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Some progress in the packing of equal circles in a square

Michael Mollard, Charles Payan
1990 Discrete Mathematics  
Packing 10 circles In a square ABCD of side s -1 let us define (whenever possible) the points 4, p2, . . . , P9 as shown in Fig. 2 where PI = A, P3 is a point of AD at distance x from D, and Pz, P4,  ...  A classical problem is to find the smallest side s of such a square.  ... 
doi:10.1016/0012-365x(90)90135-5 fatcat:ec6kkrf6cre7lfsxrybejunvgu

Page 520 of The Journal of the Operational Research Society Vol. 46, Issue 4 [page]

1995 The Journal of the Operational Research Society  
Payan (1990) Some progress in the packing of equal circles in a square. Discrete Math. 84, 303-307. . J. SCHAER (1966) On the densest packing of spheres in a cube. Canadian Math. Bull. 9, 265-280. .  ...  KerniGHaN (1993) AMPL: a Modeling Language for Mathematical Programming. The Scientific Press, South San Francisco. . M. Go_pBerc (1970) The packing of equal circles in a square. Math.  ... 

Solving the Continuous p-Dispersion Problem Using Non-linear Programming

Zvi Drezner, Erhan Erkut
1995 Journal of the Operational Research Society  
Payan (1990) Some progress in the packing of equal circles in a square. Discrete Math. 84, 303-307. . J. SCHAER (1966) On the densest packing of spheres in a cube. Canadian Math. Bull. 9, 265-280. .  ...  It is shown that this problem, in a square, is equivalent to the problem of packing the square with p equal circles of largest possible radius.  ... 
doi:10.1057/jors.1995.70 fatcat:sy47v3a6vfg4todbji4jgrovzq

Efficient deployment of connected sensing devices using circle packing algorithms

Rabie A. Ramadan, Salah Abdel-Mageid
2010 2010 International Conference on Autonomous and Intelligent Systems, AIS 2010  
We propose a novel algorithm named Sequential Packing-based Deployment Algorithm (SPDA) for the deployment of heterogeneous sensors in order to maximize the coverage of the monitored field and connectivity  ...  The algorithm is inspired from the packing theories in computational geometry where it benefits from many of the observations properties that are captured from the optimal packing solutions.  ...  Some of these forms are stated as follows: 1) Find the maximum circle radius r n , such that n equal no-overlapping circles fit in a given square. 2) Locate n points in a given square, such that the minimum  ... 
doi:10.1109/ais.2010.5547023 dblp:conf/ais2/RamadanA10 fatcat:pnejsja53jddno5lwasfubophq

Page 4784 of Mathematical Reviews Vol. , Issue 99g [page]

1999 Mathematical Reviews  
Summary: “The problem of packing n equal circles with as large a radius as possible in a given square is considered.  ...  Summary: “How must 2N non-overlapping equal circles forming antipodal pairs be packed on a sphere so that the angular diameter of the circles will be as great as possible?  ... 

Dense Packings of Congruent Circles in Rectangles with a Variable Aspect Ratio [article]

Boris D. Lubachevsky, Ronald Graham
2004 arXiv   pre-print
We use computational experiments to find the rectangles of minimum area into which a given number n of non-overlapping congruent circles can be packed.  ...  No assumption is made on the shape of the rectangles. Most of the packings found have the usual regular square or hexagonal pattern.  ...  Packings of 49 circles in a rectangle: a) the best in the class of hexagonal packings, b) a best in the class of hexagonal packings with monovacancies (one of 17 equally dense packings with the hole),  ... 
arXiv:math/0405148v2 fatcat:poxuoqaarreglcopwv6dq52meq

Dense Packings of Congruent Circles in Rectangles with a Variable Aspect Ratio [chapter]

Boris D. Lubachevsky, Ronald Graham
2003 Algorithms and Combinatorics  
We use computational experiments to find the rectangles of minimum area into which a given number n of non-overlapping congruent circles can be packed.  ...  No assumption is made on the shape of the rectangles. Most of the packings found have the usual regular square or hexagonal pattern.  ...  Packings of 49 circles in a rectangle: a) the best in the class of hexagonal packings, b) a best in the class of hexagonal packings with monovacancies (one of 17 equally dense packings with the hole),  ... 
doi:10.1007/978-3-642-55566-4_28 fatcat:23uyzpsfwbclpfltikdeps6lb4

Dense packings of congruent circles in a circle

R.L. Graham, B.D. Lubachevsky, K.J. Nurmela, P.R.J. Östergård
1998 Discrete Mathematics  
The problem of finding packings of congruent circles in a circle, or, equivalently, of spreading points in a circle, is considered.  ...  Two packing algorithms are discussed, and the best packings found of up to 65 circles are presented.  ...  In [16] it was conjectured that given a packing of n ~> 2 circles in a square, there exists a packing of n -1 circles in the square with greater value of d.  ... 
doi:10.1016/s0012-365x(97)00050-2 fatcat:jlbprce43jbkpkweyxortl3e7i

On Solving Mixed Shapes Packing Problems by Continuous Optimization with the CMA Evolution Strategy

Thierry Martinez, Lumadaiara Vitorino, Francois Fages, Abderrahmane Aggoun
2013 2013 BRICS Congress on Computational Intelligence and 11th Brazilian Congress on Computational Intelligence  
on a benchmark of circle packing problems.  ...  We then consider generalizations of this benchmark to mixed squares and circles, boxes, spheres and cylinders packing problems, and study a real-world problem for loading boxes and cylinders in containers  ...  The authors would like to thank the partners of this project, Fernando Buarque and Anthony Lins for interesting discussions on other strategies than CMA-ES and multi-modal FSS, and the reviewers for their  ... 
doi:10.1109/brics-cci-cbic.2013.91 fatcat:qzuawmbudjfiddf2wrim3wcfsu

Page 2073 of Mathematical Reviews Vol. , Issue 92d [page]

1992 Mathematical Reviews  
Florian (A-SALZ) 92d:52043 52C15 Mollard, Michel (F-INPGAM-DS); Payan, Charles (F-INPGAM-DS) Some progress in the packing of equal circles in a square. Discrete Math. 84 (1990), no. 3, 303-307.  ...  In a closed unit square, pack k congruent circles so that their radius and hence their density of packing is maximal in the square. This problem was solved for k < 9, for k = 14 [G.  ... 

Optimal packings of up to six equal circles on a triangular flat torus

William Dickinson, Daniel Guillot, Anna Keaton, Sandi Xhumari
2011 Journal of Geometry  
The study of equal circle packings on the standard triangular torus is related to this conjecture.  ...  Equal Circle Packings On Flat Tori Consider the lattice, Λ, generated by two linearly independent basis vectors v 1 and v 2 in E 2 . The quotient of the plane by this lattice is called a flat torus.  ...  The same techniques are used in [4] in the study of 1-5 equal circles on the square flat torus. There is an outstanding case of a conjecture of L.  ... 
doi:10.1007/s00022-011-0099-6 fatcat:stk5miywnjdgrfbd22s7vz72iq

Solving the problem of packing equal and unequal circles in a circular container

A. Grosso, A. R. M. J. U. Jamali, M. Locatelli, F. Schoen
2009 Journal of Global Optimization  
In this paper we propose a Monotonic Basin Hopping approach and its population-based variant Population Basin Hopping to solve the problem of packing equal and unequal circles within a circular container  ...  Different improvements with respect to the best results reported in the literature have been detected.  ...  Much literature exists about the problem of packing equal circles in a square.  ... 
doi:10.1007/s10898-009-9458-3 fatcat:65fru3k6iva43p7d7myqeou4lq

An Improved Three-Weight Message-Passing Algorithm [article]

Nate Derbinsky, José Bento, Veit Elser, Jonathan S. Yedidia
2013 arXiv   pre-print
We describe how our three-weight version of ADMM/DC can give greatly improved performance for non-convex problems such as circle packing and solving large Sudoku puzzles, while retaining the exact performance  ...  We describe how the powerful "Divide and Concur" algorithm for constraint satisfaction can be derived as a special case of a message-passing version of the Alternating Direction Method of Multipliers (  ...  CIRCLE PACKING Circle packing is the problem of positioning a given number of congruent circles in such a way that the circles fit fully in a square without overlapping.  ... 
arXiv:1305.1961v1 fatcat:2llsc2iysfatpke3lewhqmkt2e

Circle Packing for Origami Design Is Hard [article]

Erik D. Demaine, Sandor P. Fekete, Robert J. Lang
2010 arXiv   pre-print
On the positive side, we show that any set of circles of total area 1 can be packed into a square of size 4/√(pi)=2.2567...  ...  We show that deciding whether a given set of circles can be packed into a rectangle, an equilateral triangle, or a unit square are NP-hard problems, settling the complexity of these natural packing problems  ...  Acknowledgments We thank Ron Graham for several helpful hints concerning the state of the art on packing circles. We also thank Vinayak Pathak for pointing out a numerical typo related to Figure 13 .  ... 
arXiv:1008.1224v2 fatcat:pkp57wi5vrhi5nvnyxdmb7dqoy

Circle Packing for Origami Design Is Hard [chapter]

Erik Demaine, Sandor Fekete, Robert Lang
2011 Origami 5  
Acknowledgments We thank Ron Graham for several helpful hints concerning the state of the art on packing circles. We also thank Vinayak Pathak for pointing out a numerical typo related to Figure 13 .  ...  Now a recursive subdivision of S into sub-squares of progressively smaller size can be used to pack all squares S i , showing that all circles C i can be packed.  ...  Thus, several problems in origami design can be reduced to finding an optimum packing of some number of circles of specified radii within a square (or other convex polygon).  ... 
doi:10.1201/b10971-52 fatcat:jlrpzg6z25acbmczzcxm2celfq
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