A281115 Decimal expansion of the greatest minimal separation between ten points in a unit circle.

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

Offset: 0

Author

Jeremy Tan, Jan 14 2017

Keywords

• nonn
• cons

Comments

The corresponding values for two to nine points are all of the form 2*sin(Pi/k), where k is the number of points N for N <= 6 and N-1 for N > 6. The value for ten points is the first that cannot be expressed in this form with k an integer, although it is still algebraic of degree 24.

The smallest circle ten unit circles will fit into has radius r = 1 + 2/d = 3.81302563... and the maximum radius of ten non-overlapping circles in the unit circle is 1 / r = 0.26225892...

Examples

			0.71097823556124655083072597690268...
		

Links

• U. Pirl, Der Mindestabstand von n in der Einheitskreisscheibe gelegenen Punkten, Mathematische Nachrichten 40 (1969), 111-124.
• Eckard Specht, The best known packings of equal circles in a circle
• Jeremy Tan, Sympy (Python) program
• Index entries for algebraic numbers, degree 24

Crossrefs

Cf. A281065 (10 points in unit square).

Programs

PARI

p = Pol([1, 0, -32, 0, 463, 0, -3998, 0, 22899, 0, -91428, 0, 260179, 0, -529874, 0, 763206, 0, -754052, 0, 481476, 0, -176440, 0, 27556]); polrootsreal(p)[5]

Formula

d is the smallest positive root of d^24 - 32*d^22 + 463*d^20 - 3998*d^18 + 22899*d^16 - 91428*d^14 + 260179*d^12 - 529874*d^10 + 763206*d^8 - 754052*d^6 + 481476*d^4 - 176440*d^2 + 27556.