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

Last update: 11-Feb-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 ratio density contacts loose boundary symmetry group reference
1 0.500000000000000000000000000000 2.0000000000000000000000000000 0.785398163397448309615660845822 4 - 1 D4 [-]
2 0.292893218813452475599155637896 3.4142135623730950488016887242 0.539012084452647221356168169719 5 - 2 D2 [-]
3 0.254333095030249817754744760429 3.9318516525781365734994863995 0.609644808741350951447230676805 7 - 3 D1 [-]
4 0.250000000000000000000000000000 4.0000000000000000000000000000 0.785398163397448309615660845821 12 - 4 D4 [-]
5 0.207106781186547524400844362105 4.8284271247461900976033774484 0.673765105565809026695210212148 12 - 4 D4 [-]
6 0.187680601147476864319898426192 5.3282011773513747321100504007 0.663956909464133571452413649192 13 - 5 D1 [13]
7 0.174457630187009438959427204500 5.7320508075688772935274463415 0.669310826840792318436251021849 14 1 6 D1 []
8 0.170540688701054438818560595676 5.8637033051562731469989727989 0.730963825253907837357358208948 20 - 4 D4 [11]
9 0.166666666666666666666666666667 6.0000000000000000000000000000 0.785398163397448309615660845821 24 - 8 D4 [12]
10 0.148204322565228798668007362743 6.7474415232381126762112368561 0.690035785264180329182022896993 21 - 8 []
11 0.142399237695800384587114500527 7.0225095034303813492095760345 0.700741577756104830198542148982 20 2 8 D1 []
12 0.139958844038428028961026945453 7.1449575542752651213595466567 0.738468223884044498369057950513 25 - 8 C2 []
13 0.133993513499008491414263236065 7.4630478288592654598767430406 0.733264694904101752789658211573 25 1 9 []
14 0.129331793710034021408259201773 7.7320508075688772935274463415 0.735679255542682563361972309844 32 1 10 D1 []
15 0.127166547515124908877372380214 7.8637033051562731469989727989 0.762056010926688689309038346005 36 - 8 D1 []
16 0.125000000000000000000000000000 8.0000000000000000000000000000 0.785398163397448309615660845820 40 - 12 D2 []
17 0.117196742782948687473176894856 8.5326603474980966644272735286 0.733550263302331439285589240584 34 1 11 []
18 0.115521432463999509608513951182 8.6564023547027494642201008015 0.754653357875667578715912508591 38 - 10 D1 []
19 0.112265437570996304738752306983 8.9074609393260787738262975010 0.752307896742078979509898727718 37 2 11 []
20 0.111382347512479750227357863499 8.9780833528217365165192364463 0.779493686867621861721679242589 44 - 11 D1 []
21 0.106860212352064428580553201716 9.3580199588727565928284980056 0.753357702889674769521037881736 39 2 11 []
22 0.105665296756976756533092354860 9.4638450909756523911277579008 0.771680112098219822603167037094 43 1 13 []
23 0.102802323379784112346596984546 9.7274066103125462939979455978 0.763631032126130708837463558612 56 - 8 D1 []
24 0.101381800431613524388964772877 9.8637033051562731469989727989 0.774963259757823779541804629689 56 - 12 []
25 0.100000000000000000000000000000 10.0000000000000000000000000000 0.785398163397448309615660845820 60 - 16 C4 []
26 0.096362339009887092432024697394 10.3774982039134263443391268496 0.758469090483941352861819034654 56 2 13 []
27 0.095420001747936516364270819029 10.4799830400508798304005935979 0.772311456467343180199003969254 55 - 13 []
28 0.093672833832785071755016932236 10.6754536943453127881514813657 0.771854111403716173720195749794 57 1 14 []
29 0.092463144040309496841524670049 10.8151200175936906689015127153 0.778906241779735128443167917038 65 1 14 []
30 0.091671057985988438718806599233 10.9085683308339920507688052094 0.792019026460736228749276303655 65 - 14 C2 []
31 0.089338333351234633748802313039 11.1934033520469771160959973707 0.777297478729284489847888300458 55 4 16 D1 []
32 0.087858157087794923008578158871 11.3819824265232393752398264833 0.776004124474520585203569503089 63 3 15 []
33 0.087230014135567983530352112436 11.4639440324503011568460735311 0.788852303933884675540557918277 65 1 17 []
34 0.085270344350527219409280297838 11.7274066103125462939979455978 0.776649064332277077192193097006 80 - 12 []
35 0.084290712122358459903667877331 11.8637033051562731469989727989 0.781227212998714358933378085373 80 - 16 D1 []
36 0.083333333333333333333333333333 12.0000000000000000000000000000 0.785398163397448309615660845820 84 - 20 C4 []
37 0.082089766428752816962342111380 12.1817863968210698362744366706 0.783302723724208829617437864294 77 - 16 []
38 0.081709776125419800616581793806 12.2384376438023473575039603473 0.797042556834780531205882269712 77 - 16 []
39 0.081367527046974026631769391643 12.2899151085505302427190933133 0.811179027291583405559076523463 80 - 16 C2 [14]
40 0.079186752517282867898915819775 12.6283749264972757200437889959 0.787979518878382871537374468088 85 2 17 []
41 0.078450210116920559483053007213 12.7469384531873251117784855342 0.792723899305009720099449735151 100 1 18 []
42 0.077801502898165532306768693062 12.8532221454500825687745597891 0.798684278653432047162866645969 90 - 17 [14]
43 0.076339810579744074677200744991 13.0993251411778974447542878630 0.787264166427460630119036545330 84 1 16 []
44 0.075781986017897705750184232945 13.1957481262608554872478304964 0.793842807841790391912527518373 82 4 20 []
45 0.074727343686985328535051089524 13.3819824265232393752398264833 0.789444268494961791151996201485 94 3 19 []
46 0.074272199909420921796026120347 13.4639878880598071936730888487 0.797187131992481565987478372966 91 1 21 []
47 0.073113151270516836978943966705 13.6774298826215947135408970296 0.789293882075948475945003280435 92 3 19 []
48 0.072432291280497581081203202077 13.8059970535441329898933067156 0.791144033840820484633598422522 111 2 18 []
49 0.071692681703623580727975837865 13.9484250865936955971142641355 0.791216989526925284715824261840 120 1 19 []
50 0.071377103864521467530864106392 14.0100949164043850676229061557 0.800272183997614540398268557996 100 1 19 []
51 0.071043138115537018210695844258 14.0759547864243574869673926273 0.808656948184898971918843953677 97 3 19 []
52 0.070957693072547251836538973586 14.0929046125780681731433570098 0.822530842067761468539017516652 105 - 19 [14]
53 0.069947252562026650314667369460 14.2964871867302691916392131754 0.814642500051039348061532079567 108 4 20 []
54 0.068645540104688887910867652499 14.5675887825332181932543170306 0.799407623083866967669545766005 115 2 21 []
55 0.068055360558639717449689169779 14.6939196529324490658396620664 0.800271297244825744992218341275 113 2 21 []
56 0.067532596322265705546305516123 14.8076640682967035879634496450 0.802351729502980578932357268910 119 - 20 C2 []
57 0.067004873106043976620256140932 14.9242876472188700315027030559 0.803965673845865929966857781554 113 3 22 []
58 0.066232983745862232302511874085 15.0982175865279950794785486574 0.799330724318471867207813535842 125 2 20 []
59 0.065807496903601046306129157869 15.1958370558427832726382997809 0.802698827038006793401631671012 121 4 24 []
60 0.065030412648295594469592436245 15.3774204910601833698861754239 0.797139156419255784870559282621 129 3 21 []
61 0.064666268906273511710193139988 15.4640126438930255892954777349 0.801374124592793740357582679375 121 1 25 []
62 0.064252183294482911181458806768 15.5636734617524844300701130769 0.804113477816698520155376391258 128 2 21 []
63 0.064011528204864918571082299195 15.6221860037392348522231816922 0.810973780649873545826894093593 145 1 21 []
64 0.063458986813059169965842890472 15.7582093604149832660896556588 0.809685038563171020371396760572 129 2 23 []
65 0.063203957071856799974346494473 15.8217941775875915576901053101 0.815740018110047678672004161765 137 2 23 []
66 0.062862256903173650552663945791 15.9077966535674001923838046767 0.819358090828537580877710854038 141 2 22 []
67 0.062587429542203145120191597701 15.9776493029753354511388551164 0.824515655814026443272007820247 138 1 19 [27]
68 0.062520077997967509748911370819 15.9948616831941476534722680152 0.835021788301676498075386011068 156 3 20 []
69 0.061383571685582519393723042252 16.2910038067217133186173719170 0.816776573752215775394223611561 137 1 23 []
70 0.060596693631158157824224372564 16.5025505531181477757014705618 0.807506020669364260500664086986 135 5 23 []
71 0.060096531351830846614976309881 16.6398954732607026528365687555 0.805576954556086927019502619207 142 3 24 []
72 0.059801002126807327595495341667 16.7221277977835833294685943084 0.808908302137999085775684834728 180 1 23 []
73 0.059366050583080470834219283052 16.8446442062124214969120792232 0.808256206862671849525236295628 151 3 24 []
74 0.059082376336698616287994709302 16.9255209760216905870553868736 0.811516775123364681685174979476 153 3 26 []
75 0.058494535281249486795264924005 17.0956140636363263363384238859 0.806198017941952636005393179091 148 2 22 [27]
76 0.058198524936106167091962265813 17.1825660718696909505414496828 0.808699956656388340271328954092 191 1 24 []
77 0.057852577916326407945864081743 17.2853144322509594703218721469 0.809628951523275454193704449070 174 - 24 []
78 0.057702476734813469658915612215 17.3302786394378955731266815014 0.815893333309535399379288603288 183 1 24 []
79 0.057508497795177150643081806966 17.3887345060135312216405790393 0.820806922795721980813682909691 177 2 24 []
80 0.057370684146686868321824625645 17.4305050545183285645499950863 0.827217888559947854040065137730 193 - 24 C2 [14]
81 0.056869921111948347401531481988 17.5839878172417651702227896944 0.823000583646207165131312975821 198 3 26 []
82 0.056512271650043116731600865688 17.6952716782043752389211638037 0.822714696667481465122043671825 203 4 26 []
83 0.056129373021650586640744529302 17.8159837918423478438752023974 0.821501474386234485434434731555 209 2 24 []
84 0.055856665888395536222323748893 17.9029661741366901471419975275 0.823339926925100779747488690108 184 2 23 []
85 0.055680181768308624302233825464 17.9597114851584044953610605853 0.827885139567273523102982634740 203 3 23 []
86 0.055572999412015161187514147010 17.9943499645584179987825369965 0.834403272414726330623353074849 191 2 24 [27]
87 0.054695259720704525732612598175 18.2831200565897792110838339099 0.817651999581176324044528222885 175 - 26 []
88 0.054406636912959447376712741080 18.3801105295262961716797498050 0.818344762893455338659140369863 187 5 27 []
89 0.053947040858284291446539903179 18.5366979187411090776201305739 0.813720269423130443280172100705 182 5 29 []
90 0.053749948306745947917965570800 18.6046690555513310423084633158 0.816861606107565255741672170260 230 1 26 []
91 0.053496719884488436041701634807 18.6927348472808620204602434899 0.818173810485969463957034440533 182 2 24 [27]
92 0.053317085175319528829912286052 18.7557139838338322788212338613 0.821619044642187492376904130879 200 - 28 C2 [27]
93 0.052926433388240551617979070855 18.8941505403268830126971455556 0.818423476521143216019926914278 194 1 27 [27]
94 0.052795362726845093818338316930 18.9410574783592789859725041814 0.823131614711695270716775765936 224 2 26 []
95 0.052420366495943589361713667424 19.0765549126288985463205653801 0.820112788103383671049325920561 222 4 26 []
96 0.052275425469925719068554307283 19.1294473648101510739994642848 0.824168967917176961985882367601 224 4 28 []
97 0.052114795502244668305008905267 19.1884087879981873280117951648 0.827644213165413536281305095882 234 2 27 []
98 0.052032987702942098030211872968 19.2185773707446951268196647451 0.833553492453149745406906221051 204 2 27 []
99 0.051978606449505864308001821124 19.2386843031553902034492536974 0.840299937025485161853712217758 199 - 27 [14]
100 0.051401071774381815590184511455 19.4548472527842865834291824527 0.830030826635912821740802332446 216 7 28 []
101 0.051078356337468287675013092544 19.5777638848267854447349965465 0.827837458204886740849985234210 211 6 30 [27]
102 0.050774421023984229946427962168 19.6949562364804049826470402687 0.826114042027870748552195664642 252 4 29 []
103 0.050534497607782334686032855584 19.7884622849401705257411549689 0.826348041985204520696807001421 257 3 29 []
104 0.050348792811532572594881586662 19.8614493845609421072948689140 0.828249793523388576412378912432 213 2 27 [27]
105 0.050242948365384132560891118091 19.9032905618446879938853799268 0.832701612019226131524236043106 232 1 27 [27]
106 0.050015200416806532090506982216 19.9939216811369911786035320446 0.833028317432244445190333516957 246 4 25 [27]
107 0.049506166856097478668626121301 20.1995036882326096944459152817 0.823857790657640543813394281503 219 9 30 [27]
108 0.049243520787042327172316177682 20.3072401001663263129466670596 0.822757444973615405175483101189 221 5 29 [27]
109 0.049060056454336484181017952109 20.3831807843671704330002390822 0.824199710946182236760458317486 238 3 30 [27]
110 0.048779469320530920341849009163 20.5004280269836257772527740355 0.822274269043788233219044672315 285 1 30 []
111 0.048643936344712632841290942957 20.5575468422940508604125053146 0.825145003926722239389734741481 226 3 27 [27]
112 0.048445405902771448593353411940 20.6417921651223554747773961460 0.825796605618670452719327830572 260 1 29 []
113 0.048257588493549543124104737775 20.7221295389359790112102660456 0.826722100560516881572374636295 278 1 29 []
114 0.048167755522460786854554765216 20.7607763565752327263802781870 0.830935940404435353549195715343 280 1 29 []
115 0.047914247536590393009947438372 20.8706188954827236356670742290 0.829424877813220596562781585276 282 5 30 []
116 0.047775960993939870390957868549 20.9310284753214016021781164433 0.831814956318629673062000464849 249 4 30 []
117 0.047643526600358961715281019719 20.9892103157717480356010526685 0.834340904498932290987139503332 262 2 30 []
118 0.047572362680372825614856108900 21.0206082619599455807412376481 0.838960130138630368263454714311 245 2 30 []
119 0.047544646218204690832216618999 21.0328623628942525050684892224 0.845084379636190758907117056070 284 2 30 [27]
120 0.047529921591019828363001359062 21.0393782805847764767908357685 0.851658165016251855319555118511 241 - 30 C2 [14]
121 0.046891999625074798386140685493 21.3255994198478346417907240853 0.835858472004155186816930239790 259 9 32 [27]
122 0.046626691141762885306969731587 21.4469432746068006493711998339 0.833256858758184480156083575990 282 5 28 [27]
123 0.046348795248977138929874199757 21.5755338327174485430593533674 0.830102810046943880404118740922 264 6 33 [27]
124 0.046099752156975895127016673745 21.6920905907447107815572004626 0.827882569489952205937044175731 258 7 30 [27]
125 0.045978336543189571191865436362 21.7493731871020148446912225388 0.830168776967464070098670387584 309 4 31 [27]
126 0.045828714872503009682812228103 21.8203805797747733380020883538 0.831372731264193110404082757787 211 8 28 []
127 0.045663869681879453518190979282 21.8991514947500078507975203860 0.831953432091018222557685985657 276 2 30 [27]
128 0.045473271986833679888176503111 21.9909400909074622296108483009 0.831519142296790878206793082850 328 - 33 []
129 0.045185467608831612104164633677 22.1310092142223955019178434598 0.827441206644255651081870440120 249 6 34 []
130 0.045043162014594747473845715283 22.2009280715235546987376501815 0.828611517278489372066392262331 276 8 34 [27]
131 0.044887232367730475076783281761 22.2780498429415774060391285667 0.829214381766750148879227621814 338 2 32 []
132 0.044746370934914279977793589814 22.3481810727074930010259917432 0.830308417787970714465006006575 347 1 32 []
133 0.044602141236136064962501063613 22.4204482629146339073722017375 0.831214155962611458743650949790 257 2 31 []
134 0.044548122181145672249684002826 22.4476352995016942185807271535 0.835436557218761601501304303651 156 26 27 []
135 0.044430426948941603084710623941 22.5070985959503916881912326876 0.837229677160991126103728588392 243 7 33 []
136 0.044353299130704909616642392922 22.5462371367932769294128170197 0.840505655546313099184930044001 272 3 33 [27]
137 0.044293023545787681269568493800 22.5769188903136231781833500721 0.844386136710202177869379241432 308 - 32 C2 []
138 0.044032132019454350330761647753 22.7106877213708008691454110783 0.840559358467954410601720011108 326 6 33 []
139 0.043945742387968420766368204241 22.7553329551620589174004959142 0.843331425575420809020868982387 323 5 34 [27]
140 0.043804839097573616473156129456 22.8285280941801419494270256393 0.843960434882289276826864877339 355 5 33 [27]
141 0.043658469168967682652713052143 22.9050633023751825283650358015 0.844317892155816320237596014665 321 4 33 [27]
142 0.043576632732555169816354740508 22.9480787590299834948168386303 0.847121206056549661879903943053 341 2 33 []
143 0.043530511915109298389250325442 22.9723923750343783320100724424 0.851282016761247835788454445055 297 - 33 [14]
144 0.043002745898269098612524241494 23.2543289762398858880728207951 0.836574727484787381287702681585 300 11 37 [27]
145 0.042871345178979340468168525262 23.3256035196749452583441896322 0.837244100791533818442332232078 342 6 36 [27]
146 0.042646910193997457525221727652 23.4483575820869142906926432599 0.834214763956568443878128654976 275 7 36 []
147 0.042439928742539257695892878255 23.5627162822650910828112276054 0.831795371705419835550636106619 378 3 34 [27]
148 0.042329005698661792692457808386 23.6244623159578356953461324463 0.833081944918336601496910192450 352 2 32 []
149 0.042251859906775125523353939389 23.6675971710217911219134467523 0.835656514978912952927948536022 220 13 31 []
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307 0.029820950476428174990100459714 33.5334717379462847028058893293 0.857691707535787552498032720425 501 28 48 []
308 0.029767249618081149680466051945 33.5939669546286350598118190615 0.857389198363027869419848846573 667 19 52 []
309 0.029706288064074234802338833590 33.6629065820366051636404104713 0.856653371303694728588686563483 698 14 48 []
310 0.029656518475696174601061057366 33.7193996935112394500430810549 0.856548379880856261719406759518 698 14 48 []
311 0.029583285518711008228569742965 33.8028715359392449003595237153 0.855072760983037385526074540727 585 12 47 []
312 0.029572069875278565950405575745 33.8156917732692223216437009752 0.857171877644628101282242743707 553 6 43 []
313 0.029568593938052276561897792912 33.8196669782489954286391344818 0.859717083374448052182842053915 355 62 42 []
314 0.029530610813536368152826276452 33.8631668106782168972303015807 0.860249404878198546094947408812 530 14 48 []
315 0.029422061562329064058122883968 33.9881009997057301907750791383 0.856656325824769353300327536259 476 26 46 []
316 0.029319864182216755741006137425 34.1065699958639462277222910534 0.853416162867336677143576008583 834 8 51 []
317 0.029276005887819067701010914445 34.1576649434980542946366412515 0.853557507537954985915568155034 834 9 51 []
318 0.029213599496236318541198773256 34.2306328985181438652633008225 0.852603546283592545811085329732 816 11 52 []
319 0.029173220386765374105055657078 34.2780120515476812400398278609 0.852921970292297567880695389020 841 7 52 []
320 0.029151530425642118415286301075 34.3035163299826335088698452371 0.854323928265512430254898114093 499 16 47 []
321 0.029120786271087519361619541253 34.3397321312318699434092123501 0.855187016751047296948568907521 845 8 52 []
322 0.029045303347491187988349728555 34.4289742143931102739840003410 0.853409705382312009565714433595 794 5 52 []
323 0.029018830063112566098561619543 34.4603830624844901116686761371 0.854500248766461663214987588822 791 7 55 []
324 0.028998428367843685575237764686 34.4846274879123630135110565257 0.855940950987282659558915671129 637 3 51 []
325 0.028985678591978504194357821574 34.4997960571031781663225821056 0.857827921686103530810665888167 482 22 47 []
326 0.028953020606938246326144518627 34.5387106090188710053695091407 0.858529517932905369590390712400 833 5 52 []
327 0.028893699346404522202981066868 34.6096215652786643224065765825 0.857637819927309971349180404801 664 28 51 []
328 0.028877258989449583336392590030 34.6293254621345402763187678862 0.859281876950629637952286322657 684 8 51 []
329 0.028866100388198633869950968788 34.6427119199249832549203128138 0.861235664308605839264064589300 531 16 50 []
330 0.028847001355832402330188044251 34.6656481782920564389078339600 0.862710656197346821958568953557 783 4 51 []
331 0.028769555882845487426314942600 34.7589654867169190640919952151 0.860684896438454362459331240066 572 17 55 []
332 0.028742782823331200209363092180 34.7913424439987148316952160021 0.861679147571684888267154269805 544 33 46 []
333 0.028715442878125320052297460849 34.8244672472655497639280208925 0.862631164139171184362066767419 705 7 57 []
334 0.028695106059794111686735731340 34.8491480713201111031230960722 0.863996549502931970942786388073 559 18 55 []
335 0.028662406222935577541439096582 34.8889061240016474352028007548 0.864609440860300049153713554259 703 2 54 []
336 0.028634189510812885337061605932 34.9232863609559652102468509845 0.865483793504162268012015410299 850 5 55 []
337 0.028618729401824516677483055243 34.9421522513940629548067723016 0.867122529453443122424330279103 858 3 53 []
338 0.028592061739494067090919854783 34.9747426090195226779232099922 0.868075539402315333288355046264 862 2 52 []
339 0.028581764393693076300770801073 34.9873431963725270920736916955 0.870016803610931843776635735881 669 2 49 []
340 0.028577667513014682232203108524 34.9923589650759834585643647504 0.872333092426712558911558687900 697 - 52 C2 [14]
341 0.028333227924298013635093898281 35.2942489529200397848615875825 0.859995863320519207559617546867 775 17 56 []
342 0.028286101346286162302762086372 35.3530515838053269113501479605 0.859650984535727987866363031001 776 12 51 []
343 0.028212960870886447875413311673 35.4447023329593594308306614878 0.857711683200454646144065098284 708 26 56 []
344 0.028167710917871075969291208774 35.5016423917339854472417458201 0.857455173841992805343988960707 808 8 55 []
345 0.028114552563282267697884444166 35.5687680872433483057963377621 0.856705036660797690451477484854 824 14 55 []
346 0.028058655901919817634587085606 35.6396259142113224917452378929 0.855775202082306915361541698301 636 5 51 []
347 0.028031077106124179159679212292 35.6746904949122273986058299964 0.856562228471600457324671624452 741 9 51 []
348 0.028022303432320904502782306441 35.6858601012292488264373055668 0.858493041719361617076667433059 406 97 50 []
349 0.027981299239883111631091113268 35.7381546663369793763286900422 0.858442185607246805053600139697 229 177 54 []
350 0.027955208807340184852260399770 35.7715088766366328267672734506 0.859297202769774402150208497477 563 19 49 []
351 0.027896974684150346428931569060 35.8461808608999293565379835780 0.858165805832524843073871400842 898 12 53 []
352 0.027810212157221782215327452683 35.9580140685952679257401621760 0.855265867311641269854890255511 921 9 56 []
353 0.027772625907504946056777005206 36.0066780624359982992347099783 0.855378769233573949861191700882 894 9 55 []
354 0.027720781199118604021691233142 36.0740194447260200466505528722 0.854602314442041272525980328192 954 5 53 []
355 0.027685205849154016676113728734 36.1203743778758009262268814835 0.854818159734924718660789535135 935 6 53 []
356 0.027672738596821843657735078015 36.1366474988076501917496291192 0.856454216813396703158742732327 921 5 53 []
357 0.027666544055267733481684937934 36.1447384972392007461931817843 0.858475518566455481255261486590 431 92 51 []
358 0.027656743521373677246323263945 36.1575468647341023101423942540 0.860270407410508095896519585791 480 45 45 []
359 0.027578247508017814649252885822 36.2604621526175778705323082507 0.857783427209014194423208168144 964 6 53 []
360 0.027538031695477305812147667795 36.3134159717099343507661286644 0.857665941674113043330293429138 637 11 54 []
361 0.027525745932065714842681580721 36.3296239988564537504804783007 0.859281117687863417913508943402 958 8 54 []
362 0.027507356961354668107337323553 36.3539107521274743557588783247 0.860510491765324361578892507338 860 9 56 []
363 0.027498835150288865410102626224 36.3651767260219878190579712628 0.862353028236045191917417568136 550 16 49 []
364 0.027477631374641584717234310272 36.3932387899662582532794832046 0.863395622249087857653861180291 469 43 51 []
365 0.027439565947660875192421723190 36.4437251634166764802747412983 0.863370512221136606510567789734 847 8 55 []
366 0.027435113116587444886448786376 36.4496401290721707733044971141 0.865454954328495440813262664107 412 74 49 []
367 0.027426519504400928340023131257 36.4610609756566991099589210588 0.867276009173595656005909506004 646 15 54 []
368 0.027414937101253402928789731898 36.4764652315864803369121236959 0.868904805256578696352588189824 779 3 54 []
369 0.027340881072318725157450956749 36.5752660770120533063995215832 0.866565212030118853196870540887 860 13 58 []
370 0.027309127069990295925715337179 36.6177943892937234700447792724 0.866896467459628065885931960885 865 11 60 []
371 0.027299476517260438876379256483 36.6307390314879249772574498395 0.868625192374827085921879247157 619 13 59 []
372 0.027251924371867310593946025048 36.6946563609401239298980048914 0.867934921547842054878522313592 892 11 57 []
373 0.027186381476784496408825401141 36.7831224929267880806115007254 0.866086995616240389271960138331 986 9 56 []
374 0.027126961138613369533062248300 36.8636942004008164956641462376 0.864616990096925883934706915434 993 5 56 []
375 0.027094239318767516056815017003 36.9082146294959640853456099930 0.864838600832974427302771795383 945 4 56 []
376 0.027074800579973301986207093927 36.9347134079975588683727897550 0.865901018808749114747918266617 731 14 52 []
377 0.027058518216190496317891903348 36.9569387359005054266366372498 0.867160012358847219081862395097 721 19 47 []
378 0.027052326542664962591171144236 36.9653973540084511667953080632 0.869062308100927467134297854689 982 1 55 []
379 0.026903380084200303900006612639 37.1700506356550901968156951925 0.861792632995806222213631478029 766 22 55 []
380 0.026832471789417316957880534436 37.2682773263698488810404554284 0.859517715159439312889671930153 725 20 56 []
381 0.026788876126865026486604508829 37.3289269495392301577035451075 0.858981551391919134385380290482 738 19 57 []
382 0.026733626338819392807478690181 37.4060738085472125957137853765 0.857687306456410654744023017839 649 33 51 []
383 0.026699835736386722509955209191 37.4534139413147407445242321345 0.857760070638882191321242774375 726 21 54 []
384 0.026659712014863401542458745536 37.5097825303768112904109840315 0.857416831260931544886586368968 729 16 54 []
385 0.026643368682338708795194762395 37.5327914395028268081686152667 0.858596020226225382775366708199 739 21 55 []
386 0.026602082386969425400374961156 37.5910421392361704312653856447 0.858160351793152212827135156698 719 12 56 []
387 0.026591657618921388460669423214 37.6057790127549794764796293511 0.859709366666620558171811535499 681 15 50 []
388 0.026557922158179378869820051745 37.6535481218743373761334920678 0.859745250996175327579214371058 711 11 53 []
389 0.026546764852293296290366040614 37.6693734835117946784332744974 0.861237000743175341636817979920 737 12 51 []
390 0.026442350327990230011286793931 37.8181208400927242496733011721 0.856672033369193036357947742487 1022 6 57 []
391 0.026395019912199627612545261432 37.8859346697369114419948490139 0.855796720757173553196167423290 845 11 59 []
392 0.026377703600881293299261074184 37.9108058506878309465195934392 0.856860074923491137147254045294 947 7 58 []
393 0.026363102465002138414777865221 37.9318026521169873271460444492 0.858095171675910279631227483105 801 17 56 []
394 0.026343787890580113718318762932 37.9596132550693380035660288014 0.859018538460119734910027079718 640 25 53 []
395 0.026337326490696358978769898542 37.9689259786200872463930277015 0.860776384053067470406591656572 717 24 55 []
396 0.026332973438690692065978117609 37.9752025470360719731513284606 0.862670328530336251125072691714 594 35 55 []
397 0.026269974698195532049502969292 38.0662719126520273223287835477 0.860715627875316352946266614038 1045 10 61 []
398 0.026253578455636563467676505233 38.0900455794932991545956322941 0.861806886414646684654071598513 1042 15 57 []
399 0.026207111721201613179289756061 38.1575814472908023014721245145 0.860916613328249586775277387842 1015 10 56 []
400 0.026202276101088931514508963721 38.1646234144689948834108354109 0.862755827170608199424224441428 157 273 37 []
401 0.026179011231126183558433433149 38.1985397069170145516570530157 0.863377495244150362312508278084 1034 7 57 []
402 0.026169800265054708425394272154 38.2119844198936794025698367980 0.864921597496359848168971686511 1077 5 57 []
403 0.026163864089183780551440441866 38.2206541278206296184668552015 0.866679826663807802594265977691 487 70 39 []
404 0.026144393618061678496904030862 38.2491181325068761778647064401 0.867537755876161884737431574088 719 14 58 []
405 0.026133773812474668757848699713 38.2646611689376842615124633976 0.868978740991200194321705600971 665 21 56 []
406 0.026132108259174107361742413119 38.2671000013530678744625861943 0.871013334345541772808411433804 674 23 56 []
407 0.026124303718886219599635628363 38.2785321576652485370785175902 0.872637215143982842429067056505 859 - 57 []
408 0.025994878967184940182836539566 38.4691154462525609171769798688 0.866135092984195892829461039799 663 23 56 []
409 0.025951755436606539867593386340 38.5330388321030020872151621689 0.865379614473234026079120079078 1081 5 61 []
410 0.025930217810949359183425238895 38.5650443544572725277425783610 0.866056167942486900450302817525 751 16 52 []
411 0.025905882907868864616991020484 38.6012707444243031944033442418 0.866539753015466453814962600141 650 89 54 []
412 0.025843855322148040151825481976 38.6939172787817600293609362230 0.864493417904423619541800343566 989 7 60 []
413 0.025779347877415608327799942340 38.7907407415866125210545058316 0.862270996937400338245435050721 1054 4 58 []
414 0.025752362139890942422153338383 38.8313893136420012017960281980 0.862550151320414615367872143074 1026 4 59 []
415 0.025724778364373036113830974579 38.8730268473343942832977496216 0.862782351435721358038971571180 1040 3 58 []
416 0.025703562783662356971302539170 38.9051124319472878251918649738 0.863435407052222113864195929439 837 5 58 []
417 0.025702953280603302529658392704 38.9060350023920632774880450465 0.865469926069930555221804006000 429 119 46 []
418 0.025697890872699222569752637530 38.9136993753201063528980134062 0.867203686781415581393111585633 1088 1 58 []
419 0.025549077679796327229351118040 39.1403561620848234386720082957 0.859239728397450953898323027794 1002 15 62 []
420 0.025491968936157507913636789937 39.2280408980732673336757518256 0.857444313068108287162182974124 1021 15 63 []




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 [10]; 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 (reference [22], see below).
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.
01-Jul-2005: Improved packing for N = 108 found by Bernardetta Addis, Marco Locatelli and Fabio Schoen.
08-Jul-2005: Further improvements for N = 111, 130 by Bernardetta Addis, Marco Locatelli and Fabio Schoen.
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.
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.
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.



References

[1]   L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd ed., Springer, New York, 1972.
[2]   H.T. Croft, K.J. Falconer, R.K. Guy, Unsolved Problems in Geometry, Springer, New York, 1991.
[3]   J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups, 2nd ed., Springer, New York, 1993.
[4]   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.
[11]   J. Schaer, A. Meir, On a geometric extremum problem, Canadian Mathematical Bulletin, 8 (1965), p. 21-27.
[12]   J. Schaer, The densest packing of nine circles in a square, Canadian Mathematical Bulletin, 8 (1965), p. 273-277.
[13]   B.L. Schwartz, Separating points in a square, J. of Recreational Mathematics, 3 (1970), p. 195-204.
[14]   M. Goldberg, The Packing of Equal Circles in a Square, Math. Mag. (1970), p. 24-30.
[4]   R. Peikert, Dichteste Packungen von gleichen Kreisen in einem Quadrat, El. Math. 49 (1994), p. 16-25.
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[5]   C.D. Maranas, C.A. Floudas, P.M. Pardalos, New results in the packing of equal circles in a square, Discrete Mathematics 142 (1995), p. 287-293.
[6]   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.
[7]   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.
[8]   B.D. Lubachevsky, R.L. Graham, F.H. Stillinger, Patterns and Structures in Disk Packings, Period. Math. Hung. 34 (1997) 1--2, p. 123-142.
[9]   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.
[10]   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.
[11]   Erich Friedman
[12]   Dave Boll
[13]   Péter Gábor Szabó
[14]   Jerry Donovan
[15]   Patric Östergård
[16]   Kari J. Nurmela
[17]   Torsten Sillke
[18]   Ken Stephenson
[19]   Eric Weisstein
[20]   Douglas Hanson,   private communication
[21]   David Bass,   private communication
[22]   Bernardetta Addis, Marco Locatelli and Fabio Schoen
[23]   Bernardetta Addis, Marco Locatelli and Fabio Schoen,   private communication
[24]   www.spacefillingdesigns.nl, Edwin van Dam, Dick den Hertog, Bart Husslage, and Gijs Rennen,   Department of Econometrics and Operations Research at Tilburg University, The Netherlands.
[25]   J. Akiyama, R. Mochizuki, N. Mutoh, G. Nakamura, Maximin Distance for n Points in a Unit Square or a Unit Circle, in: J. Akiyama, M. Kano (Eds.), Discrete and Computational Geometry, Japanese Conference, JCDCG 2002, Tokyo, Japan, December 6-9, 2002, Springer, Berlin, Heidelberg, 2003, p. 9-13.
[26]   E.G. Birgin, F.N.C. Sobral, Minimizing the object dimensions in circle and sphere packing problems, Computers & Operations Research 35, pp. 2357-2375, 2008.
[27]   David W. Cantrell,   private communication



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E. Specht     11-Feb-2010