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Some progress in the packing of equal circles in a square

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*. ...

##
###
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

1995
*
Journal of the Operational Research Society
*

Payan (1990)

doi:10.1057/jors.1995.70
fatcat:sy47v3a6vfg4todbji4jgrovzq
*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. ...##
###
Efficient deployment of connected sensing devices using circle packing algorithms

2010
*
2010 International Conference on Autonomous and Intelligent Systems, AIS 2010
*

We propose

doi:10.1109/ais.2010.5547023
dblp:conf/ais2/RamadanA10
fatcat:pnejsja53jddno5lwasfubophq
*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 ...##
###
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]

2004
*
arXiv
*
pre-print

We use computational experiments to find

arXiv:math/0405148v2
fatcat:poxuoqaarreglcopwv6dq52meq
*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), ...##
###
Dense Packings of Congruent Circles in Rectangles with a Variable Aspect Ratio
[chapter]

2003
*
Algorithms and Combinatorics
*

We use computational experiments to find

doi:10.1007/978-3-642-55566-4_28
fatcat:23uyzpsfwbclpfltikdeps6lb4
*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), ...##
###
Dense packings of congruent circles in a circle

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. ...

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

2013
*
2013 BRICS Congress on Computational Intelligence and 11th Brazilian Congress on Computational Intelligence
*

on

doi:10.1109/brics-cci-cbic.2013.91
fatcat:qzuawmbudjfiddf2wrim3wcfsu
*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 ...##
###
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

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. ...

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

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*. ...

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

2013
*
arXiv
*
pre-print

We describe how our three-weight version

arXiv:1305.1961v1
fatcat:2llsc2iysfatpke3lewhqmkt2e
*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. ...##
###
Circle Packing for Origami Design Is Hard
[article]

2010
*
arXiv
*
pre-print

On

arXiv:1008.1224v2
fatcat:pkp57wi5vrhi5nvnyxdmb7dqoy
*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 . ...##
###
Circle Packing for Origami Design Is Hard
[chapter]

2011
*
Origami 5
*

Acknowledgments We thank Ron Graham for several helpful hints concerning

doi:10.1201/b10971-52
fatcat:jlrpzg6z25acbmczzcxm2celfq
*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). ...
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