The Densest Packing of 9 Circles in a Square

J. Schaer

doi:10.4153/CMB-1965-018-9

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Packing problems of this kind are obviously equivalent to the problems of placing k (here 9) points in a unit square such that the minimum distance between any two of them be as large as possible. The solutions of these problems are known for 2 ≤ k ≤ 9. The largest possible minimum distances mk are given in table 1, and the corresponding "best" configurations shown in figure 1.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Goldberg, Michael 1970. The Packing of Equal Circles in a Square. Mathematics Magazine, Vol. 43, Issue. 1, p. 24.

Schaer, J. 1971. On the Packing of Ten Equal Circles in a Square. Mathematics Magazine, Vol. 44, Issue. 3, p. 139.

Goldberg, Michael 1971. On the Densest Packing of Equal Spheres in a Cube. Mathematics Magazine, Vol. 44, Issue. 4, p. 199.

Baranovskii, E. P. 1971. Progress in Mathematics. p. 209.

Byrnes, J. S. 1971. The Massivity of the Square. The American Mathematical Monthly, Vol. 78, Issue. 4, p. 376.

Schwartz, Benjamin L. 1973. Separating Points in a Rectangle. Mathematics Magazine, Vol. 46, Issue. 2, p. 62.

Saaty, Thomas L. and Alexander, Joyce M. 1975. Optimization and the Geometry of Numbers: Packing and Covering. SIAM Review, Vol. 17, Issue. 3, p. 475.

Tóth, G. Fejes 1983. Convexity and Its Applications. p. 318.

Moser, W. O. J. 1985. PROBLEMS ON EXTREMAL PROPERTIES OF A FINITE SET OF POINTS. Annals of the New York Academy of Sciences, Vol. 440, Issue. 1, p. 52.

Valette, Guy 1989. A better packing of ten equal circles in a square. Discrete Mathematics, Vol. 76, Issue. 1, p. 57.

Mollard, Michael and Payan, Charles 1990. Some progress in the packing of equal circles in a square. Discrete Mathematics, Vol. 84, Issue. 3, p. 303.

Peikert, R. Würtz, D. Monagan, M. and de Groot, C. 1992. System Modelling and Optimization. Vol. 180, Issue. , p. 45.

Melissen, Hans (J. B. M.) 1993. Densest Packings of Congruent Circles in an Equilateral Triangle. The American Mathematical Monthly, Vol. 100, Issue. 10, p. 916.

Melissen, Hans 1994. Densest packings of eleven congruent circles in a circle. Geometriae Dedicata, Vol. 50, Issue. 1, p. 15.

Melissen, J.B.M. and Schuur, P.C. 1995. Packing 16, 17 or 18 circles in an equilateral triangle. Discrete Mathematics, Vol. 145, Issue. 1-3, p. 333.

Maranas, Costas D. Floudas, Christodoulos A. and Pardalos, Panos M. 1995. New results in the packing of equal circles in a square. Discrete Mathematics, Vol. 142, Issue. 1-3, p. 287.

Xu, Yinfeng 1996. On the minimum distance determined byn (≤ 7) points in an isoscele right triangle. Acta Mathematicae Applicatae Sinica, Vol. 12, Issue. 2, p. 169.

Melisseny, J. B. M. 1998. How different can colours be? Maximum separation of points on a spherical octant. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Vol. 454, Issue. 1973, p. 1499.

Szabó, P. G. Csendes, T. Casado, L. G. and García, I. 2001. Optimization Theory. Vol. 59, Issue. , p. 191.

Wenqi, Huang and Yan, Kang 2002. A heuristic quasi-physical strategy for solving disks packing problem. Simulation Modelling Practice and Theory, Vol. 10, Issue. 3-4, p. 195.

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The Densest Packing of 9 Circles in a Square

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This article has been cited by the following publications. This list is generated based on data provided by Crossref.

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