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

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

Author

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

Keywords

• cons
• nonn

Comments

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

Examples

			0.517638090205041524697797675248096656698137802639861027628006414630113....
		

References

• 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.

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Formula

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)