OFFSET

1,1

COMMENTS

1 + csc(Pi/8) is the radius of the smallest circle into which 9 unit circles can be packed ("r=3.613+ Proved by Pirl in 1969", according to the Friedman link, which has a diagram).

csc(Pi/8) is the distance between the center of the larger circle and the center of each unit circle that touches the larger circle.

A rectangle of length L and width W is a called a silver rectangle if L=rW, where r is the silver ratio; i.e., r = 1+sqrt(2). The diagonal has length D = sqrt(W^2+L^2), so that D/W = sqrt(4+2*sqrt(2)) = csc(Pi/8). - Clark Kimberling, Apr 04 2011

This algebraic integer of degree 4 also gives the length ratio diagonal/side of the longest diagonal in the regular octagon. The minimal polynomial is x^4 - 8*x + 8. In the power basis of Gal(Q(rho(8))/Q), with rho(8) = sqrt(2 + sqrt(2)) = A179260 it is -2*rho(8) + 1*rho(8)^3 which equals sqrt(2)*rho(8). - Wolfdieter Lang, Oct 28 2020

REFERENCES

D. Mumford et al., Indra's Pearls, Cambridge 2002; see p. 362. - N. J. A. Sloane, Nov 22 2009

LINKS

FORMULA

Equals 2*sqrt(2)*cos(Pi/8).

Equals Product_{k >= 0} (8*k + 4)^2/((8*k + 1)*(8*k + 7)). - Antonio GraciĆ” Llorente, Mar 11 2024

EXAMPLE

2.6131259297527530557132863468543743071675223766985390550977...

MAPLE

evalf(1/sin(Pi/8), 120); # Muniru A Asiru, Nov 02 2018

MATHEMATICA

RealDigits[Csc[Pi/8], 10, 130][[1]] (* corrected by Harvey P. Dale, Jul 28 2012 *)

PROG

(PARI) 1/sin(Pi/8)

(Magma) SetDefaultRealField(RealField(100)); R:=RealField(); 1/Sin(Pi(R)/8); // G. C. Greubel, Nov 02 2018

CROSSREFS

KEYWORD

cons,nonn

AUTHOR

Rick L. Shepherd, Aug 09 2006

STATUS

approved