A367480 Decimal expansion of the radius of a common circle surrounded by seven tangent unit circles.

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

Offset: 1

Author

Thomas Otten, Dec 23 2023

Keywords

• nonn
• cons
• easy
• changed

Comments

The radius of a common circle surrounded by n tangent unit circles (n > 2) is r = 1/sin(Pi/n) - 1.

n=7 is the smallest number for which the radius cannot be expressed using square roots, since the regular heptagon formed by the centers of the tangent circles is non-constructible (see A246724, A188582, and A121570 for n=3, 4, 5).

Examples

			1.3047648709624865052...
		

Links

• Andrew M. Gleason, Angle Trisection, the Heptagon, and the Triskaidecagon, The American Mathematical Monthly 95, no. 3 (March 1988), pp. 185-194.
• Index entries for algebraic numbers, degree 6.

Crossrefs

Cf. A121598.

Cf. A121570, A188582, A246724.

Programs

Mathematica

RealDigits[Csc[Pi/7] - 1, 10, 120][[1]] (* Amiram Eldar, Dec 28 2023 *)

PARI

1/sin(Pi/7) - 1

Formula

Equals 1 / sin(Pi/7) - 1.

Equals A121598 - 1.

Largest of the 6 real-valued roots of 7*x^6+ 42*x^5 +49*x^4 -84*x^3 -119*x^2 +42*x-1=0. - R. J. Mathar, Aug 29 2025

Extensions

More digits from Jon E. Schoenfield, Dec 24 2023

Comments edited by Michal Paulovic, Dec 26 2023