OFFSET
1,2
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
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.
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
EXAMPLE
1.701301616704079864363080994126...
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 *)
PROG
(PARI) 1/sin(Pi/5)
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); 1/Sin(Pi(R)/5); // G. C. Greubel, Nov 02 2018
(Sage) numerical_approx(1/sin(pi/5), digits=100) # G. C. Greubel, Dec 12 2018
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Rick L. Shepherd, Aug 08 2006
STATUS
approved