The best known packings of equal circles in a square (complete up to N = 420)

Last update: 11-Mar-2010

This page is still under construction!

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Overview

1-12   13-24   25-36   37-48   49-60   61-72   73-84   85-96   97-108   109-120   121-132   133-144   145-156   157-168   169-180   181-192   193-204   205-216   217-228   229-240   241-252   253-264   265-276   277-288   289-300   301-312   313-324   325-336   337-348   349-360   361-372   373-384   385-396   397-408   409-420  


Download

You may download ASCII files which contain all the values of radius, distance 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 PostScript figures here.
All packings are stored as nice PDF files here.
All contact graphs of all packings are stored as nice PostScript figures 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:
N
the number of circles
radius
of the circles in the square
ratio
= 1/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


N radius distance ratio density contacts loose boundary symmetry group reference
1 0.500000000000000000000000000000 0.053722945171243479357856334105 2.0000000000000000000000000000 0.785398163397448309615660845822 4 1 D4 []
2 0.292893218813452475599155637896 1.414213562373095048801688724217 3.4142135623730950488016887242 0.539012084452647221356168169719 5 2 D2 []
3 0.254333095030249817754744760429 1.035276180410083049395595350499 3.9318516525781365734994863995 0.609644808741350951447230676805 7 3 D1 []
4 0.250000000000000000000000000000 1.000000000000000000000000000000 4.0000000000000000000000000000 0.785398163397448309615660845821 12 4 D4 []
5 0.207106781186547524400844362105 0.707106781186547524400844362106 4.8284271247461900976033774484 0.673765105565809026695210212148 12 4 D4 []
6 0.187680601147476864319898426192 0.600925212577331548853203544579 5.3282011773513747321100504007 0.663956909464133571452413649192 13 5 D1 [3]
7 0.174457630187009438959427204500 0.535898384862245412945107316989 5.7320508075688772935274463415 0.669310826840792318436251021849 14 1 6 D1 [13]
8 0.170540688701054438818560595676 0.517638090205041524697797675249 5.8637033051562731469989727989 0.730963825253907837357358208948 20 4 D4 [1]
9 0.166666666666666666666666666667 0.500000000000000000000000000000 6.0000000000000000000000000000 0.785398163397448309615660845821 24 8 D4 [2]
10 0.148204322565228798668007362743 0.421279543983903432768821760651 6.7474415232381126762112368561 0.690035785264180329182022896993 21 8 [9]
11 0.142399237695800384587114500527 0.398207310236844165221512929748 7.0225095034303813492095760345 0.700741577756104830198542148982 20 2 8 D1 [9]
12 0.139958844038428028961026945453 0.388730126323020031391610191837 7.1449575542752651213595466567 0.738468223884044498369057950513 25 8 C2 [9]
13 0.133993513499008491414263236065 0.366096007696425085295389370603 7.4630478288592654598767430406 0.733264694904101752789658211574 25 1 9 [9]
14 0.129331793710034021408259201773 0.348915260374018877918854408999 7.7320508075688772935274463415 0.735679255542682563361972309844 32 1 10 D1 [6]
15 0.127166547515124908877372380214 0.341081377402108877637121191352 7.8637033051562731469989727989 0.762056010926688689309038346005 36 8 D1 [10]
16 0.125000000000000000000000000000 0.333333333333333333333333333333 8.0000000000000000000000000000 0.785398163397448309615660845820 40 12 D4 [5]
17 0.117196742782948687473176894856 0.306153985300332915214516914060 8.5326603474980966644272735286 0.733550263302331439285589240584 34 1 11 D1 [10]
18 0.115521432463999509608513951182 0.300462606288665774426601772289 8.6564023547027494642201008015 0.754653357875667578715912508591 38 10 D2 [10]
19 0.112265437570996304738752306983 0.289541991994981660261698764510 8.9074609393260787738262975010 0.752307896742078979509898727718 37 2 11 [10]
20 0.111382347512479750227357863499 0.286611652351681559449894454737 8.9780833528217365165192364463 0.779493686867621861721679242589 44 11 D1 [10]
21 0.106860212352064428580553201716 0.271812255359307094464880424464 9.3580199588727565928284980056 0.753357702889674769521037881736 39 2 11 [13]
22 0.105665296756976756533092354860 0.267958401550716769338023770982 9.4638450909756523911277579008 0.771680112098219822603167037094 43 1 13 [13]
23 0.102802323379784112346596984546 0.258819045102520762348898837624 9.7274066103125462939979455978 0.763631032126130708837463558612 56 8 D2 [13]
24 0.101381800431613524388964772877 0.254333095030249817754744760429 9.8637033051562731469989727989 0.774963259757823779541804629689 56 12 D1 [13]
25 0.100000000000000000000000000000 0.250000000000000000000000000000 10.0000000000000000000000000000 0.785398163397448309615660845820 60 16 D4 [7]
26 0.096362339009887092432024697394 0.238734757241216609476898102079 10.3774982039134263443391268496 0.758469090483941352861819034654 56 2 13 [13]
27 0.095420001747936516364270819029 0.235849528301415095283016509441 10.4799830400508798304005935979 0.772311456467343180199003969254 55 13 D1 [13]
28 0.093672833832785071755016932236 0.230535493642667484673265824903 10.6754536943453127881514813657 0.771854111403716173720195749794 57 1 14 [17]
29 0.092463144040309496841524670049 0.226882900744209089183505212940 10.8151200175936906689015127153 0.778906241779735128443167917038 65 1 14 [18]
30 0.091671057985988438718806599233 0.224502964531088275573348086074 10.9085683308339920507688052094 0.792019026460736228749276303655 65 14 C2 [18]
31 0.089338333351234633748802313039 0.217547291619124453229498484724 11.1934033520469771160959973707 0.777297478729284489847888300458 55 4 16 D1 [12]
32 0.087858157087794923008578158871 0.213174562589876532673591443375 11.3819824265232393752398264833 0.776004124474520585203569503089 63 3 15 [14]
33 0.087230014135567983530352112436 0.211328384143263135492079924874 11.4639440324503011568460735311 0.788852303933884675540557918277 65 1 17 [12]
34 0.085270344350527219409280297838 0.205604646759568224693193969093 11.7274066103125462939979455978 0.776649064332277077192193097006 80 12 [11]
35 0.084290712122358459903667877331 0.202763600863227048777929545753 11.8637033051562731469989727989 0.781227212998714358933378085373 80 16 D1 [12]
36 0.083333333333333333333333333333 0.200000000000000000000000000000 12.0000000000000000000000000000 0.785398163397448309615660845820 84 20 D4 [8]
37 0.082089766428752816962342111380 0.196429184629568997838626302563 12.1817863968210698362744366706 0.783302723724208829617437864294 77 16 D1 [14]
38 0.081709776125419800616581793806 0.195342304126905900273112801034 12.2384376438023473575039603473 0.797042556834780531205882269712 77 16 [12]
39 0.081367527046974026631769391643 0.194365063161510015695805095918 12.2899151085505302427190933133 0.811179027291583405559076523463 80 16 D2 [4]
40 0.079186752517282867898915819775 0.188175522018315453787532891212 12.6283749264972757200437889959 0.787979518878382871537374468088 85 2 17 [11]
41 0.078450210116920559483053007213 0.186099511848124556123865658327 12.7469384531873251117784855342 0.792723899305009720099449735152 100 1 18 [12]
42 0.077801502898165532306768693062 0.184277072117098926478785467479 12.8532221454500825687745597891 0.798684278653432047162866645969 90 17 D1 [4]
43 0.076339810579744074677200744991 0.180191135457425956798065671241 13.0993251411778974447542878630 0.787264166427460630119036545331 85 1 16 [11]
44 0.075781986017897705750184232945 0.178639245671200889655388983438 13.1957481262608554872478304964 0.793842807841790391912527518373 82 4 20 [11]
45 0.074727343686985328535051089524 0.175716314175589846017156317742 13.3819824265232393752398264833 0.789444268494961791151996201485 94 3 19 [11]
46 0.074272199909420921796026120347 0.174459360872413203234903861303 13.4639878880598071936730888487 0.797187131992481565987478372966 91 1 21 [12]
47 0.073113151270516836978943966705 0.171270563822987220137903144197 13.6774298826215947135408970296 0.789293882075948475945003280435 92 3 19 [22]
48 0.072432291280497581081203202077 0.169405429370288096853431066357 13.8059970535441329898933067156 0.791144033840820484633598422522 111 2 18 [14]
49 0.071692681703623580727975837865 0.167386076868325407948667740525 13.9484250865936955971142641355 0.791216989526925284715824261840 120 1 19 [12]
50 0.071377103864521467530864106392 0.166526577343550707602077799846 14.0100949164043850676229061557 0.800272183997614540398268557996 100 1 19 D1 [14]
51 0.071043138115537018210695844271 0.165618374312594797328771838599 14.0759547864243574869673926248 0.808656948184898971918843953969 99 3 19 [11]
52 0.070957693072547251836538973586 0.165386237969640529862292226679 14.0929046125780681731433570098 0.822530842067761468539017516652 105 19 D1 [4]
53 0.069947252562026650314667369460 0.162648077424770238765761023446 14.2964871867302691916392131754 0.814642500051039348061532079567 108 4 20 [19]
54 0.068645540104688887910867652499 0.159139516307189745750491914975 14.5675887825332181932543170306 0.799407623083866967669545766005 115 2 21 [11]
55 0.068055360558639717449689169779 0.157555747529721900779385929688 14.6939196529324490658396620664 0.800271297244825744992218341275 113 2 21 [11]
56 0.067532596322265705546305516123 0.156156500462147178018025515378 14.8076640682967035879634496450 0.802351729502980578932357268910 119 20 C2 [11]
57 0.067004873106043976620256140932 0.154747406943575157196228417844 14.9242876472188700315027030559 0.803965673845865929966857781554 113 3 22 [22]
58 0.066232983745862232302511874085 0.152692531391223377922003488297 15.0982175865279950794785486574 0.799330724318471867207813535842 125 2 20 [22]
59 0.065807496903601046306129157869 0.151562950613614202473276994896 15.1958370558427832726382997809 0.802698827038006793401631671012 121 4 24 [19]
60 0.065030412648295594469592436244 0.149505654048667538884669306102 15.3774204910601833698861754239 0.797139156419255784870559282621 129 3 21 [11]
61 0.064666268906273511710193139988 0.148544126695183636847514175879 15.4640126438930255892954777349 0.801374124592793740357582679375 121 1 25 [11]
62 0.064252183294482911181458806768 0.147452679809765301918531485006 15.5636734617524844300701130769 0.804113477816698520155376391258 128 2 21 [22]
63 0.064011528204864918571082299195 0.146819313688053305488424768330 15.6221860037392348522231816922 0.810973780649873545826894093593 145 1 21 [22]
64 0.063458986813059169965842890472 0.145367754451708300975695503458 15.7582093604149832660896556588 0.809685038563171020371396760572 129 2 23 [22]
65 0.063203957071856799974346494473 0.144699014780805622215620325415 15.8217941775875915576901053101 0.815740018110047678672004161765 137 2 23 D1 [22]
66 0.062862256903173650552663945791 0.143804230807975809694486872230 15.9077966535674001923838046766 0.819358090828537580877710854049 141 2 22 [19]
67 0.062587429542203145120191597701 0.143085575882510688941819936205 15.9776493029753354511388551164 0.824515655814026443272007820249 138 1 19 [24]
68 0.062520077997967509748911370819 0.142909593912008257586957737282 15.9948616831941476534722680150 0.835021788301676498075386011080 156 3 20 [19]
69 0.061383571685582519393723042252 0.139948181880639374117066567102 16.2910038067217133186173719170 0.816776573752215775394223611561 137 1 23 [22]
70 0.060596693631158157824224372564 0.137906776651089576180187528643 16.5025505531181477757014705618 0.807506020669364260500664086986 135 5 23 [22]
71 0.060096531351830846614976309881 0.136612997248029233193793849708 16.6398954732607026528365687555 0.805576954556086927019502619207 142 3 24 [22]
72 0.059801002126807327595495341667 0.135849927909272737389808677495 16.7221277977835833294685943084 0.808908302137999085775684834728 180 1 23 [22]
73 0.059366050583080470834219283052 0.134728725876973755094131088759 16.8446442062124214969120792232 0.808256206862671849525236295628 151 3 24 [19]
74 0.059082376336698616287994709302 0.133998672690424785486208280518 16.9255209760216905870553868736 0.811516775123364681685174979476 153 3 26 [22]
75 0.058494535281249486795264924005 0.132488813742117320487412003691 17.0956140636363263363384238859 0.806198017941952636005393179092 148 2 22 [24]
76 0.058198524936106167091962265813 0.131730037632150119187637713078 17.1825660718696909505414496828 0.808699956656388340271328954092 191 1 24 [22]
77 0.057852577916326407945864081743 0.130844544210365598221438004803 17.2853144322509594703218721469 0.809628951523275454193704449070 174 24 [19]
78 0.057702476734813469658915612215 0.130460772895210245048599607141 17.3302786394378955731266815013 0.815893333309535399379288603296 183 1 24 [20]
79 0.057508497795177150643081806966 0.129965202740904406115442669788 17.3887345060135312216405790393 0.820806922795721980813682909695 177 2 24 [24]
80 0.057370684146686868321824625645 0.129613385494103716195210421865 17.4305050545183285645499950863 0.827217888559947854040065137730 193 24 C2 [22]
81 0.056869921111948347401531481988 0.128336855973876344193090313500 17.5839878172417651702227896944 0.823000583646207165131312975821 198 3 26 [22]
82 0.056512271650043116731600865688 0.127426911811749595536595821291 17.6952716782043752389211638037 0.822714696667481465122043671825 203 4 26 [22]
83 0.056129373021650586640744529302 0.126454353160855578690844392900 17.8159837918423478438752023974 0.821501474386234485434434731555 209 2 24 [22]
84 0.055856665888395536222323748893 0.125762702259446400412319775749 17.9029661741366901471419975275 0.823339926925100779747488690109 184 2 23 [22]
85 0.055680181768308624302233825464 0.125315548583687285482636439087 17.9597114851584044953610605854 0.827885139567273523102982634735 203 3 23 [19]
86 0.055572999412015161187514147010 0.125044156494747378054229049817 17.9943499645584179987825369965 0.834403272414726330623353074852 191 2 24 [24]
87 0.054695259720704525732612598175 0.122826583176275233296982422389 18.2831200565897792110838339099 0.817651999581176324044528222879 175 26 [21]
88 0.054406636912959447376712741080 0.122099298194286293682894289304 18.3801105295262961716797498050 0.818344762893455338659140369863 187 5 27 [20]
89 0.053947040858284291446539903179 0.120943129627674435260681788143 18.5366979187411090776201305739 0.813720269423130443280172100705 182 5 29 [22]
90 0.053749948306745947917965570800 0.120448049479875240508744907406 18.6046690555513310423084633158 0.816861606107565255741672170260 230 1 26 [22]
91 0.053496719884488436041701634807 0.119812602206749063475830637822 18.6927348472808620204602434898 0.818173810485969463957034440537 182 2 24 [24]
92 0.053317085175319528829912286052 0.119362266623172872359260303157 18.7557139838338322788212338612 0.821619044642187492376904130881 200 28 C2 [24]
93 0.052926433388240551617979070855 0.118384170617275808035564716013 18.8941505403268830126971455556 0.818423476521143216019926914278 194 1 27 [24]
94 0.052795362726845093818338316930 0.118056384765521590764646103637 18.9410574783592789859725041814 0.823131614711695270716775765936 224 2 26 [22]
95 0.052420366495943589361713667424 0.117119642119436392986518895312 19.0765549126288985463205653801 0.820112788103383671049325920561 222 4 26 [20]
96 0.052275425469925719068554307283 0.116757999099766426161703680386 19.1294473648101510739994642848 0.824168967917176961985882367601 224 4 28 D1 [22]
97 0.052114795502244668305008905267 0.116357483968876788965964658911 19.1884087879981873280117951648 0.827644213165413536281305095882 234 2 27 [22]
98 0.052032987702942098030211872968 0.116153614606866964947161412450 19.2185773707446951268196647452 0.833553492453149745406906221039 204 2 27 [20]
99 0.051978606449505864308001821124 0.116018134843035411576847053431 19.2386843031553902034492536974 0.840299937025485161853712217758 199 27 D1 [4]
100 0.051401071774381815590184511455 0.114581352161702399226953857052 19.4548472527842865834291824527 0.830030826635912821740802332446 216 7 28 [20]
101 0.051078356337468287675013092544 0.113780115212857654792340271126 19.5777638848267854447349965464 0.827837458204886740849985234223 211 6 30 [24]
102 0.050774421023984229946427962168 0.113026558148629120932399074586 19.6949562364804049826470402687 0.826114042027870748552195664642 252 4 29 [22]
103 0.050534497607782334686032855584 0.112432427714295079274739675573 19.7884622849401705257411549689 0.826348041985204520696807001421 257 3 29 D1 [22]
104 0.050348792811532572594881586662 0.111972995972474527584456427713 19.8614493845609421072948689142 0.828249793523388576412378912418 207 4 27 [24]
105 0.050242948365384132560891118091 0.111711307655497688129274682613 19.9032905618446879938853799268 0.832701612019226131524236043108 232 1 27 [24]
106 0.050015200416806532090506982216 0.111148644272282114987435759297 19.9939216811369911786035320444 0.833028317432244445190333516970 246 4 25 [24]
107 0.049506166856097478668626121301 0.109893106661647871088786677442 20.1995036882326096944459152817 0.823857790657640543813394281499 219 9 30 [24]
108 0.049243520787042327172316177682 0.109246395909879908646940060945 20.3072401001663263129466670597 0.822757444973615405175483101187 221 5 29 [24]
109 0.049060056454336484181017952109 0.108795100448600019374100010791 20.3831807843671704330002390822 0.824199710946182236760458317482 238 3 30 [24]
110 0.048779469320530920341849009163 0.108105606912603252132735697199 20.5004280269836257772527740355 0.822274269043788233219044672315 285 1 30 [22]
111 0.048643936344712632841290942957 0.107772865508378992465546268040 20.5575468422940508604125053145 0.825145003926722239389734741481 226 3 27 [24]
112 0.048445405902771448593353411940 0.107285822215198642210909771476 20.6417921651223554747773961460 0.825796605618670452719327830573 261 1 29 [31]
113 0.048257588493549543124104737776 0.106825454649303987557168953930 20.7221295389359790112102660450 0.826722100560516881572374636346 287 1 29 [20]
114 0.048167755522460786854554765216 0.106605396385904078477789735564 20.7607763565752327263802781870 0.830935940404435353549195715343 280 1 29 [22]
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120 0.047529921591019828363001359062 0.105045446890431334697158751407 21.0393782805847764767908357685 0.851658165016251855319555118511 241 30 C2 [4]
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128 0.045473271986833679888176503111 0.100045320075250780804194884910 21.9909400909074622296108483011 0.831519142296790878206793082839 328 33 D1 [31]
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137 0.044293023545787681269568493800 0.097196281457933909040235291947 22.5769188903136231781833500722 0.844386136710202177869379241425 308 32 C2 [22]
138 0.044032136193155772597254590328 0.096568507757398747999703151775 22.7106855686787482389801884359 0.840559517817220290171235730530 327 6 36 [25]
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144 0.043003765459860945384237740878 0.094100918584448037664921248538 23.2537776472942735911704396859 0.836614397024687736727561866517 328 7 37 [24]
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161 0.041010347768823710694417606611 0.089349177197067395884325097883 24.3840897335722043300283482589 0.850673008399595623546819277639 323 35 D1 [23]
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352 0.027810212157221782215327452683 0.058896259244135870764546159161 35.9580140685952679257401621756 0.855265867311641269854890255529 921 9 56 []
353 0.027772625907504946056777005206 0.058811977939392240706412922077 36.0066780624359982992347099777 0.855378769233573949861191700908 896 9 55 []
354 0.027720781199118604021691233142 0.058695746278021220922240488526 36.0740194447260200466505528719 0.854602314442041272525980328205 954 5 53 []
355 0.027685205849154016676113728734 0.058616003970250459524468074249 36.1203743778758009262268814836 0.854818159734924718660789535130 933 6 53 []
356 0.027672738596821843657735078015 0.058588061410244150770466419320 36.1366474988076501917496291199 0.856454216813396703158742732296 921 5 53 []
357 0.027666544055267733481684937934 0.058574178278205631818876505629 36.1447384972392007461931817846 0.858475518566455481255261486578 428 92 50 []
358 0.027656744704473911992258001863 0.058552216834704677654212876792 36.1575453179865497471670195422 0.860270481011808403376051771724 472 67 52 [31]
359 0.027578247508017814649252885822 0.058376328698975107486535836632 36.2604621526175778705323082504 0.857783427209014194423208168158 964 6 53 []
360 0.027538042702510963432688673556 0.058286264697437721324810573119 36.3134014571347286357643385304 0.857666627297535811186013136361 642 14 51 [31]
361 0.027525745932065714842681580721 0.058258721390791275265591085294 36.3296239988564537504804783011 0.859281117687863417913508943380 958 8 54 []
362 0.027507356961354668107337323553 0.058217534953459226976975812010 36.3539107521274743557588783241 0.860510491765324361578892507364 860 9 56 []
363 0.027498835150288865410102626224 0.058198449434585932384428670567 36.3651767260219878190579712627 0.862353028236045191917417568138 549 16 49 []
364 0.027477631374641584717234310272 0.058150964269857358063251691478 36.3932387899662582532794832041 0.863395622249087857653861180315 469 43 51 []
365 0.027439565955878242223113095749 0.058065728696440718535547788562 36.4437251525028206957125418694 0.863370512738246271947108902533 502 30 49 [31]
366 0.027435113116587444886448786376 0.058055758855727292900587117037 36.4496401290721707733044971147 0.865454954328495440813262664076 409 76 50 []
367 0.027426519504400928340023131257 0.058036518417491569120504095474 36.4610609756566991099589210583 0.867276009173595656005909506028 648 15 54 []
368 0.027414937103823514264257613074 0.058010587418515970562069730856 36.4764652281668636361893081994 0.868904805419495874042193254111 893 1 54 []
369 0.027340882685807322520742648691 0.057844822376784711210528921764 36.5752639185676665996942850617 0.866565314308690347677825959485 543 30 58 [31]
370 0.027309127069990295925715337179 0.057773755817861747019912311633 36.6177943892937234700447792728 0.866896467459628065885931960870 864 11 60 []
371 0.027299487078281464064664599738 0.057752184167404086557911424536 36.6307248605987807815736743544 0.868625864444175251843510710002 525 34 55 [31]
372 0.027251924371867310593946025048 0.057645764788483001912778574284 36.6946563609401239298980048909 0.867934921547842054878522313614 892 11 57 []
373 0.027186381476784496408825401141 0.057499150641426849405819022278 36.7831224929267880806115007252 0.866086995616240389271960138338 986 9 56 []
374 0.027126961138613369533062248300 0.057366267283775299227411890953 36.8636942004008164956641462372 0.864616990096925883934706915450 993 5 56 []
375 0.027094239318767516056815017003 0.057293104824389518842990768441 36.9082146294959640853456099935 0.864838600832974427302771795359 945 4 56 []
376 0.027074800650203223396389772618 0.057249646852033108656266738307 36.9347133121917927072904906423 0.865901023300908560146613113251 669 17 47 [31]
377 0.027058518216190496317891903348 0.057213247850734008497695389015 36.9569387359005054266366372502 0.867160012358847219081862395078 720 19 47 []
378 0.027052326542664962591171144236 0.057199407166774810571467496977 36.9653973540084511667953080628 0.869062308100927467134297854709 983 1 55 []
379 0.026903380084200303900006612639 0.056866565838049077639404989191 37.1700506356550901968156951922 0.861792632995806222213631478041 764 22 55 []
380 0.026832471789417316957880534436 0.056708185134537708982008480647 37.2682773263698488810404554285 0.859517715159439312889671930150 726 20 56 []
381 0.026788880496011489020621459485 0.056610843219599568287169569356 37.3289208613583837780709260874 0.858981831584077620080123735559 560 33 41 [31]
382 0.026733626338819392807478690181 0.056487483215865337544350504734 37.4060738085472125957137853760 0.857687306456410654744023017860 650 33 51 []
383 0.026699835736386722509955209191 0.056412056771473578729781687596 37.4534139413147407445242321348 0.857760070638882191321242774362 726 21 54 []
384 0.026659712014863401542458745536 0.056322507700211957428574557585 37.5097825303768112904109840321 0.857416831260931544886586368941 729 16 54 []
385 0.026643368682338708795194762395 0.056286036615084016277879560555 37.5327914395028268081686152670 0.858596020226225382775366708185 739 21 55 []
386 0.026602082386969425400374961156 0.056193915091774342012286452612 37.5910421392361704312653856454 0.858160351793152212827135156666 718 13 56 []
387 0.026591657618921388460669423214 0.056170656995976535798751724420 37.6057790127549794764796293509 0.859709366666620558171811535505 682 15 50 []
388 0.026557922158179378869820051745 0.056095398785091751407657355140 37.6535481218743373761334920680 0.859745250996175327579214371048 710 11 53 []
389 0.026546764852293296290366040614 0.056070511048491000231688823680 37.6693734835117946784332744972 0.861237000743175341636817979928 737 12 51 []
390 0.026442350327990230011286793931 0.055837658511702717651867421275 37.8181208400927242496733011719 0.856672033369193036357947742496 1022 6 57 []
391 0.026395019912199627612545261432 0.055732141810050909531551707504 37.8859346697369114419948490134 0.855796720757173553196167423312 846 11 59 []
392 0.026377703600881293299261074184 0.055693542726825002669528746858 37.9108058506878309465195934397 0.856860074923491137147254045276 947 7 58 []
393 0.026363102465002138414777865221 0.055660998123682124040494394316 37.9318026521169873271460444499 0.858095171675910279631227483075 801 17 56 []
394 0.026343787890580113718318762932 0.055617950777544967295857291686 37.9596132550693380035660288018 0.859018538460119734910027079702 639 25 53 []
395 0.026337326490696358978769898542 0.055603550720107658015328616320 37.9689259786200872463930277018 0.860776384053067470406591656559 715 24 55 []
396 0.026332973438690692065978117609 0.055593849607520171390171618486 37.9752025470360719731513284604 0.862670328530336251125072691722 593 35 55 []
397 0.026269974698195532049502969292 0.055453472009631279827847984787 38.0662719126520273223287835479 0.860715627875316352946266614028 1044 10 61 []
398 0.026253578455636563467676505233 0.055416943034741376790619978544 38.0900455794932991545956322939 0.861806886414646684654071598526 1042 15 57 []
399 0.026207111721201613179289756061 0.055313434138716571389104459966 38.1575814472908023014721245143 0.860916613328249586775277387853 1016 10 56 []
400 0.026202276126322286340628403706 0.055302663575708196614803610128 38.1646233777156426257766243803 0.862755828832312994976393376359 248 220 41 [31]
401 0.026179011231126183558433433149 0.055250847580954462737378107404 38.1985397069170145516570530150 0.863377495244150362312508278114 1034 7 57 []
402 0.026169800265054708425394272154 0.055230334157032980467736633794 38.2119844198936794025698367984 0.864921597496359848168971686494 1077 5 57 []
403 0.026163874452515016638282485040 0.055217137406491459645479412192 38.2206389888816507838415496040 0.866680513236086845892122201603 394 130 43 [31]
404 0.026144396976212074698686520855 0.055173763503857113479914749452 38.2491132195501410152294071500 0.867537978740190653460427049472 701 23 56 [31]
405 0.026133773812474668757848699713 0.055150108550113521667079225390 38.2646611689376842615124633975 0.868978740991200194321705600975 665 21 56 []
406 0.026132108259174107361742413119 0.055146399903090767607524759515 38.2671000013530678744625861936 0.871013334345541772808411433837 674 23 56 []
407 0.026124303718886219599635628363 0.055129022070354694442196288512 38.2785321576652485370785175899 0.872637215143982842429067056521 859 57 D1 []
408 0.025994878967184940182836539566 0.054840924314371135182381113412 38.4691154462525609171769798695 0.866135092984195892829461039770 661 24 56 []
409 0.025951755436606539867593386340 0.054744966855659491834230840390 38.5330388321030020872151621683 0.865379614473234026079120079106 1081 5 61 []
410 0.025930225254505962250078288965 0.054697064938229169734428487443 38.5650332839367608597834377716 0.866056665164614365025510368485 742 18 52 [31]
411 0.025905882907868864616991020484 0.054642911552590624394981699966 38.6012707444243031944033442414 0.866539753015466453814962600157 651 89 54 []
412 0.025843862506202623711095452903 0.054504962527329296637101601876 38.6939065226800466813934541511 0.864493898526891679488126453660 986 8 60 [31]
413 0.025779347877415608327799942340 0.054361504000360862456765877231 38.7907407415866125210545058322 0.862270996937400338245435050695 1054 4 58 []
414 0.025752362139890942422153338383 0.054301508503216270360450030470 38.8313893136420012017960281978 0.862550151320414615367872143083 1026 4 59 []
415 0.025724778364373036113830974579 0.054240190486140764487710501990 38.8730268473343942832977496212 0.862782351435721358038971571196 1040 3 58 []
416 0.025703564163671267281439013865 0.054193036720480670614420650815 38.9051103431551860439108768452 0.863435499766885619847909878663 764 11 52 [31]
417 0.025702953280603302529658392704 0.054191678945472470148828321392 38.9060350023920632774880450465 0.865469926069930555221804006003 429 119 46 []
418 0.025697890872699222569752637530 0.054180427154292944692574118207 38.9136993753201063528980134062 0.867203686781415581393111585632 1084 1 58 []
419 0.025549077679796327229351118040 0.053849779772487047603018818139 39.1403561620848234386720082956 0.859239728397450953898323027799 1002 15 62 []
420 0.025491968936157507913636789937 0.053722945171243479357856334105 39.2280408980732673336757518262 0.857444313068108287162182974099 1021 15 63 []
986 0.016915289170718942474147539201 0.035015161505901386262973318253 59.1181143820491570295416750014 0.886309982398534423886360400502 2081 90 C2 [31]



Applications

Suppose, you have to cut circles with a fixed diameter from a square-shaped plate of any material with fixed side length.
How many circles can you get in order to minimize the waste?
Please use the following form for this calculation:

Side length of square-shaped plate:       (enter only numbers, no units and/or text please)
Diameter of small circles:       (no units please)

Please enter your name and your e-mail address in the following forms. I can assure you that your data won't be misused for any purposes.

Name:       e-mail-address:




Updates

Please note that the results are taken from a running search. For updates look at the list below. All listed improvements are sorted by decreasing relative gain.

29-Nov-1999: First complete presentation from N=1 to N=200
17-Jan-2000: Improvements for N = 197, 183, 194, 174, 173, 192, 186, 198, 189, 177, 182, 196, 176, 193, 178, 179, 199.
05-May-2000: Improvements for N = 127, 98, 135, 198, 151, 184, 134, 136, 121, 100, 147, 162, 92, 150, 166, 183, 152, 84, 89, 185, 192, 95.
17-May-2000: Improvements for N = 84, 183, 134, 109, 198, 169, 151, 165, 187, 192, 197, 173, 196, 125, 121, 164, 144, 140, 103, 78, 178, 186, 176, 116, 106, 111, 179, 105, 175, 112, 181, 96, 193.
06-Jul-2000: Improvements for N = 47, 65. Many thanks to P. G. Szabó and L. G. Casado (see [7]) who suggested these improvements, and to Jerry Donovan (see [8]) whose 'Pack.exe' is great! After myriads of CPU cycles I never thought that better values are possible. And the story goes on, see below.
09-Jul-2000: Improvements for N = 86, 145, 101, 138, 134, 119, 103, 128, 106, 84, 140, 111, 136, 91, 196, 125, 124, 191, 96, 73, 182, 197, 165, 146, 139, 108, 115, 95, 166, 180, 186, 169, 123, 85.   50 yellow entries remaining...
12-Jul-2000: Improvements for N = 75, 139, 91, 145, 144, 146, 181, 182, 130, 107, 154.   3 yellow entries were blown away, 47 remaining...
13-Jul-2000: Improvements for N = 71, 89, 87.   1 yellow entry was blown away, 46 remaining...
14-Jul-2000: Improvements for N = 107, 106, 121, 140, 129, 102.   4 yellow entries were gone, 42 remaining...
18-Jul-2000: Improvements for N = 165, 115, 151, 192, 134, 133, 170, 181, 176, 183, 184, 122, 185.   7 yellow entries were gone, 35 remaining...
19-Aug-2000: Improvements for N = 140, 122, 166, 138, 124, 91, 147, 89, 135, 133, 146.   2 yellow entries were gone, 33 remaining...
23-Aug-2000: Improvements for N = 44, 51.   The lowest 2 yellow entries were gone, 31 remaining. This was hard to verify.
28-Aug-2000: Improvements for N = 68, 124, 182, 186, 123, 130, 125, 73, 76.   The lowest 3 yellow entries vanished, 28 remaining. The case N = 76 looks very easy, but was so diffucult to guess.
31-Aug-2000: Improvements for N = 131, 89.   Only 26 yellow entries remaining.
04-Sep-2000: New packings for N = 208, 255, 270, 295, 304, 340, 378.
12-Sep-2000: New packings for N = 449, 986.
15-Sep-2000: Improvement for N = 97.    This packing was found by Péter Gábor Szabó.
18-Sep-2000: New packings for N = 407, 437, 621.
06-Oct-2000: Boll et. al. confirmed N = 32, 37, 48, 50 in [14]; missing my last green entry N = 47 below 50 :-)
11-Oct-2000: Improvements for N = 186, 136, 103, 95, 96, 108; erasing 2 yellow entries, 24 remain. Note that the new packings for N = 96, 103, 136, 186 have symmetries while the former were not symmetric.
12-Oct-2000: Density plots added (see above).
18-Oct-2000: Improvements for N = 180, 131. New packings for N = 261, 418, 559, 658.
28-Oct-2002: Improvements for N = 112, 85, 139, 148, 156, 178, 200, 116. These improvements are due to Douglas Hanson, an 8th grade student(!) from Texas. They were found by running Jerry Donovan's program.
13-Dec-2002: New packings found by Douglas Hanson for N = 201 to 250.
31-Dec-2002: Improvements for N = 217, 211, 242, 226, 227, 210, 202, 213, 206, 235, 249, 215, 248, 218, 203, 233, 239, 201, 229, 204, 225, 221, 234, 228, 207, 243, 231, 209, 205, 246, 230, 236, 220, 224, 237, 245, 244, 250, 185, 223.
31-Dec-2002: New packings found for N = 251 to 300 (except 255, 261, 270, 295).
31-Dec-2002: Improvements for N = 197, 107, 193, 171, 155, 182, 194, 125, 122, 152, 158. These packings are due to David Bass. Many thanks to him for reporting them.
05-Jan-2003: Improvement for N = 97. It seems to be very unlikely to find still better packings for N < 100, but it is possible :-)
06-Jan-2003: Improvement for N = 199 (slightly better than Donovan's packing).
02-Mar-2005: Improved packings found by Bernardetta Addis, Marco Locatelli and Fabio Schoen for N = 86, 59, 66, 68, 53, 73, 85, 77. More about the method used for this achievement can be found in a short report [20].
23-Jun-2005: Slightly improved packing for N = 78, 86, 88, 100, 101, 106, 108, 113, 115, 116, 130, 133, 134, 135, 146, 155, 157 found by Bernardetta Addis, Marco Locatelli and Fabio Schoen [19].
01-Jul-2005: Improved packing for N = 108 found by Bernardetta Addis, Marco Locatelli and Fabio Schoen [19].
08-Jul-2005: Further improvements for N = 111, 130 by Bernardetta Addis, Marco Locatelli and Fabio Schoen [19].
31-Aug-2005: Further improvements for N = 88, 95, 98, 104, 105, 106, 107, 109, 111, 115, 117, 118, 119, 121, 122, 123 by Bernardetta Addis, Marco Locatelli and Fabio Schoen [19].
04-Sep-2009: Reorganization of the page, all numerical values are now given with 30 decimal places. Attempt to clearify the authorship of each packing.
10-Feb-2010: Astonishing improvements for such small values of N = 67, 75, 86, 91, 92 and 93 by David W. Cantrell [24].
11-Feb-2010: Further improvements for N = 101, 104, 105, 106, 107, 108, 109, 111, 119, 121, 122, 123, 124, 125, 127, 130, 136, 139, 140, 141, 144, 145, 147 and 150 by David W. Cantrell [24].
13-Feb-2010: Further improvements for N = 151, 156, 157, 160, 162, 164, 166, 169, 170, 173, 175, 178, 179, 182, 184, 186, 187, 193, 196, 197 and 199 by David W. Cantrell [24].
15-Feb-2010: Further improvements for N = 201, 211, 213, 214, 215, 216, 217, 218, 220, 225, 226, 228, 231, 232, 248 and 249 by David W. Cantrell [24].
22-Feb-2010: Some improvements for N = 157, 171, 244, 287, 289 and 290 by Eckard Specht [31].
25-Feb-2010: Overwhelming many more better packings than before for N = 129, 133, 134, 135, 136, 138, 144, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 162, 166, 167, 168, 170, 171, 173, 175, 176, 177, 179, 182, 184, 185, 186, 187, 192, 193, 194, 196, 198, 199 and 200 by Wenqi Huang and Tao Ye [25]. Totally, this are 42 better packings in the range 129 <= N <= 200.
01-Mar-2010: Further improvements for N = 129 and 145 by David W. Cantrell [24].
08-Mar-2010: Some slightly refinements of previous packings for N = 126, 134, 136, 144, 146, 147, 148, 150, 152, 154, 162, 166, 173, 182, 184 and 200 by David W. Cantrell [24].


References

[1]   J. Schaer, A. Meir, On a geometric extremum problem, Canadian Mathematical Bulletin, 8 (1965), p. 21-27.
[2]   J. Schaer, The densest packing of nine circles in a square, Canadian Mathematical Bulletin, 8 (1965), p. 273-277.
[3]   B.L. Schwartz, Separating points in a square, J. of Recreational Mathematics, 3 (1970), p. 195-204.
[4]   M. Goldberg, The Packing of Equal Circles in a Square, Math. Mag. (1970), p. 24-30.
[5]   G. Wengerodt, Die dichteste Packung von 16 Kreisen in einem Quadrat, In German. Beiträge zur Algebra und Geometrie, Contributions to Algebra and Geometry, 16 (1983), p. 173-190.
[6]   G. Wengerodt, Die dichteste Packung von 14 Kreisen in einem Quadrat, In German. Beiträge zur Algebra und Geometrie, Contributions to Algebra and Geometry, 25 (1987), p. 25-46.
[7]   G. Wengerodt, Die dichteste Packung von 25 Kreisen in einem Quadrat, In German. Ann. Univ. Sci. Budapest Eötvös Sect. Math., 30 (1987), p. 3-15.
[8]   K. Kirchner, G. Wengerodt, Die dichteste Packung von 36 Kreisen in einem Quadrat, In German. Beiträge zur Algebra und Geometrie, Contributions to Algebra and Geometry, 25 (1987), p. 147-159.
[9]   C. de Groot, R. Peikert, D. Würtz, The Optimal Packing of Ten Equal Circles in a Square, IPS Research Report, Eidgenössische Technische Hochschule, Zürich, No. 90-12, August, 1990.
[10]   R. Peikert, D. Würtz, M. Monagan, C. de Groot, Packing Circles in a Square: A Review and New Results, In: P. Kall (ed.): System Modelling and Optimization, Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, 180 (1992), p. 45-54.
[11]   R.L. Graham, B.D. Lubachevsky, Repeated Patterns of Dense Packings of Equal Disks in a Square, The Electronic Journal of Combinatorics 3 (1996) #R16, p. 211-227.
[12]   K.J. Nurmela, P.R.J. Östergård, Packing up to 50 Equal Circles in a Square, Discrete Comput. Geom. 18 (1997) 1, p. 111-120.
[13]   K.J. Nurmela, P.R.J. Östergård, More Optimal Packings of Equal Circles in a Square, Discrete Comput. Geom. 22 (1999) 1, p. 439-457.
[14]   D.W. Boll, J. Donovan, R.L. Graham, B.D. Lubachevsky, Improving Dense Packings of Equal Disks in a Square, The Electronic Journal of Combinatorics (2000) #R46.
[15]   Douglas Hanson,   private communication, October–December 2002.
[16]   David Bass,   private communication, November 2002–January 2003.
[17]   M.Cs. Markót, Optimal Packing of 28 Equal Circles in a Unit Square – the First Reliable Solution, Numerical Algorithms 37 (2004), p. 253-261.
[18]   M.Cs. Markót, T. Csendes, A New Verified Optimization Technique for the "Packing Circles in a Unit Square" Problems, SIAM J. Optimization 16 (2005), 193-219.
[19]   B. Addis, M. Locatelli, F. Schoen, Packing circles in a square: new putative optima obtained via global optimization, DSI report 01-2005, Università di Firenze.
[20]   Bernardetta Addis, Marco Locatelli and Fabio Schoen,   private communication
[21]   P.G. Szabó, M.Cs. Markót, T. Csendes, E. Specht, L.G. Casado, I. García, New Approaches to Circle Packing in a Square - With Program Codes, Springer, New York, 2007.
[22]   P.G. Szabó, E. Specht, Packing up to 200 Equal Circles in a Square, In: A. Törn, J. Žilinskas (ed.): Models and Algorithms for Global Optimization, Springer-Verlag, Berlin, 2007, p. 141-156.
[23]   Simon Gravel, Using Symmetries to Solve Asymmetric Problems, PhD Thesis, Faculty of the Graduate School of Cornell University, January 2009.
[24]   David W. Cantrell, Packing unit circles in squares: new results and Packing unit circles in squares: new results (sci.math forum) and private communication, February–March 2010.
[25]   Wenqi Huang, Tao Ye, private communication, February 2010.
[31]   Eckard Specht, program csq, 1999–2010.


©  E. Specht     11-Mar-2010