A343851 Decimal expansion of the solution to the Heilbronn triangle problem for seven points in a unit square.

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

Offset: 0

Author

Jeremy Tan, May 03 2021

Keywords

• nonn
• cons

Comments

The Heilbronn triangle problem: find an arrangement of n points in a convex region such that the minimum area among triangles formed by three of the points is maximized.

The seven-point configuration in the square was found by Comellas and Yebra and proved optimal by Chen and Zeng.

Examples

			0.08385900900751340663796674354476...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.16, p. 527.

Links

• Liangyu Chen and Zhenbing Zeng, On the Heilbronn Optimal Configuration of Seven Points in the Square, Automated Deduction in Geometry, Springer-Verlag, 2011, pp. 196-224.
• Francesc Comellas and J. Luis A. Yebra, New Lower Bounds for Heilbronn Numbers, Electronic Journal of Combinatorics, 9 (2002).
• Erich Friedman, The Heilbronn Problem for Squares.
• Eric Weisstein's World of Mathematics, Heilbronn Triangle Problem.
• Index entries for algebraic numbers, degree 3.

Crossrefs

Cf. A248866 (discrete Heilbronn triangle problem).

Programs

Mathematica

First@ RealDigits@ N[Root[152x^3+12x^2-14x+1, x, 2], 105]

PARI

polrootsreal(152*x^3+12*x^2-14*x+1)[2]

Formula

This is the smallest positive root of 152x^3 + 12x^2 - 14x + 1.