OFFSET
0,1
COMMENTS
The corresponding values for two to nine points have simple expressions:
N ... d_min
2 ... sqrt(2) (A002193)
3 ... sqrt(6) - sqrt(2) (A120683)
4 ... 1 (A000007)
5 ... sqrt(2) / 2 (A010503)
6 ... sqrt(13) / 6
7 ... 4 - 2*sqrt(3)
8 ... sqrt(2 - sqrt(3)) (A101263)
9 ... 1 / 2 (A020761)
In contrast, the value for ten points has a minimal polynomial of degree 18.
The smallest square ten unit circles will fit into has side length s = 2 + 2/d = 6.74744152... and the maximum radius of ten non-overlapping circles in the unit square is 1 / s = 0.14820432...
LINKS
C. de Groot, R. Peikert, D. Würtz, The Optimal Packing of Ten Equal Circles in a Square, IPS Research Report, ETH Zürich, No. 90-12, August 1990.
Eckard Specht, The best known packings of equal circles in a square
Jeremy Tan, Sympy (Python) program
FORMULA
d is the smallest real root of 1180129*d^18 - 11436428*d^17 + 98015844*d^16 - 462103584*d^15 + 1145811528*d^14 - 1398966480*d^13 + 227573920*d^12 + 1526909568*d^11 - 1038261808*d^10 - 2960321792*d^9 + 7803109440*d^8 - 9722063488*d^7 + 7918461504*d^6 - 4564076288*d^5 + 1899131648*d^4 - 563649536*d^3 + 114038784*d^2 - 14172160*d + 819200.
EXAMPLE
0.421279543983903432768821760650298...
PROG
(PARI) p = Pol([1180129, -11436428, 98015844, -462103584, 1145811528, -1398966480, 227573920, 1526909568, -1038261808, -2960321792, 7803109440, -9722063488, 7918461504, -4564076288, 1899131648, -563649536, 114038784, -14172160, 819200]); polrootsreal(p)[1]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jeremy Tan, Jan 14 2017
STATUS
approved