A182007 Decimal expansion of 2*sin(Pi/5).

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

Offset: 1

Author

Stanislav Sykora, Apr 06 2012

Keywords

• nonn
• cons
• easy

Comments

The golden ratio phi is the real part of 2*exp(i*Pi/5), while this constant c is the corresponding imaginary part. It is handy, for example, in simplifying metric expressions for Platonic solids (particularly for regular icosahedron and dodecahedron).

Note that c^2+A001622^2 = 4; c*A001622 = A188593 = 2*A019881; c = 2*A019845.

Edge length of a regular pentagon with unit circumradius. - Stanislav Sykora, May 07 2014

This is a constructible number (see A003401 for more details). Moreover, since phi is also constructible, (2^k)*exp(i*Pi/5), for any integer k, is a constructible complex number. - Stanislav Sykora, May 02 2016

rms(c, phi) := sqrt((c^2+phi^2)/2) = sqrt(2) = A002193.

Examples

			1.1755705045849462583374119...
		

Links

• Ivan Panchenko, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Pentagon.
• Wikipedia, Platonic solid.
• Index entries for algebraic numbers, degree 4.

Crossrefs

Cf. A001622, A002193, A003401, A019881, A019845, A063226, A102769, A131595, A188593, A300074.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); 2*Sin(Pi(R)/5); // G. C. Greubel, Nov 02 2018

Maple

evalf(2*sin(Pi/5),100); # Muniru A Asiru, Nov 02 2018

Mathematica

RealDigits[2*Sin[Pi/5],10,120][[1]] (* Harvey P. Dale, Sep 29 2012 *)

PARI

2*sin(Pi/5) \\ Stanislav Sykora, May 02 2016

Formula

Equals sqrt(3-phi).

Equals sqrt((5-sqrt(5))/2). - Jean-François Alcover, May 21 2013

Equals Product_{k>=0} ((10*k + 4)*(10*k + 6))/((10*k + 3)*(10*k + 7)). - Antonio Graciá Llorente, Mar 25 2024

Equals Product_{k>=1} (1 - (-1)^k/A063226(k)). - Amiram Eldar, Nov 23 2024

Equals 2*A019845 = 1/A300074. - Hugo Pfoertner, Nov 23 2024