A281115 - OEIS

%I #10 Oct 27 2023 10:37:47

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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...

%H U. Pirl, <a href="http://dx.doi.org/10.1002/mana.19690400110">Der Mindestabstand von n in der Einheitskreisscheibe gelegenen Punkten</a>, Mathematische Nachrichten 40 (1969), 111-124.

%H Eckard Specht, <a href="http://hydra.nat.uni-magdeburg.de/packing/cci/">The best known packings of equal circles in a circle</a>

%H Jeremy Tan, <a href="https://gist.github.com/Parclytaxel/fe51678ea07cc448c56c3927afc44ac1">Sympy (Python) program</a>

%H <a href="/index/Al#algebraic_24">Index entries for algebraic numbers, degree 24</a>

%F 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.

%e 0.71097823556124655083072597690268...

%o (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]

%Y Cf. A281065 (10 points in unit square).

%K nonn,cons

%O 0,1

%A _Jeremy Tan_, Jan 14 2017