A272534 Decimal expansion of the edge length of a regular 15-gon with unit circumradius.

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

Offset: 0

Author

Stanislav Sykora, May 02 2016

Keywords

• nonn
• cons
• easy
• changed

Comments

15-gon is the first m-gon with odd composite m which is constructible (see A003401) in virtue of the fact that 15 is the product of two distinct Fermat primes (A019434). The next such case is 51-gon (m=3*17), followed by 85-gon (m=5*17), 771-gon (m=3*257), etc.

From Wolfdieter Lang, Apr 29 2018: (Start)

This constant appears in a historic problem posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593, solved by Viète. See the Havil reference, problem 4, pp. 69-74. See also the comments in A302711 with a link to Romanus' book, Exemplum quaesitum.

This problem is equivalent to R(45, 2*sin(Pi/675)) = 2*sin(Pi/15), with a special case of monic Chebyshev polynomials of the first kind, named R, given in A127672. For the constant 2*sin(Pi/675) see A302716. (End)

Examples

			0.415823381635518674203484568810250332433169521255447672814363947...
		

References

• Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..2000
• Mauro Fiorentini, Construibili (numeri)
• Eric Weisstein's World of Mathematics, Constructible Number
• Wikipedia, Constructible number
• Wikipedia, Regular polygon
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for algebraic numbers, degree 8.

Crossrefs

Cf. A003401, A019434, A127672, A302711, A302716.

Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272535 (16), A228787 (17), A272536 (20).

Programs

Mathematica

RealDigits[N[2Sin[Pi/15], 100]][[1]] (* Robert Price, May 02 2016*)

PARI

2*sin(Pi/15)

Formula

Equals 2*sin(Pi/m) for m=15, 2*A019821.

Also equals (sqrt(3) - sqrt(15) + sqrt(10 + 2*sqrt(5)))/4.

Also equals sqrt(7 - sqrt(5) - sqrt(30 - 6*sqrt(5)))/2. This is the rewritten expression of the Havil reference on top of p. 70. - Wolfdieter Lang, Apr 29 2018