A272487 - OEIS

%I #30 Aug 21 2023 12:29:53

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

%T 0,9,2,1,9,9,8,1,4,5,5,5,7,4,9,1,9,7,5,2,8,8,9,0,9,4,6,0,7,0,6,4,4,0,

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

%N Decimal expansion of the edge length of a regular heptagon with unit circumradius.

%C The edge length e(m) of a regular m-gon is e(m) = 2*sin(Pi/m). In this case, m = 7, and the constant, a = e(7), is the smallest m for which e(m) is not constructible using a compass and a straightedge (see A004169). With an odd m, in fact, e(m) would be constructible only if m were a Fermat prime (A019434).

%H Stanislav Sykora, <a href="/A272487/b272487.txt">Table of n, a(n) for n = 0..2000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Constructible_number">Constructible number</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Heptagon">Heptagon</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Regular_polygon">Regular polygon</a>

%H <a href="/index/Al#algebraic_06">Index entries for algebraic numbers, degree 6</a>

%F Equals 2*sin(Pi/7) = 2*cos(Pi*5/14).

%F Equals i^(-5/7) + i^(5/7). - _Gary W. Adamson_, Feb 12 2022

%e 0.8677674782351162409515366656967175092199814555749197528890946...

%t N[2*Sin[Pi/7], 25] (* _G. C. Greubel_, May 01 2016 *)

%t RealDigits[2*Sin[Pi/7],10,120][[1]] (* _Harvey P. Dale_, Mar 07 2020 *)

%o (PARI) 2*sin(Pi/7)

%Y Cf. A004169, A019434.

%Y Cf. A160389.

%Y Edge lengths of nonconstructible n-gons: A272488 (n=9), A272489 (n=11), A272490 (n=13), A255241 (n=14), A130880 (n=18), A272491 (n=19).

%K nonn,cons,easy

%O 0,1

%A _Stanislav Sykora_, May 01 2016