OFFSET
0,1
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...
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
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.
EXAMPLE
0.71097823556124655083072597690268...
PROG
(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]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jeremy Tan, Jan 14 2017
STATUS
approved