A281065 Decimal expansion of the greatest minimum separation between ten points in a unit square.

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

Offset: 0

Author

Jeremy Tan, Jan 14 2017

Keywords

• nonn
• cons

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 (A381485)

7 ... 4 - 2*sqrt(3) (A379338)

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...

Examples

			0.421279543983903432768821760650298...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

Links

• C. de Groot, R. Peikert, and 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.
• Index entries for algebraic numbers, degree 18.

Crossrefs

Cf. A281115 (10 points in unit circle), A000007, A002193, A010503, A020761, A101263, A120683, A379338, A381485.

Programs

Mathematica

RealDigits[x /. FindRoot[x^Range[18, 0, -1].{1180129, -11436428, 98015844, -462103584, 1145811528, -1398966480, 227573920, 1526909568, -1038261808, -2960321792, 7803109440, -9722063488, 7918461504, -4564076288, 1899131648, -563649536, 114038784, -14172160, 819200}, {x, 2/5}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Feb 24 2025 *)

PARI

my(p = Pol([1180129, -11436428, 98015844, -462103584, 1145811528, -1398966480, 227573920, 1526909568, -1038261808, -2960321792, 7803109440, -9722063488, 7918461504, -4564076288, 1899131648, -563649536, 114038784, -14172160, 819200])); polrootsreal(p)[1]

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.