A121570 - OEIS

%I #52 Jan 25 2024 07:41:54

%S 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,

%T 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,

%U 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

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

%C 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).

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

%C 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

%C Longer diagonal of golden rhombus with unit edge length. - _Eric W. Weisstein_, Dec 11 2018

%C The length of the longer side of a golden rectangle inscribed in a unit circle. - _Michal Paulovic_, Sep 01 2022

%C The radius of a common circle surrounded by 5 tangent unit circles is A121570 - 1. - _Thomas Otten_, Dec 27 2023

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

%H G. C. Greubel, <a href="/A121570/b121570.txt">Table of n, a(n) for n = 1..10000</a>

%H E. Friedman, <a href="https://erich-friedman.github.io/packing/cirincir/">Erich's Packing Center: "Circles in Circles"</a>

%H I. C. Karpinski, <a href="http://www.jstor.org/stable/2972073">The Algebra of Abu Kamil</a>, Amer. Math. Month. XXI,2 (1914), 37-48.

%H MacTutor History of Mathematics, <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Abu_Kamil.html">Abu Kamil Shuja</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldenRhombus.html">Golden Rhombus</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ab%C5%AB_K%C4%81mil_Shuj%C4%81%CA%BF_ibn_Aslam">Abu Kamil</a>.

%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>

%F Equals 1/A019845.

%F Equals 2*(2*phi - 1)*sqrt(2 + phi)/5, with the golden ratio phi = A001622. - _Wolfdieter Lang_, Mar 01 2018

%F Equals sqrt(2 + 2 / sqrt(5)). - _Michal Paulovic_, Sep 01 2022

%F The minimal polynomial is 5*x^4 - 20*x^2 + 16. - _Joerg Arndt_, Sep 09 2022

%e 1.701301616704079864363080994126...

%p evalf(1/sin(Pi/5),130); # _Muniru A Asiru_, Nov 02 2018

%t RealDigits[Csc[Pi/5], 10, 100][[1]] (* _G. C. Greubel_, Nov 02 2018 *)

%o (PARI) 1/sin(Pi/5)

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); 1/Sin(Pi(R)/5); // _G. C. Greubel_, Nov 02 2018

%o (Sage) numerical_approx(1/sin(pi/5), digits=100) # _G. C. Greubel_, Dec 12 2018

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

%Y Cf. A179290 (shorter golden rhombus diagonal).

%K cons,nonn

%O 1,2

%A _Rick L. Shepherd_, Aug 08 2006