017

Equals Sum_{k>=1} binomial(2*k,k)/(6^k*(k+1)). - Amiram Eldar, Jul 11 2020
Equals exp(-arccosh(2)). - Amiram Eldar, Jul 06 2023
tan(Pi/12) = A019824 / A019884. - R. J. Mathar, Aug 31 2025

A232736 Decimal expansion of sin(Pi/14), or the imaginary part of (-1)^(1/7).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding real part is in A232735.
Root of the equation 1 - 4*x - 4*x^2 + 8*x^3 = 0. - Vaclav Kotesovec, Apr 04 2021
The other 2 roots are -A362922 and A073052. - R. J. Mathar, Aug 29 2025

			0.222520933956314404288902564496794759466355568764544955311987...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 3.

Cf. A232735 (real part), A010503 (imag(I^(1/2))), A182168 (imag(I^(1/4))), A019827 (imag(I^(1/5))), A019824 (imag(I^(1/6))), A232738 (imag(I^(1/8))), A019819 (imag(I^(1/9))), A019818 (imag(I^(1/10))).

A384473 Decimal expansion of the middle interior angle (in degrees) in Albrecht Dürer's approximate construction of the regular pentagon.

Original entry on oeis.org

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

Offset: 3

Stefano Spezia, May 30 2025

			108.366120162561467008046935277164429896133431...

Alfred S. Posamentier and Herbert A. Hauptman, 101 great ideas for introducing key concepts in mathematics: a resource for secondary school teachers, Corwin Press, Inc., 2001. See pages 144-145.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 181-182.

Alexander Bogomolny, Approximate Construction of Regular Pentagon by A. Dürer.
Stefano Spezia, Albrecht Dürer's approximate construction of the regular pentagon.
Wikipedia, Albrecht Dürer.

Cf. A228719, A384474 (in radians).
Cf. A002194, A019824, A072097, A177870.
Cf. A384475, A384476, A384477, A384478.

RealDigits[(3Pi/4-ArcSin[Sqrt[3]Sin[Pi/12]])180/Pi,10,100][[1]] (* or *)
RealDigits[(Pi+ArcTan[(3-Sqrt[3]+Sqrt[6Sqrt[3]-4])/(3-Sqrt[3]-Sqrt[6Sqrt[3]-4])])180/Pi,10,100][[1]]

Equals 135 - 180*arcsin(sqrt(3)*sin(Pi/12))/Pi.
Equals (Pi + arctan((3 - sqrt(3) + sqrt(6*sqrt(3) - 4))/(3 - sqrt(3) - sqrt(6*sqrt(3) - 4))))*180/Pi.
Equals (540 - 2*A384475 - A384477)/2.
A384475 < this constant < A384477.

A384474 Decimal expansion of the middle interior angle (in radians) in Albrecht Dürer's approximate construction of the regular pentagon.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, May 30 2025

			1.891345594448510418687173478952739199024779225...

Alfred S. Posamentier and Herbert A. Hauptman, 101 great ideas for introducing key concepts in mathematics: a resource for secondary school teachers, Corwin Press, Inc., 2001. See pages 144-145.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 181-182.

Alexander Bogomolny, Approximate Construction of Regular Pentagon by A. Dürer.
Stefano Spezia, Albrecht Dürer's approximate construction of the regular pentagon.
Wikipedia, Albrecht Dürer.

Cf. A228719, A384473 (in degrees).
Cf. A002194, A019824, A177870.
Cf. A384475, A384476, A384477, A384478.

RealDigits[3Pi/4-ArcSin[Sqrt[3]Sin[Pi/12]],10,100][[1]] (* or *)
RealDigits[Pi+ArcTan[(3-Sqrt[3]+Sqrt[6Sqrt[3]-4])/(3-Sqrt[3]-Sqrt[6Sqrt[3]-4])],10,100][[1]]

Equals 3*Pi/4 - arcsin(sqrt(3)*sin(Pi/12)).
Equals Pi + arctan((3 - sqrt(3) + sqrt(6*sqrt(3) - 4))/(3 - sqrt(3) - sqrt(6*sqrt(3) - 4))).
Equals (3*Pi - 2*A384476 - A384478)/2.
A384476 < this constant < A384478.

A232738 Decimal expansion of the imaginary part of I^(1/8), or sin(Pi/16).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding real part is in A232737.

			0.195090322016128267848284868477022240927691617751954807754502...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 8

Cf. A232737 (real part), A010503 (imag(I^(1/2))), A182168 (imag(I^(1/4))), A019827 (imag(I^(1/5))), A019824 (imag(I^(1/6))), A232736 (imag(I^(1/7))), A019819 (imag(I^(1/9))), A019818 (imag(I^(1/10))).

R:= RealField(116); Sin(Pi(R)/16); // G. C. Greubel, Sep 20 2022

RealDigits[Sin[Pi/16],10,120][[1]] (* Harvey P. Dale, Sep 01 2018 *)

imag(I^(1/8)) \\ Seiichi Manyama, Apr 04 2021

sin(Pi/16) \\ Seiichi Manyama, Apr 04 2021

sqrt(2-sqrt(2+sqrt(2)))/2 \\ Seiichi Manyama, Apr 04 2021

numerical_approx(sin(pi/16), digits=116) # G. C. Greubel, Sep 20 2022

Equals (1/2) * sqrt(2-sqrt(2+sqrt(2))). - Seiichi Manyama, Apr 04 2021
This^2 + A232737^2 = 1.
Smallest positive of the 8 real-valued roots of 128*x^8-256*x^6+160*x^4-32*x^2+1=0.

A280819 Decimal expansion of 12*sin(Pi/12).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 08 2017

The ratio of the perimeter of a regular 12-gon (dodecagon) to its diameter (greatest diagonal).

A quartic integer: the least positive root of x^4 - 144x^2 + 1296.

			3.105828541230249148186786051488579940188826815839166165768038487780683696...

Index entries for algebraic numbers, degree 4

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280533 (n=7), A280585 (n=8), A280633 (n=9), A280725 (n=11).

First@ RealDigits@ N[12 Sin[Pi/12], 120] (* Michael De Vlieger, Feb 05 2017 *)

PARI
```
12*sin(Pi/12)
```

polrootsreal(x^4-144*x^2+1296)[3] \\ Charles R Greathouse IV, Feb 05 2017

12*A019824.

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