The best known packings of equal circles in a 1x0.30000 rectangle (complete up to N = 300)

Last update: 11-Aug-2010


Overview    Download    Results    History of updates    References

Overview

1-10   11-20   21-30   31-40   41-50   51-60   61-70   71-80   81-90   91-100   101-110   111-120   121-130   131-140   141-150   151-160   161-170   171-180   181-190   191-200   201-210   211-220   221-230   231-240   241-250   251-260   261-270   271-280   281-290   291-300  


Download

You may download ASCII files which contain all the values of radius, ratio etc. by using the links given in the table header below.
All coordinates of all packings are packed as ASCII files here.
All packings are stored as nice PDF files here.
All contact graphs of all packings are stored as nice PDF files here.
  For industrial applications, for instance if a machine has to do an important job at every circle center,
it is useful to know a tour visiting each of the circle centers once which is of minimal length.
This problem is known as the "Traveling Salesman Problem" (TSP). Thus (very near) optimal tours are provided for every packing.
All optimal TSP tours of all packings are stored as nice PDF files here.


Results

The table below summarizes the current status of the search.
Please use the links in the following table to view a picture for a certain configuration.
Furthermore, note that for certain values of N several distinct optimal configurations exist; however, only one is shown here.
Proven optimal packings are indicated by a radius in bold face type.

Legend:
Please note that all packings (including their coordinates, of course) are normalized such that their width (i.e. the horizontal dimension) is equal 1.
N
the number of circles; colors correspond to active researchers in the past, see "References" at the bottom of the page
radius
of the circles in the rectangle
ratio
= 0.3/radius
density
ratio of total area occupied by the circles to container area (for an infinite hexagonal packing you get the well-known value ρ = Pi/(2*sqrt(3))=0.90689968211)
contacts
number of contacts between circles and container and between the circles themselves, respectively
loose
number of circles that have still degrees of freedom for a movement inside the container (so called "rattlers")
boundary
number of circles that have contact to the container (rattlers too if possible)
symmetry group
of the packing (Schönfliess notation); if field is empty then the packing has symmetry element C1
reference
for the best known packing so far
records
the sequence of N 's that establish density records

N radius ratio density contacts loose boundary symmetry group reference
1 0.150000000000000000000000000000 2.0000000000000000000000000000 0.235619449019234492884698253746 2 1 D2
2 0.150000000000000000000000000000 2.0000000000000000000000000000 0.471238898038468985769396507492 4 2 D2
3 0.150000000000000000000000000000 2.0000000000000000000000000000 0.706858347057703478654094761238 6 3 D2
4 0.126631206038591402044816559535 2.3690842832893316671526078058 0.671691875906921931431584910096 9 4 C2 [1]
5 0.107151991246211766023215091783 2.7997613157805515904803860204 0.601172511776718163029248121325 11 5 D1 [1]
6 0.096688068540962573707868627546 3.1027613285387215112463254298 0.587388768238511304576222294961 13 6 C2 [1]
7 0.090524980688874672231101900327 3.3140023639559788663496881414 0.600708171406763018587647279192 15 7 D1 [1]
8 0.086624445513407986563879928173 3.4632256313094104281189175760 0.628636423075487506937142195570 17 8 C2 [1]
9 0.084012779560868469248747528918 3.5708853054033961297191572434 0.665214695103852694072700467999 19 9 D1 [1]
10 0.082183960607952947104467913022 3.6503473157141684515650951151 0.707298524108643363690586820416 21 10 C2 [1]
11 0.080856054890725549128789228176 3.7102972734131129341045452926 0.753089163096126437546995574419 23 11 D1 [1]
12 0.078183050093750876264961776380 3.8371488403210661883978451930 0.768130628331484219334563343000 30 12 C2 [1]
13 0.074383118578882528398855021422 4.0331731948271717180228843292 0.753218078836032467933426466402 28 10 [1]
14 0.071690963196452521131547349910 4.1846278334679269424513223188 0.753503865045371163820006560389 30 9 D1 [1]
15 0.067052525214119914224462306330 4.4741044284611973008712421020 0.706236490404541389612390329163 29 1 10 [1]
16 0.065161636802461890269909506198 4.6039359156899756539859651940 0.711430647944008023113897188614 34 10 C2 [1]
17 0.062488544823889879508873858776 4.8008799187992541827172410404 0.695149735764049314568706309577 35 11 [1]
18 0.061300900367034192835166155455 4.8938922300288285413277394078 0.708328685142876666919736075889 38 11 D1 [1]
19 0.059550656751250132811711587320 5.0377278163888960160000263023 0.705594732211097883045482727655 40 1 14 [1]
20 0.058578634153027236565466859064 5.1213211836981103331035768656 0.718682543474401499780975412963 40 2 14 C2 [1]
21 0.057556164173945465572147860585 5.2123000951443538392927781138 0.728503425355062055622697339789 44 16 D1 [1]
22 0.056829196367008116655910085286 5.2789766383915184383988386961 0.744036648957186424891885477988 43 1 16 [1]
23 0.056480789058860864653297028022 5.3115405255290986113230386893 0.768347999198918965716160493634 48 16 D2 [1]
24 0.055687549391540451928672059783 5.3872006090749844079573914448 0.779392231743551838545342848753 49 17 D1 [1]
25 0.055133307173835925009539287161 5.4413568744225100523672098692 0.795786771492610205163603343204 51 18 D1 [1]
26 0.055029721700425140817596972087 5.4515994399019885022771044381 0.824511275003472964377601005413 54 18 D2 [1]
27 0.053129454470372796800891154001 5.6465853638172123893146032956 0.798110630349891639514728923503 63 19 D1 [1]
28 0.051263389719084686712968074972 5.8521295927552356163950056383 0.770550823872884630055186213020 64 20 [1]
29 0.050574671762540061910281662280 5.9318229766981052651394519790 0.776770567638421702475857729140 69 18 D1 [1]
30 0.050000000000000000000000000000 6.0000000000000000000000000000 0.785398163397448309615660845820 73 22 D2
31 0.048237334081921615156521944228 6.2192491709950028094676468032 0.755365086143441348935730427083 63 15 [2]
32 0.047506172350128100070079679517 6.3149688800215673878410926522 0.756273140296828387333693693272 79 18 C2 [1]
33 0.046528041427494177311067302354 6.4477246579892596768602108978 0.748121439624874994753121564869 69 15 D1 [1]
34 0.045842255778321503803911267635 6.5441805798280108590624654287 0.748237542575895367440475021428 75 2 19 [1]
35 0.045326929298257786901628748190 6.6185820359891659441343403083 0.753024765147721771495203254018 84 20 [1]
36 0.044696399788238293082900053513 6.7119499875008722424041389664 0.753140850753165463943023063450 84 1 21 [1]
37 0.044189570871487661105934798261 6.7889321865664084833240261923 0.756606225221181564484798534660 86 1 21 [1]
38 0.043924639199845338127647393716 6.8298796635546712009579433065 0.767765548375689522427832873678 94 21 D1 [1]
39 0.043492666398085474449307216485 6.8977145998389708027369496586 0.772547679565261364501535204744 92 1 22 [1]
40 0.043101962030921619401738162276 6.9602399952182712221570046104 0.778184702623336372176557403688 90 2 22 [1]
41 0.042734400222033722159271602277 7.0201055459138266990950547635 0.784093223886261195830590407502 97 1 23 [1]
42 0.042538652292561095623883942424 7.0524096047222963072276647050 0.795875913496332003071775012153 100 1 23 [1]
43 0.042216386991065198825363988063 7.1062452611942582511139584749 0.802526160211313159910420692516 101 1 24 [2]
44 0.042084159922576466823992128235 7.1285728538224190965320248232 0.816053476015515185248945222046 109 24 C2 [1]
45 0.041738765654977466112815057130 7.1875628158214149212752135008 0.820956857146701261374804443997 107 1 25 [1]
46 0.041666698686612742275306743112 7.1999944669575701601126954573 0.836304885277459735934542054621 112 1 22 [2]
47 0.040705379222986533781442836620 7.3700332910935879645095877940 0.815511493374632800052033624739 122 1 26 [1]
48 0.040264246406544206363518328956 7.4507789608410643173318009643 0.814908796266958075889362242054 108 26 C2 [1]
49 0.039445405839191235339383348207 7.6054484322717521803082482899 0.798394534784522923867755084005 90 5 23 [2]
50 0.038949256048498339628656535561 7.7023294007579968354274108775 0.794322667196681577652169450249 124 1 23 [2]
51 0.038658295201252175494057881338 7.7603008212913315816643084811 0.798149397693304042633027000344 121 23 D1 [2]
52 0.037963722246073354949049414283 7.9022809738059721600933426823 0.784819050086844519371463081638 119 3 24 [1]
53 0.037538613523303451425815861700 7.9917709217940123046293540009 0.782097582890552431195270025096 119 24 [1]
54 0.037377672840456481195489451151 8.0261818674620353626575638497 0.790036023917505352488340483020 115 1 21 [2]
55 0.036821146016561448902344535603 8.1474922009506636276802728347 0.780882895412762963672969909667 109 1 21 [2]
56 0.036489585424092138070358922411 8.2215239365785150897107644821 0.780826426532308953799405212628 138 1 26 [1]
57 0.036134102581187336287994831898 8.3024062746805389666991920119 0.779359834374849115598465887800 134 3 26 [1]
58 0.035946228253989012564529538442 8.3457991163984917517360641728 0.784807720849160400931982817523 142 2 26 D1 [1]
59 0.035623233954159996204811152461 8.4214701109405233069682284689 0.784056421049869244213866298529 136 3 27 [1]
60 0.035410163893910687467572142897 8.4721437861401576662420943181 0.787835855195293366398628939425 150 1 27 [1]
61 0.035194861912490522255389203742 8.5239715031679423704565651730 0.791255941532064124831479184623 145 2 27 [1]
62 0.034979567404528373120441297937 8.5764353952862979227920408911 0.794418182959927733100527380330 148 1 28 [1]
63 0.034784473616478034030693163406 8.6245375827071471397272913183 0.798252043155626568480763168215 142 3 28 [1]
64 0.034577764524454160060602771758 8.6760958704497325986151449811 0.801313421185712723105663387873 150 3 29 [1]
65 0.034416597359089708480898899210 8.7167245753528134500999900781 0.806265054248585779358137014164 158 1 29 [1]
66 0.034249166970938659383062853915 8.7593371323325346856261127629 0.810723158801635444221644250745 142 2 29 [1]
67 0.034148410253936484471474255671 8.7851820266045113292849880148 0.818171601779443892459876171444 150 1 30 [1]
68 0.034081602735789718686733126686 8.8024029364371551440461044334 0.827137194403976067676248879090 168 2 30 C2 [1]
69 0.033847639555750483296400771786 8.8632473028398237132949615366 0.827817272542820129829826965141 152 2 31 [1]
70 0.033796001733098026542546115369 8.8767896974687327874226015531 0.837254143319168819699134071348 176 1 31 [1]
71 0.033606813648998757248906552800 8.9267611959081950912101823744 0.839733815577387776340934028570 157 1 31 [1]
72 0.033409172025378879884142596682 8.9795700346033290103854095087 0.841574445936160970919219879542 136 6 26 [1]
73 0.033338463053847396170619977166 8.9986151885720707326549971639 0.849655018516798368363554535051 129 2 26 [1]
74 0.032616987204617423820257206424 9.1976612713491361260679317969 0.824419060751393266072677958816 187 34 D1 [1]
75 0.032421427171979613680023527331 9.2531398574359044094803471362 0.825570446829484761202996852613 165 33 D1 [1]
76 0.031955171771317619131510424843 9.3881516940952432163483282676 0.812689264890399754652791055415 147 9 28 [1]
77 0.031714333544939433857008311529 9.4594451929723791429015067848 0.811018053799491200338332965305 122 4 29 [1]
78 0.031637269812226431326163115735 9.4824870091686300877064262050 0.817562978720551434475508570280 205 30 C2 [2]
79 0.031278191336827302451429038418 9.5913474270098395168000950566 0.809354849530974994440619576793 177 1 29 [2]
80 0.031009454065161705448802783935 9.6744689335579757379121217063 0.805576605822966972506368109056 202 1 32 [1]
81 0.030740788187739453010506263583 9.7590210819529649063046578040 0.801574022182938964005223154575 193 3 31 [1]
82 0.030529136629875429149799646930 9.8266781546132481523885599931 0.800334457638670755323516968034 207 1 32 [1]
83 0.030441831476039112921251952482 9.8548604158764625446941305521 0.805467951605716097995442332238 205 1 31 [1]
84 0.030195297504174257180073645041 9.9353218811150117431832843719 0.802022458815374489032920394157 204 3 34 [1]
85 0.030038382028080899980426001461 9.9872223383919216565439564161 0.803157309826013485882324318010 218 1 33 [1]
86 0.030002462118380845680673398462 9.9991793612233786382477813125 0.810663951621468408994845407358 204 3 27 D1 [2]
87 0.029751438200420344250648227372 10.0835461458720822226145887087 0.806424660024167896774677799442 127 8 22 [1]
88 0.029583740266262674826277857186 10.1407055801568228899281335239 0.806524290734714669840124211503 217 1 33 [1]
89 0.029461270081064462565428471088 10.1828603849912748334301356373 0.808949769278770411527475962289 216 4 35 [1]
90 0.029322141118449119925149337381 10.2311764610955984552534026962 0.810331061417747143819031444546 233 1 34 [1]
91 0.029174331835118965340937691174 10.2830118508102819071572418784 0.811095229546335329029347907975 226 3 35 [1]
92 0.029046029161450878611969158788 10.3284341667656266916833272381 0.812811769216057567882438737455 231 1 35 [1]
93 0.029003588941959269180908743440 10.3435475037364188404010551662 0.819247357648244697702941830199 245 35 D1 [1]
94 0.028836696680915642828031246174 10.4034107415133442242618920425 0.818554292391424968234646806096 238 1 36 [1]
95 0.028757878783588562214959261885 10.4319237958260978676037129952 0.822746267737508302901168392551 236 4 37 [1]
96 0.028665307614145275329611862981 10.4656124412898688212556783029 0.826062798578996677983281074283 236 2 36 [1]
97 0.028545553751987484012836369512 10.5095176154749669817360365349 0.827708274383706852466632980726 237 3 36 [1]
98 0.028484248696577589182156016260 10.5321366624651503779572027338 0.832653345942273261809473438967 246 2 37 [1]
99 0.028435554212868030849212417760 10.5501724268922480064144595241 0.838276336681570672037977532163 232 1 37 [1]
100 0.028302338820544108590585131176 10.5998307031161653323633860472 0.838828677630589860547541263581 248 1 38 [2]
101 0.028257754633513118549048815243 10.6165547790625282416036955128 0.844549854196118482541809547002 218 1 38 [1]
102 0.028238375042450805414717962642 10.6238407680686088628602623124 0.851742255835606116975938899693 205 38 C2 [1]
103 0.028011659857520984298839134413 10.7098258912868945046942810122 0.846337396032587513590157429487 274 2 39 [1]
104 0.027852988007340577203932745090 10.7708372229556066951919772370 0.844900450362476724795948872234 275 2 39 [1]
105 0.027794892978654206271859994434 10.7933497074585830026793477821 0.849469772161175331548868215479 277 1 35 [2]
106 0.027535211845764605436590231747 10.8951404361955260776774616398 0.841610853404411599681127498595 211 1 40 [2]
107 0.027251205775747549592704313366 11.0086871923659113895901610575 0.832115940951936300484378337042 260 1 41 [1]
108 0.027164475846542418237686506455 11.0438354008654639242017459430 0.834555132644782721317544757576 260 3 36 [1]
109 0.026889123500713840534251190570 11.1569274465951161610351773211 0.825293467656541610410851454142 270 5 37 [1]
110 0.026707061442037960273116045794 11.2329842296984517811232289556 0.821624752074302956307658888077 272 3 37 [1]
111 0.026642297475135009163974702072 11.2602901562820177781941757049 0.825077878062464380667913954435 276 3 36 [1]
112 0.026484179830352499698437282841 11.3275171034815864122978044730 0.822658703712847917502327557995 284 37 D1 [2]
113 0.026284900321440956351333974035 11.4133969058762553160856758564 0.817560179385860464200803055777 284 3 37 [1]
114 0.026202601459959226245404598275 11.4492448567916671545670827756 0.819638390658566954326521581502 287 1 37 [1]
115 0.026025255837879467477427504126 11.5272642032342049260424815220 0.815673725981816278057584104677 280 4 37 [1]
116 0.025906232973915120234114437482 11.5802247398172003625561884090 0.815258134492937974497965852114 286 5 39 [1]
117 0.025830027267520138953571387912 11.6143895975376596008130860691 0.817455667147216701340873428613 300 3 38 [1]
118 0.025704786885211698621329765502 11.6709779131681476326004622121 0.816467008034346254507959305207 91 61 31 [1]
119 0.025618955161162966842588054692 11.7100794358227670160794288931 0.817896606969926421083208579425 271 4 38 [1]
120 0.025527181881409762752561895220 11.7521785755158421107618893042 0.818871223409407579195527220822 315 1 40 [1]
121 0.025446335088338135215489550163 11.7895169956119831604950775790 0.820473336430273626879487735425 285 4 40 [1]
122 0.025314649299144289656061160803 11.8508455896381266935218935463 0.818714117158310154734487086217 307 3 40 [1]
123 0.025276716365927278477504335494 11.8686302309581846126544991530 0.822953013305400893575499541209 323 2 40 D1 [1]
124 0.025162433941115291176539359240 11.9225350259062777248815409159 0.822158589852398698690080812353 316 3 41 [1]
125 0.025055482306722672207139970751 11.9734274649946200379816385667 0.821758424823183678491451396481 317 3 41 [1]
126 0.025021618168217231919147180378 11.9896322445309992381964964519 0.826094913262342408758596556099 334 1 41 [1]
127 0.024952538805520807957127023933 12.0228247048602643071439323744 0.828060022879222952867234735359 256 32 D2 [1]
128 0.024860119793920455816095854460 12.0675202889957522564758275699 0.828409406715038966864538801149 315 6 42 [1]
129 0.024797971801845074130131553691 12.0977635750710358254052298517 0.830712321030196657598723245418 312 2 42 [1]
130 0.024740658660292697545650788795 12.1257887317883885068371719648 0.833286767358104055018195469077 320 42 D1 [1]
131 0.024652676016254139593923331381 12.1690643158658447416615314277 0.833735032113784713086557416490 271 6 40 [1]
132 0.024603396159233857947114469542 12.1934385829660154554730876869 0.836744118516707683241274673152 335 4 43 [1]
133 0.024554836403489723605882524555 12.2175523823634238933540638035 0.839758384909051007456954643285 330 4 43 [1]
134 0.024484059315114268721657102388 12.2528701690739177246250466952 0.841201933501860174266205556451 326 4 43 [1]
135 0.024438434008531799027103622513 12.2757456511029227648935508087 0.844323997527474695568681631779 334 5 45 [1]
136 0.024411983584887020764007097872 12.2890464413438367088769414887 0.848738034899727207249611708710 308 2 44 [1]
137 0.024392030974825139716361666048 12.2990988454232488901062743476 0.853581729875034133971598573930 330 2 44 D1 [1]
138 0.024286083557267434973110196064 12.3527533491593859025538505957 0.852359241323120985740138944923 328 2 45 [1]
139 0.024260736169346817115476302966 12.3656593891428084009718587622 0.856744585058161955717109918767 291 2 45 [1]
140 0.024250222592084895513672836264 12.3710204663407056557731754848 0.862160481349883233617098496274 344 45 D1 [1]
141 0.024055693968047485002728752397 12.4710598828901684540847549995 0.854443815849968888075981613494 228 10 36 [1]
142 0.023917579859034299820297531778 12.5430750840236914280617031671 0.850651021350765265203183128543 265 8 46 C2 [1]
143 0.023860432934142405927444215640 12.5731163733715643108618026274 0.852552818137163856000759561727 239 8 31 [1]
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294 0.016713233394864897815163425572 17.9498480582562979095018477099 0.859996936893575708897782250733 654 7 62 [1]
295 0.016701578334441598116319080439 17.9623742135404727661161238417 0.861718989741823240266563355585 510 10 60 [1]
296 0.016670946905219811520356849343 17.9953785292224472168393970694 0.861471403631194466867549987526 333 62 50 [1]
297 0.016548318307044348044457018319 18.1287303297939896377764906967 0.851712064785853977037701494725 566 28 69 [1]
298 0.016524083016780614879194909260 18.1553190997250850276303812782 0.852078522181213097124552904521 700 7 62 [1]
299 0.016498917570028840864565116502 18.1830110203693554329198037575 0.852335764052726721071201904042 610 8 63 [1]
300 0.016458728976061140249536135425 18.2274099316140031206006102656 0.851025278401262355880145955315 580 3 51 [1]





Updates

Please note that the results are taken from a running search. For updates look at the list below.

09-Jul-2010: First complete presentation from N=1 to N=225
12-Jul-2010: Some better packings for N=31, 43, 46, 47, 49, 50 and 200 by David W. Cantrell [2].
15-Jul-2010: Many improvements for N=54, 59, 71, 89, 91, 112, 118, 119, 131, 136, 149, 150, 152, 153, 154, 158, 160, 163, 167, 175, 179, 182, 187, 190, 192, 193, 194, 195, 196, 198, 199, 200 and 201–225 by Eckard Specht [1].
17-Jul-2010: Further improvements for N=47, 51, 52, 80, 81, 85, 109, 147, 151, 152, 153, 154, 155, 197, 198, 199, 201, 202, 203, 204, 205, 206, 207, 209–225 and new packings for N=226–239 by Eckard Specht [1].
19-Jul-2010: More better packings for N=51, 79, 112 and 151 by David W. Cantrell [2].
23-Jul-2010: Some improvements which are difficult to find for N=46, 55, 105, 144 and 208 by David W. Cantrell [2].
24-Jul-2010: Many improvements for N=59, 73, 100, 108, 109, 110, 114, 124, 130, 131, 142, 145, 146, 147, 148, 149, 150, 153, 154, 155, 157, 178, 179, 190, 191, 192, 193, 194, 195, 196, 201, 203, 205, 207, 209, 213, 214, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 229, 230, 231, 232, 236, 237, 238 and 239 by Eckard Specht [1].
26-Jul-2010: Some better packings for N=78, 100, 106 and 230 by David W. Cantrell [2].
04-Aug-2010: Two improvements for N=180 and 225 and one correction for N=230 by David W. Cantrell [2].
08-Aug-2010: Some improvements for N=54, 76, 86, 87, 98, 115, 127, 155, 178, 179, 189, 203, 205, 214, 218, 223, 226, 237, 238 and 239 and extension up to N=300 by Eckard Specht [1].
11-Aug-2010: Three better packings for N=54, 86 and 285 by David W. Cantrell [2].

References

[1]   , program crc, 1999–2010.
[2]   , private communication, July–August 2010.


©  E. Specht     11-Aug-2010