A121570 Decimal expansion of cosecant of 36 degrees = csc(Pi/5) = 1/sin(Pi/5).

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

Offset: 1

Author

Rick L. Shepherd, Aug 08 2006

Keywords

• cons
• nonn

Comments

1 + csc(Pi/5) is the radius of the smallest circle into which 5 unit circles can be packed ("r=2.701+ Proved by Graham in 1968.", according to the Friedman link, which has a diagram).

csc(Pi/5) = 1/A019845 is the distance between the center of the larger circle and the center of each unit circle.

The problem of finding the diameter d of the circumscribing circle of a regular pentagon of side s = 10 (in some length units) appears as an example in Abū Kāmil's treatise on the pentagon and decagon (see the Havil reference) and Abū Kāmil links. The answer is d/s = 1/sin(Pi/5). - Wolfdieter Lang, Mar 01 2018

Longer diagonal of golden rhombus with unit edge length. - Eric W. Weisstein, Dec 11 2018

The length of the longer side of a golden rectangle inscribed in a unit circle. - Michal Paulovic, Sep 01 2022

The radius of a common circle surrounded by 5 tangent unit circles is A121570 - 1. - Thomas Otten, Dec 27 2023

Examples

			1.701301616704079864363080994126...
		

References

• Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 58.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• E. Friedman, Erich's Packing Center: "Circles in Circles"
• I. C. Karpinski, The Algebra of Abu Kamil, Amer. Math. Month. XXI,2 (1914), 37-48.
• MacTutor History of Mathematics, Abu Kamil Shuja.
• Eric Weisstein's World of Mathematics, Golden Rhombus
• Wikipedia, Abu Kamil.
• Index entries for algebraic numbers, degree 4

Crossrefs

Cf. A001622, A019845 (inverse), A182007 (2/A121570).

Cf. A179290 (shorter golden rhombus diagonal).

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); 1/Sin(Pi(R)/5); // G. C. Greubel, Nov 02 2018

Maple

evalf(1/sin(Pi/5),130); # Muniru A Asiru, Nov 02 2018

Mathematica

RealDigits[Csc[Pi/5], 10, 100][[1]] (* G. C. Greubel, Nov 02 2018 *)

PARI

1/sin(Pi/5)

Sage

numerical_approx(1/sin(pi/5), digits=100) # G. C. Greubel, Dec 12 2018

Formula

Equals 1/A019845.

Equals 2*(2*phi - 1)*sqrt(2 + phi)/5, with the golden ratio phi = A001622. - Wolfdieter Lang, Mar 01 2018

Equals sqrt(2 + 2 / sqrt(5)). - Michal Paulovic, Sep 01 2022

The minimal polynomial is 5*x^4 - 20*x^2 + 16. - Joerg Arndt, Sep 09 2022