Best known packings of equal circles in a square

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Numbers

$n$

normalisation

$d_n$ or $S_n$

$d_n$:

1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

comment: The exact container side is $S_{2}=2+\sqrt2$ [8]; $d_{2}=2/(S_{2}-2)$. Optimality is proven; see [1] and [2].

$S_n$:

3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

comment: The exact container side is $S_{2}=2+\sqrt2$ [8]; $d_{2}=2/(S_{2}-2)$. Optimality is proven; see [1] and [2].

$d_n$:

1.035276180410083049395595350496193313396275605279722055256012829260227898995207987689471898776998662

comment: The exact container side is $S_{3}=2+(\sqrt2+\sqrt6)/2$ [8]; $d_{3}=2/(S_{3}-2)$. Optimality is proven; see [1] and [2].

$S_n$:

3.931851652578136573499486399457794735267809678016809100804686152620846427959711032695123483716140904

comment: The exact container side is $S_{3}=2+(\sqrt2+\sqrt6)/2$ [8]; $d_{3}=2/(S_{3}-2)$. Optimality is proven; see [1] and [2].

$d_n$:

comment: The $2\times2$ square grid gives $S_{4}=4$ and $d_{4}=1/1$. Optimality is proven; see [1] and [2].

$S_n$:

comment: The $2\times2$ square grid gives $S_{4}=4$ and $d_{4}=1/1$. Optimality is proven; see [1] and [2].

$d_n$:

0.7071067811865475244008443621048490392848359376884740365883398689953662392310535194251937671638207864

comment: The exact container side is $S_{5}=2+2\sqrt2$ [8]; $d_{5}=2/(S_{5}-2)$. Optimality is proven; see [1] and [2].

$S_n$:

4.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: The exact container side is $S_{5}=2+2\sqrt2$ [8]; $d_{5}=2/(S_{5}-2)$. Optimality is proven; see [1] and [2].

$d_n$:

0.6009252125773315488532035445784159910418827623075410354517421760378611580488350742007698470030817863

comment: The exact container side is $S_{6}=2+12/\sqrt{13}$ [8]; $d_{6}=2/(S_{6}-2)$. Optimality is proven; see [1] and [2].

$S_n$:

5.328201177351374732110050400741996258078119914318688811732725898055846413808932718650417614170914509

comment: The exact container side is $S_{6}=2+12/\sqrt{13}$ [8]; $d_{6}=2/(S_{6}-2)$. Optimality is proven; see [1] and [2].

$d_n$:

0.5358983848622454129451073169882552661143894923792387438883860410961339661823999258377076264855028486

comment: The exact container side is $S_{7}=4+\sqrt3$ [8]; $d_{7}=2/(S_{7}-2)$. Optimality is proven; see [1] and [2].

$S_n$:

5.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: The exact container side is $S_{7}=4+\sqrt3$ [8]; $d_{7}=2/(S_{7}-2)$. Optimality is proven; see [1] and [2].

$d_n$:

0.5176380902050415246977976752480966566981378026398610276280064146301139494976039938447359493884993310

comment: The exact container side is $S_{8}=2+\sqrt2+\sqrt6$ [8]; $d_{8}=2/(S_{8}-2)$. Optimality is proven; see [1] and [2].

$S_n$:

5.863703305156273146998972798915589470535619356033618201609372305241692855919422065390246967432281808

comment: The exact container side is $S_{8}=2+\sqrt2+\sqrt6$ [8]; $d_{8}=2/(S_{8}-2)$. Optimality is proven; see [1] and [2].

$d_n$:

1/2

comment: The $3\times3$ square grid gives $S_{9}=6$ and $d_{9}=1/2$. Optimality is proven; see [1] and [2].

$S_n$:

comment: The $3\times3$ square grid gives $S_{9}=6$ and $d_{9}=1/2$. Optimality is proven; see [1] and [2].

$d_n$:

0.4212795440

comment: Optimality is proven; see [1] and [2].

$S_n$:

6.747441523

comment: Optimality is proven; see [1] and [2].

$d_n$:

0.3982073102

comment: Optimality is proven; see [1] and [2].

$S_n$:

7.022509503

comment: Optimality is proven; see [1] and [2].

$d_n$:

0.3887301263230200313916101918363722051014265556590647969633337829911873374664475085110160217635356932

comment: The exact container side is $S_{12}=2+15\sqrt{2/17}$ [8]; $d_{12}=2/(S_{12}-2)$. Optimality is proven; see [1] and [2].

$S_n$:

7.144957554275265121359546656657867420460057354311151724514711833706891231173569965586976758635031234

comment: The exact container side is $S_{12}=2+15\sqrt{2/17}$ [8]; $d_{12}=2/(S_{12}-2)$. Optimality is proven; see [1] and [2].

$d_n$:

0.3660960077

comment: Optimality is proven; see [1] and [2].

$S_n$:

7.463047829

comment: Optimality is proven; see [1] and [2].

$d_n$:

0.3489152603740188779188544089990965589318761147984029802991066185458564589371076866029005866527309884

comment: The exact container side is $S_{14}=6+\sqrt3$ [8]; $d_{14}=2/(S_{14}-2)$. Optimality is proven; see [1] and [2].

$S_n$:

7.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: The exact container side is $S_{14}=6+\sqrt3$ [8]; $d_{14}=2/(S_{14}-2)$. Optimality is proven; see [1] and [2].

$d_n$:

0.3410813774021088776371211913519128558134333107832837225604363792693997307766535008846206737851964985

comment: The exact container side is $S_{15}=4+\sqrt2+\sqrt6$ [8]; $d_{15}=2/(S_{15}-2)$. Optimality is proven; see [1] and [2].

$S_n$:

7.863703305156273146998972798915589470535619356033618201609372305241692855919422065390246967432281808

comment: The exact container side is $S_{15}=4+\sqrt2+\sqrt6$ [8]; $d_{15}=2/(S_{15}-2)$. Optimality is proven; see [1] and [2].

$d_n$:

1/3

comment: The $4\times4$ square grid gives $S_{16}=8$ and $d_{16}=1/3$. Optimality is proven; see [1] and [2].

$S_n$:

comment: The $4\times4$ square grid gives $S_{16}=8$ and $d_{16}=1/3$. Optimality is proven; see [1] and [2].

$d_n$:

0.3061539853

comment: Optimality is proven; see [1] and [2].

$S_n$:

8.532660348

comment: Optimality is proven; see [1] and [2].

$d_n$:

0.3004626062886657744266017722892079955209413811537705177258710880189305790244175371003849235015408931

comment: The exact container side is $S_{18}=2+24/\sqrt{13}$ [8]; $d_{18}=2/(S_{18}-2)$. Optimality is proven; see [1] and [2].

$S_n$:

8.656402354702749464220100801483992516156239828637377623465451796111692827617865437300835228341829017

comment: The exact container side is $S_{18}=2+24/\sqrt{13}$ [8]; $d_{18}=2/(S_{18}-2)$. Optimality is proven; see [1] and [2].

$d_n$:

0.2895419920

comment: Optimality is proven; see [1] and [2].

$S_n$:

8.907460939

comment: Optimality is proven; see [1] and [2].

$d_n$:

0.2866116523516815594498944547368938700893955077889407454264575163755792200961183100718507791045224017

comment: The exact container side is $S_{20}=(130+16\sqrt2)/17$ [8]; $d_{20}=2/(S_{20}-2)$. Optimality is proven; see [1] and [2].

$S_n$:

8.978083352821736516519236446315009956300867647413598186519227988697159979729041918918011797014250892

comment: The exact container side is $S_{20}=(130+16\sqrt2)/17$ [8]; $d_{20}=2/(S_{20}-2)$. Optimality is proven; see [1] and [2].

$d_n$:

0.2718122554

comment: Optimality is proven in [2].

$S_n$:

9.358019959

comment: Optimality is proven in [2].

$d_n$:

0.2679584016

comment: Optimality is proven in [2].

$S_n$:

9.463845091

comment: Optimality is proven in [2].

$d_n$:

0.2588190451025207623488988376240483283490689013199305138140032073150569747488019969223679746942496655

comment: The exact container side is $S_{23}=2+2\sqrt2+2\sqrt6$ [8]; $d_{23}=2/(S_{23}-2)$. Optimality is proven in [2].

$S_n$:

9.727406610312546293997945597831178941071238712067236403218744610483385711838844130780493934864563615

comment: The exact container side is $S_{23}=2+2\sqrt2+2\sqrt6$ [8]; $d_{23}=2/(S_{23}-2)$. Optimality is proven in [2].

$d_n$:

0.2543330950302498177547447604286624495125095981933858805208757630505942714763599372265343880384328673

comment: The exact container side is $S_{24}=6+\sqrt2+\sqrt6$ [8]; $d_{24}=2/(S_{24}-2)$. Optimality is proven in [2].

$S_n$:

9.863703305156273146998972798915589470535619356033618201609372305241692855919422065390246967432281808

comment: The exact container side is $S_{24}=6+\sqrt2+\sqrt6$ [8]; $d_{24}=2/(S_{24}-2)$. Optimality is proven in [2].

$d_n$:

1/4

comment: The $5\times5$ square grid gives $S_{25}=10$ and $d_{25}=1/4$. Optimality is proven in [2].

$S_n$:

comment: The $5\times5$ square grid gives $S_{25}=10$ and $d_{25}=1/4$. Optimality is proven in [2].

$d_n$:

0.2387347572

comment: Optimality is proven in [2].

$S_n$:

10.37749820

comment: Optimality is proven in [2].

$d_n$:

0.2358495283014150952830165094405660179245905658353780330165745362245005727396279514635256746722596026

comment: The exact container side is $S_{27}=2+80/\sqrt{89}$ [9]; $d_{27}=2/(S_{27}-2)$. Optimality is proven in [2].

$S_n$:

10.47998304005087983040059359786304783549089674913718770396672489795957115468325218745260852754191830

comment: The exact container side is $S_{27}=2+80/\sqrt{89}$ [9]; $d_{27}=2/(S_{27}-2)$. Optimality is proven in [2].

$d_n$:

0.2305354936

comment: Optimality is proven by the interval computations in [3] and the structural verification in [4].

$S_n$:

10.67545369

comment: Optimality is proven by the interval computations in [3] and the structural verification in [4].

$d_n$:

0.2268829007

comment: Optimality is proven by the interval computations in [3] and the structural verification in [4].

$S_n$:

10.81512002

comment: Optimality is proven by the interval computations in [3] and the structural verification in [4].

$d_n$:

0.2245029645310882755733480860742304195504059314756637756418999352960987408181434903820766918882759961

comment: The exact container side is $S_{30}=(252+10\sqrt{10})/26$ [9]; $d_{30}=2/(S_{30}-2)$. Optimality is proven by the interval computations in [3] and the structural verification in [4].

$S_n$:

10.90856833083399205076880520939719943604598274589431416417596340492022863024586085436317234937665396

comment: The exact container side is $S_{30}=(252+10\sqrt{10})/26$ [9]; $d_{30}=2/(S_{30}-2)$. Optimality is proven by the interval computations in [3] and the structural verification in [4].

$d_n$:

0.2175472916191

comment: The global optimum is enclosed by the certified interval in [5], Section 7.5.

$S_n$:

11.193403352047

comment: The global optimum is enclosed by the certified interval in [5], Section 7.5.

$d_n$:

0.21317456258988

comment: The global optimum is enclosed by the certified interval in [5], Section 7.5.

$S_n$:

11.381982426523

comment: The global optimum is enclosed by the certified interval in [5], Section 7.5.

$d_n$:

0.21132838414326

comment: The global optimum is enclosed by the certified interval in [5], Section 7.5.

$S_n$:

11.463944032450

comment: The global optimum is enclosed by the certified interval in [5], Section 7.5.

$d_n$:

0.2056046468

$S_n$:

11.72740661

$d_n$:

0.2027636009

$S_n$:

11.86370331

$d_n$:

1/5

comment: The $6\times6$ square grid gives $S_{36}=12$ and $d_{36}=1/5$. Optimality is proven in [6].

$S_n$:

comment: The $6\times6$ square grid gives $S_{36}=12$ and $d_{36}=1/5$. Optimality is proven in [6].

$d_n$:

0.1964291846

$S_n$:

12.18178640

$d_n$:

0.1953423041

$S_n$:

12.23843764

$d_n$:

0.1943650631615100156958050959181861025507132778295323984816668914955936687332237542555080108817678466

comment: The exact container side is $S_{39}=2+30\sqrt{2/17}$ [9]; $d_{39}=2/(S_{39}-2)$. This is a best known configuration; global optimality is not asserted.

$S_n$:

12.28991510855053024271909331331573484092011470862230344902942366741378246234713993117395351727006247

comment: The exact container side is $S_{39}=2+30\sqrt{2/17}$ [9]; $d_{39}=2/(S_{39}-2)$. This is a best known configuration; global optimality is not asserted.

$d_n$:

0.1881755220

$S_n$:

12.62837493

$d_n$:

0.1860995118

$S_n$:

12.74693845

$d_n$:

0.1842770721170989264787854674790851060146832603637447134992249030631675969873777772627618585172604364

comment: The exact container side is $S_{42}=(434+24\sqrt3)/37$ [9]; $d_{42}=2/(S_{42}-2)$. This is a best known configuration; global optimality is not asserted.

$S_n$:

12.85322214545008256877455978908489018396290070517430094792809101369855114610300542945804076978848556

comment: The exact container side is $S_{42}=(434+24\sqrt3)/37$ [9]; $d_{42}=2/(S_{42}-2)$. This is a best known configuration; global optimality is not asserted.

$d_n$:

0.1801911355

$S_n$:

13.09932514

$d_n$:

0.1786392457

$S_n$:

13.19574813

$d_n$:

0.1757163142

$S_n$:

13.38198243

$d_n$:

0.1744593609

$S_n$:

13.46398789

$d_n$:

0.1712705638

$S_n$:

13.67742988

$d_n$:

0.1694054294

$S_n$:

13.80599705

$d_n$:

0.1673860769

$S_n$:

13.94842509

$d_n$:

0.1665265773

$S_n$:

14.01009492

$d_n$:

0.1656183743

$S_n$:

14.07595479

$d_n$:

0.1653862379696405298622922266789113622537784924086634681842429561429769002206880471341834986552507480

comment: The exact container side is $S_{52}=2+168/\sqrt{193}$ [9]; $d_{52}=2/(S_{52}-2)$. This is a best known configuration; global optimality is not asserted.

$S_n$:

14.09290461257806817314335700980723908873223878171533089645614817144917106691372912516889913483367127

comment: The exact container side is $S_{52}=2+168/\sqrt{193}$ [9]; $d_{52}=2/(S_{52}-2)$. This is a best known configuration; global optimality is not asserted.

$d_n$:

0.1626480774

$S_n$:

14.29648719

$d_n$:

0.1591395163

$S_n$:

14.56758878

$d_n$:

0.1575557475

$S_n$:

14.69391965

$d_n$:

0.1561565004621471780180255153782997987683427762947807063416173654395100605865022896817273065502270222

comment: The exact container side is $S_{56}=(688+14\sqrt{14})/50$ [9]; $d_{56}=2/(S_{56}-2)$. This is a best known configuration; global optimality is not asserted.

$S_n$:

14.80766406829670358796344964504863380449168554617804354496504873084960984376594292783350677065442628

comment: The exact container side is $S_{56}=(688+14\sqrt{14})/50$ [9]; $d_{56}=2/(S_{56}-2)$. This is a best known configuration; global optimality is not asserted.

$d_n$:

0.1547474069

$S_n$:

14.92428765

$d_n$:

0.1526925314

$S_n$:

15.09821759

$d_n$:

0.1515629506

$S_n$:

15.19583706

$d_n$:

0.1495056540

$S_n$:

15.37742049

$d_n$:

0.1485441267

$S_n$:

15.46401264

$d_n$:

0.1474526798

$S_n$:

15.56367346

$d_n$:

0.1468193137

$S_n$:

15.62218600

$d_n$:

0.1453677545

$S_n$:

15.75820936

$d_n$:

0.1446990148

$S_n$:

15.82179418

$d_n$:

0.1438042308

$S_n$:

15.90779665

$d_n$:

0.1430855759

$S_n$:

15.97764930

$d_n$:

0.1429095939

$S_n$:

15.99486168

$d_n$:

0.1399481819

$S_n$:

16.29100381

$d_n$:

0.1379067767

$S_n$:

16.50255055

$d_n$:

0.1366129972

$S_n$:

16.63989547

$d_n$:

0.1358499279

$S_n$:

16.72212780

$d_n$:

0.1347287259

$S_n$:

16.84464421

$d_n$:

0.1339986727

$S_n$:

16.92552098

$d_n$:

0.1324888137

$S_n$:

17.09561406

$d_n$:

0.1317300376

$S_n$:

17.18256607

$d_n$:

0.1308445442

$S_n$:

17.28531443

$d_n$:

0.1304607729

$S_n$:

17.33027864

$d_n$:

0.1299652027

$S_n$:

17.38873451

$d_n$:

0.1296133855

$S_n$:

17.43050505

$d_n$:

0.1283368560

$S_n$:

17.58398782

$d_n$:

0.1274269118

$S_n$:

17.69527168

$d_n$:

0.1264543532

$S_n$:

17.81598379

$d_n$:

0.1257627023

$S_n$:

17.90296617

$d_n$:

0.1253155486

$S_n$:

17.95971149

$d_n$:

0.1250441565

$S_n$:

17.99434996

$d_n$:

0.1228265832

$S_n$:

18.28312006

$d_n$:

0.1220992982

$S_n$:

18.38011053

$d_n$:

0.1209431296

$S_n$:

18.53669792

$d_n$:

0.1204480495

$S_n$:

18.60466906

$d_n$:

0.1198126022

$S_n$:

18.69273485

$d_n$:

0.1193622666

$S_n$:

18.75571398

$d_n$:

0.1183841706

$S_n$:

18.89415054

$d_n$:

0.1180563848

$S_n$:

18.94105748

$d_n$:

0.1171196421

$S_n$:

19.07655491

$d_n$:

0.1167579991

$S_n$:

19.12944736

$d_n$:

0.1163574840

$S_n$:

19.18840879

$d_n$:

0.1161536146

$S_n$:

19.21857737

$d_n$:

0.1160181348430354115768470534308694341160733412320537204518471982988654022495278829154476731850606988

comment: The exact container side is $S_{99}=2+340/\sqrt{389}$ [9]; $d_{99}=2/(S_{99}-2)$. This is a best known configuration; global optimality is not asserted.

$S_n$:

19.23868430315539020344925369744023982495896946841312350158552200944581041136943864399196789227894188

comment: The exact container side is $S_{99}=2+340/\sqrt{389}$ [9]; $d_{99}=2/(S_{99}-2)$. This is a best known configuration; global optimality is not asserted.

100

$d_n$:

0.1145813522

100

$S_n$:

19.45484725

Definition

For $n\geq2$, $d_n$ is the best known minimum Euclidean distance among $n$ points in $[0,1]^2$ [7]. The equivalent square side for $n$ non-overlapping circles of radius 1 is $S_n=2+2/d_n$. These are two invertible normalisations of the same packing.

Parameters

$n$

— number of circles ($n\geq2$)

normalisation

— whether the row holds the point separation or the container side length (either separation or container)

Formulas

(1)

$S_n=2+2/d_n$, equivalently $d_n=2/(S_n-2)$. Scaling a unit-square point configuration by $2/d_n$ makes the minimum separation 2; a margin of 1 on each side then contains the unit circles.

(2)

If $r_n$ is the radius of the equal circles in a square of side 1, then $r_n=1/S_n=d_n/(2+2d_n)$ and $d_n=2r_n/(1-2r_n)$.

(3)

For $k=2,3,4,5,6$, the $k\times k$ square grid is optimal: $d_{k^2}=1/(k-1)$ and $S_{k^2}=2k$ [1] [6].

Comments

(4)

The separation $d_n$ concerns points in a square of side 1. The container side length $S_n$ concerns circles of radius 1, not diameter 1. The table starts at $n=2$ because the minimum pairwise separation is undefined for one point; the elementary one-circle value $S_1=2$ is not listed.

(5)

The Packomania index [7] is dated 10-Sep-2026. Its HTML values and coordinate files were copied on 25-Oct-2026. The attached source file preserves those values, all coordinates for the listed configurations, and file hashes. Radical formulas are recorded in the entry comments from the collection [8] and the constructions [9]. The certified bounds for $n=31,32,33$ come from [5], Section 7.5.

(6)

Optimality is proven for $2\leq n\leq20$ [1], for $21\leq n\leq27$ [2], for $28\leq n\leq30$ [3] [4], and for $n=36$ [6]. For $n=31,32,33$, [5] gives certified enclosures of the global optimum, which supply the corresponding rows. For $n=34,35$ and $37\leq n\leq100$, this table makes no claim of proven global optimality.

(7)

Where optimality is unproved, the listed decimals describe a best known configuration. Their numerical uncertainty does not bound the possible improvement from a better packing. In particular, a radical expression may specify the known configuration exactly without proving that configuration optimal.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.real_arb import RealBallField

RBF = RealBallField(numberdb.bits(100, losing=64))
S6 = RBF(2) + RBF(12) / RBF(13).sqrt()
d6 = RBF(2) / (S6 - 2)
d6, S6

References

[1]

R. Peikert, Dichteste Packungen von gleichen Kreisen in einem Quadrat, Elemente der Mathematik 49 (1994), no. 1, 16-26. (doi) (MR)

[2]

K. J. Nurmela and P. R. J. Östergård, More optimal packings of equal circles in a square, Discrete & Computational Geometry 22 (1999), no. 3, 439-457. (doi)

[3]

M. Cs. Markót and T. Csendes, A new verified optimization technique for the packing circles in a unit square problems, SIAM Journal on Optimization 16 (2005), no. 1, 193-219. (doi)

[4]

M. Cs. Markót, Interval methods for verifying structural optimality of circle packing configurations in the unit square, Journal of Computational and Applied Mathematics 199 (2007), no. 2, 353-357. (doi)

[5]

M. Cs. Markót, Improved interval methods for solving circle packing problems in the unit square, Journal of Global Optimization 81 (2021), 773-803. (doi)

[6]

K. Kirchner and G. Wengerodt, Die dichteste Packung von 36 Kreisen in einem Quadrat, Beiträge zur Algebra und Geometrie 25 (1987), 147-159.

Links

[7]

Packomania: The best known packings of equal circles in a square

[8]

Erich Friedman: Circles in Squares

[9]

David W. Cantrell: Packing unit circles in squares, shift and compression constructions

Similar tables

Best known packings of equal circles in a circle — the same finite equal-circle packing problem with a circular container

Best known packings of equal circles in an equilateral triangle — the same finite equal-circle packing problem with a triangular container

Global minimum energies of Lennard-Jones clusters — another optimisation problem for finite configurations indexed by the number of objects, with pair-potential energy as the objective

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds both normalisations for every $2\leq n\leq100$, covering the small configurations in the classical square-packing problem, including all cases with proven optimality cited here; the source contains larger configurations)

How they were obtained:

The 25 configurations with an explicit radical or rational formula are evaluated in Sage real ball arithmetic at 100 significant digits with 64 guard bits, and repeated at 160 digits. The widest resulting interval at the first precision is less than $4\cdot10^{-118}$. The square-grid cases $n=4,9,16,25,36$ are stored exactly as integers and rationals. For $n=31,32,33$, the certified separation intervals in [5], Section 7.5, are converted by interval arithmetic and written to 13 or 14 significant digits. The remaining 71 configurations use the Packomania HTML columns rounded to 10 significant digits.

Every configuration is independently checked from all published center coordinates. If $D$ is the least pairwise center distance and $L$ is the side length of the smallest axis-aligned square containing the centers, the check computes $d=D/L$ and $S=2+2L/D$, without using the printed circle radius. Each coordinate is allowed one unit in its twelfth decimal place to account for rounding. Every retained 10-digit interval contains the corresponding coordinate-derived interval; 11 digits fail this test for four container rows. Every formula and certified interval agrees with the coordinate-derived enclosure and with the printed radius, separation and container columns. These checks establish numerical agreement with the source, not global optimality. Exact square-grid and two-point controls, a perturbed control, both normalisation conversions, and monotonicity over the complete range are also checked. The attached checks can be rerun against database values.