The best known packings of equal circles in a 1x0.60000 rectangle (complete up to N = 307)

Last update: 24-Jun-2013


Overview    Download    Results    History of updates    References

Overview

1-8   9-16   17-24   25-32   33-40   41-48   49-56   57-64   65-72   73-80   81-88   89-96   97-104   105-112   113-120   121-128   129-136   137-144   145-152   153-160   161-168   169-176   177-184   185-192   193-200   201-208   209-216   217-224   225-232   233-240   241-248   249-256   257-264   265-272   273-280   281-288   289-296   297-304   305-307  


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
= 0.6/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 0.300000000000000000000000000000 2.0000000000000000000000000000 0.471238898038468985769396507497 2 1 D2
2 0.252277442494833886543030217200 2.3783339250091166708064172638 0.666477445975256506910981042672 5 2 C2 [1]
3 0.190033112945850060552663038611 3.1573444790696556781449216014 0.567255143232324319601403348765 7 3 D1 [1]
4 0.170148518491376613990984224879 3.5263310272690321052881221369 0.606338238331151146255689099161 9 4 C2 [1]
5 0.161906968533948391953240608944 3.7058318454909058963479619140 0.686277419103816026262609324873 11 5 D1 [1]
6 0.152859547920896831706610375860 3.9251718859622267905420705010 0.734065839761241645526554185378 15 6 C2 [1]
7 0.138202193662700008063251246929 4.3414650961646538109953151706 0.700045930791898814677459768005 15 6 D1 [3]
8 0.129823202479241967529606948497 4.6216699984422027787760408163 0.705981377834001606858663742348 18 6 C2 [1]
9 0.119859330293026045975108487226 5.0058681166760256301913349321 0.676994008756503038945299958829 20 8 D1 [1]
10 0.115181843761790566266666927050 5.2091543285317580332286654032 0.694651015054415128086164757550 21 8 D1 [1]
11 0.113391171522144578074331661858 5.2914172412692975641128055896 0.740542166139035561560888183950 24 8 D2 [1]
12 0.110087864570056706926546414921 5.4501920111110657031973534936 0.761480459868437546403710499007 25 9 D1 [1]
13 0.103387978507317515916280049437 5.8033826433460410016533204529 0.727582194423531245153515281637 29 10 D1 [1]
14 0.101761192204379056241227668568 5.8961573366293599242617058669 0.759086085795673917738766389965 34 8 D2 [1]
15 0.100000000000000000000000000000 6.0000000000000000000000000000 0.785398163397448309615660845821 38 12 D2 [1]
16 0.093797838338222718713266723881 6.3967359017004058498249080286 0.737062412766190895889822735164 31 1 10 [1]
17 0.090595211729214143885671491673 6.6228665792335537179242113180 0.730563604088401423412279600997 35 10 [1]
18 0.088816840445305969363688135661 6.7554756169184400378956843357 0.743467120163417816652458551730 39 10 C2 [1]
19 0.086139465043560085744089332046 6.9654484120209531127960500373 0.738170293797819194571251082531 37 1 12 [1]
20 0.085313228856734899660154480474 7.0329069482010828545104636802 0.762186717396847269738527248033 49 12 C2 [1]
21 0.083632003811217052960148270432 7.1742870271814011284613215184 0.769064778624038388773767905152 47 1 13 [1]
22 0.083333390926571692920705530966 7.1999950239476447752836530432 0.799943679541916364984616664097 45 1 12 [2]
23 0.080963851409875205430096646174 7.4107146529200026472380439067 0.789421184364948951618446463501 45 1 14 [1]
24 0.078738362897184921216107645667 7.6201736729460422809512779408 0.779081022717477447315238393050 54 14 C2 [1]
25 0.076111210904624479971168700325 7.8832013427281353801436304939 0.758290986866519664381481205407 49 2 13 [1]
26 0.075269601455117943710372487386 7.9713455153309185265601827005 0.771278461007163279579581904476 61 12 [1]
27 0.074365852672261350381143856416 8.0682191952301987832194920155 0.781824922087306772433975058104 53 1 13 [1]
28 0.072185619195103696372984993568 8.3119048737161433115556077616 0.763937846173858495626584865218 60 13 D1 [1]
29 0.071013064611104942093870916333 8.4491495091196597077823944699 0.765725536476782364124988990285 54 2 14 [1]
30 0.069873184169852096555429375186 8.5869852237087486475866952211 0.766903900561711420089682026678 72 1 15 [1]
31 0.068983877503286077022884143746 8.6976844694098086746830976917 0.772423574322901583009720302224 60 2 16 [1]
32 0.068531951302623509104816842223 8.7550403657779852517834623368 0.786927633010363303073106793661 63 1 17 [1]
33 0.068238280906633925850685147161 8.7927185742111764857303289243 0.804579044121247582777658405751 76 16 C2 [1]
34 0.067224848345629393909876512775 8.9252711574024524292816298207 0.804520660852904311523298709334 64 2 17 [1]
35 0.066777988620622636333178554191 8.9849965893507862092595146935 0.817209363799249316729472707930 77 17 D1 [1]
36 0.064864187234285720686962087628 9.2500966339535627524345199849 0.793069201086986528649490850099 72 2 18 [1]
37 0.063795381290717125360470501751 9.4050695812254057630119946877 0.788458467021915957924164246910 83 1 17 [2]
38 0.063164281791970610657152125222 9.4990393776039338350527296858 0.793826044806376552820495889286 85 16 C2 [1]
39 0.062549676213039062207347093133 9.5923757935444662671446933495 0.798938520829680284649191943262 81 2 17 [1]
40 0.060979899611387310535536259518 9.8393077690137218097277135667 0.778810904722743793715699432956 88 19 D1 [1]
41 0.060365631801331908669269450468 9.9394304688907701681817478171 0.782279554707617708677103172031 97 1 15 [1]
42 0.060050889756071894912618890029 9.9915255616896585078937146779 0.793024867248271573807948819859 93 2 18 D2 [1]
43 0.059249487499123673151910721080 10.1266698721887556021600327464 0.790380604012616103764141930782 85 2 17 [2]
44 0.058460541101100379843314762725 10.2633329883548825936950969960 0.787366549724677645213486559315 102 3 19 [1]
45 0.058007177883918538361817486879 10.3435475037364188404010551663 0.792820023530559384873814674373 113 1 19 [1]
46 0.057373989202189921721680286809 10.4577005772695766176044262150 0.792841817953716259551143904296 95 5 21 [1]
47 0.057130592273447163141442690262 10.5022541535748202139150251230 0.803218926030172398642594434479 97 3 21 [1]
48 0.056939492662112365819745661873 10.5375016872821736450342460101 0.814830067262634974809346265401 125 20 C2 [1]
49 0.056298951605230106563868615085 10.6573920631278807653127520938 0.813196156647588996579416867587 107 2 21 [1]
50 0.055830318092897474239042328996 10.7468490328434967685576193690 0.816035084480107180537294129901 125 2 21 [1]
51 0.055583903409206094373719803296 10.7944919877761749284446665957 0.825024570225582547792773649382 98 1 18 [1]
52 0.055040033495100533998187846034 10.9011561566983756281723363597 0.824820307551995864855597980664 111 1 22 [2]
53 0.053790817406067476784201543400 11.1543201783046602536771574291 0.802954210286649045310064752470 125 1 23 [1]
54 0.053311482080480786983761514150 11.2546111378824553860088211533 0.803588836062339683525595159055 117 22 C2 [1]
55 0.052616338387560684439075272058 11.4033020614343865626865018174 0.797264736877447427159487240022 110 4 20 [1]
56 0.052137768714640547575382499042 11.5079723354466640903009619421 0.797060948537532052278352659773 105 4 21 [1]
57 0.051582534356042388682237612219 11.6318441404714709720745486665 0.794106645058201165034054097513 137 1 21 [1]
58 0.051237910320536906490482141180 11.7100794362316371364972126631 0.797277364721731660299335398975 126 3 21 [1]
59 0.050706174315801995478246356597 11.8328785023921023245202996749 0.794277616955828645584970778215 117 5 22 [1]
60 0.050553432731854556955008670987 11.8686302309581846126544991532 0.802880988590635018122438576765 152 2 22 C2 [1]
61 0.050182346715695772179223575945 11.9563958098503021040258823047 0.804322820324720811246048441463 148 19 [1]
62 0.050019265294996293430818188740 11.9953781100423590637793444703 0.812203634326804041334995988080 159 22 C2 [2]
63 0.049588519874224096144942710287 12.0995746903080580532605306874 0.811150542459531102150346341495 163 1 23 [1]
64 0.049178971206793468597089855416 12.2003365519187070094832064629 0.810470991173140113505618284944 145 2 25 D1 [1]
65 0.048996637465208928326848777752 12.2457383004301541896812516823 0.817042281548939168342028289415 144 2 23 [1]
66 0.048877198579304509519441441418 12.2756626287917175980848346363 0.825572408875644649570886479558 131 1 24 [1]
67 0.048808032286899533489909394351 12.2930585784144549956196574046 0.835710817147001408036077219546 164 24 C2 [1]
68 0.048203585880325583732822235349 12.4472067594641652406471030236 0.827306103764661501895951633291 154 2 25 [1]
69 0.047936328230153371927565323054 12.5166032141481835067632541341 0.830189514532243616612169667917 158 2 25 [1]
70 0.047738743414358588960235352626 12.5684079028258501885650803886 0.835292590224620485689816305142 150 25 D1 [1]
71 0.046845939657103390984717011752 12.8079403336084046742235143171 0.815832271162844162693183646624 133 4 24 [1]
72 0.046454400273506725972324693338 12.9158916371197726225727203884 0.813551095371944279073122976749 146 3 28 [1]
73 0.046115119350366066638652143470 13.0109172100676160837287136574 0.812845781868861568867078123908 145 4 24 [1]
74 0.045943929309031043069390760402 13.0593967260449363808947766772 0.817874395183719778790499958143 139 4 23 [1]
75 0.045529788975339611234628761837 13.1781853925345266810271798532 0.814050149758008857758949498735 156 24 [1]
76 0.045062666008299022987615217020 13.3147914481912863232760797799 0.808064408561022005574432339947 97 16 20 [1]
77 0.044832403115893020873674900225 13.3831773070246346822163820636 0.810351396243239942533647022945 200 1 26 [1]
78 0.044593362449119819230429272407 13.4549172129504389814633757598 0.812145170188491528313596389170 208 1 23 [1]
79 0.044210741815710147985137170899 13.5713624191393207529350271760 0.808502407884047750927849828616 189 3 26 [1]
80 0.044024472311550027429922689820 13.6287834582991055458981325143 0.811852117045192315968864963597 173 2 26 [1]
81 0.043770785252999991499400659697 13.7077732677614413308018976646 0.812554158230991079328590221795 188 3 27 [1]
82 0.043622023044679100922527684288 13.7545202657258788962327848706 0.817003807309582927883179650611 171 1 27 [1]
83 0.043400610809484700317996902222 13.8246902246103171485879147890 0.818593700314981557732077674698 181 5 27 [1]
84 0.043305433574897591283199039157 13.8550743052205748613331971046 0.824826661663238490025851403661 222 1 27 [1]
85 0.042982324827260661373803909779 13.9592263194535782360726314823 0.822237636228075117083428392965 207 4 28 [1]
86 0.042857267414152390057784885164 13.9999593114951401458368120883 0.827077159157083343356419806107 225 26 C2 [2]
87 0.042772277821076229840796827272 14.0277775831790593930291154757 0.833379154291536742256516507736 194 2 28 [1]
88 0.042734604206597389425263933056 14.0401440738597434338220839683 0.841473931980054379292951875909 237 28 C2 [1]
89 0.042391428293754711239213251551 14.1538047702062116612791440321 0.837422697711826836579536065376 152 8 27 [1]
90 0.042044712409535343135846332539 14.2705221564061804478170083461 0.833036257274698652646952427722 232 3 28 [1]
91 0.041823077456194908129888354384 14.3461465892469120505072784484 0.833435484457533549581383624242 169 7 25 [1]
92 0.041724924554608747131330656711 14.3798941856619056228585152135 0.838643856339394945010353735051 159 10 27 [1]
93 0.041666672490105979936457489951 14.3999979874196546173083974487 0.845394092743561202808496283953 36 66 32 [1]
94 0.040837872228762735480219514075 14.6922444107509318373065804578 0.820828992818453943640425854494 210 2 31 [1]
95 0.040614330036223259545060730052 14.7731108568052155765216408807 0.820504212034797526659017531943 226 4 28 [1]
96 0.040464655607444877605067759248 14.8277550121941277260317203603 0.823041155553025901225033522681 234 2 28 [1]
97 0.040222406127193383326113173166 14.9170588677029617109941523442 0.821687064574837409842961329125 216 1 26 [1]
98 0.040145258829918742473304634167 14.9457250367219628323704852396 0.826976603065564161940007512966 203 26 C2 [1]
99 0.039705133723128588452475142309 15.1113960271211664587933788279 0.817197713755921461676528389717 249 2 29 [1]
100 0.039601343582990219035127383959 15.1510010952687779519952200880 0.821142373961397452563527687406 264 1 30 [1]
101 0.039395550373002270829281261777 15.2301464078841596270844684710 0.820756518056847598786353016751 187 3 29 [1]
102 0.039219929578305156839739422587 15.2983446541396950361474267310 0.821509165413231780079263954892 128 22 32 [1]
103 0.039024466504370088593760029126 15.3749699546260297798532526948 0.821315078803178083233163769356 246 4 30 [1]
104 0.038824656938777344709555126466 15.4540966310698077994407591394 0.820818649746633498563002048013 255 5 31 [1]
105 0.038695258580272363135736101717 15.5057756948519866515834375242 0.823196334067610556437563031782 209 4 32 [1]
106 0.038591710769278615170651086770 15.5473802026324536587601162874 0.826594573939870299395644084442 255 2 30 [1]
107 0.038465300822719144657169962077 15.5984741355672957504318393377 0.828935361323197092488771167372 270 3 31 [1]
108 0.038436768318894983827801838815 15.6100532443839274696837628072 0.835441625053791965847869887428 290 1 31 [1]
109 0.038161714003952677975162625306 15.7225642416861508528890418243 0.831152786448319827022591979608 217 3 34 [1]
110 0.038072360058739145232236021172 15.7594643219990180910208270923 0.834854717417244722497940304275 199 5 32 [1]
111 0.038049094852954012781849863304 15.7691004823842182836252719426 0.841415020909019786345644580110 218 2 32 D1 [2]
112 0.038015912104341849825999159480 15.7828647739185290079517405897 0.847515158527759493186427069603 254 3 32 [1]
113 0.037992860696830608783020003566 15.7924407111058215967521529693 0.854045593637706042677068946162 288 32 C2 [1]
114 0.037644408547753311119541989070 15.9386220463227101205524489725 0.845871572702167286554376301698 232 3 34 [1]
115 0.037363867647885350339594772989 16.0582947583039537813300891732 0.840620763044207412449527102020 257 2 33 [1]
116 0.037230168485884040237819061331 16.1159625218320534457693177984 0.841873064872864559333953788120 265 33 [1]
117 0.037131308721004909317314016113 16.1588702544325995069330907918 0.844627071604705340684073583412 247 33 D1 [1]
118 0.036698886764217093168933752324 16.3492697709028165118126498354 0.832120862334302624026251827194 232 8 34 [1]
119 0.036476444068007089709547099981 16.4489718044158388593822690664 0.829030621427754660979549371010 280 5 33 [1]
120 0.036290083585114096902612492052 16.5334422168736811261200102552 0.827476760086650326279277948280 219 6 30 [1]
121 0.036207859355308014814897247452 16.5709879203902992022941826437 0.830595725820216665747257442410 309 4 33 [1]
122 0.036070333100163751699094648406 16.6341685377248741470291775247 0.831110480407110664602272338457 223 3 32 [1]
123 0.035842470183765791375281322662 16.7399176709578268919351475709 0.827369675119470914341912877531 301 4 33 [1]
124 0.035653641555662961401830719874 16.8285755345150831107811008141 0.825330881467237518440173105660 312 5 34 [1]
125 0.035551647694401039812005557853 16.8768549114108384559500389163 0.827233478971749918464662108396 323 3 33 [1]
126 0.035468718119521284135424937731 16.9163147644112859257702725105 0.829965716196109943961222776356 247 8 32 [1]
127 0.035407730637512828290468244251 16.9454520014995868942635528255 0.833678360308068997565118965982 291 1 30 [1]
128 0.035366537453999741400704335201 16.9651892210370633923009369031 0.838288825341203894541828720580 264 30 C2 [1]
129 0.035019926001753872856333353554 17.1331030216897293176888901398 0.828359354000533404860881934567 336 1 34 [1]
130 0.034988991949928848400978368331 17.1482505371584483572451814464 0.833306626682924522318278657172 169 30 33 [1]
131 0.034816261951563169578967496568 17.2333262207966983861467559468 0.831446291308794515591882775350 322 2 35 [1]
132 0.034738961664461883772389843411 17.2716733964389827059215428608 0.834077143624103223698150536858 322 6 35 [1]
133 0.034703820292364330195742789904 17.2891628340991132493046192906 0.838696508005937189829622076244 321 5 35 D1 [1]
134 0.034575768752801840849303857736 17.3531933386550982771102827000 0.838778153631462633045408259275 338 3 35 [1]
135 0.034553219759794081253661293713 17.3645178125529242933491831726 0.843935849019885283962319604027 367 1 35 [1]
136 0.034361179216360126382916849962 17.4615660371261814201349391329 0.840763113874115052594620771734 332 3 37 [1]
137 0.034300868397485173257673891079 17.4922685060647043158534228946 0.843974684591323035565880013561 350 4 36 [1]
138 0.034243125032239252670297028295 17.5217653013593642875710759515 0.847275194582551992850749693158 271 8 37 [1]
139 0.034218031188992641572308503719 17.5346149135842096971776140173 0.852164539383797393390379614617 308 3 36 [1]
140 0.034205856530511793869479645526 17.5408558901250439610028988157 0.857684571167928020792367523674 385 36 C2 [1]
141 0.033857171698861394672044431856 17.7215038910110084224693365688 0.846289761714508503101623532373 262 12 34 [1]
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293 0.023535547828115300347485082499 25.4933517750220692964351809307 0.849796359011783585561928514838 415 34 51 [1]
294 0.023521522943535559824997292009 25.5085523773407925014584421559 0.851680743020564558191003158938 432 25 52 [1]
295 0.023500163555372985952056435514 25.5317371977531762626764970438 0.853026273947137389034833811400 690 8 52 [1]
296 0.023461206880485300286640076146 25.5741319300615990494931427923 0.853082497305002293474006977834 753 15 55 [1]
297 0.023405600922652805048250173422 25.6348897848334173411795961073 0.851911857823721622790567130821 766 6 54 [1]
298 0.023362017568086337936568413192 25.6827133295041021658071092354 0.851599854865223970845826397223 788 5 54 [1]
299 0.023346372477131727916619360498 25.6999240711897686110418694138 0.853313528305728579484689328681 499 30 46 [1]
300 0.023275893755896650785991783252 25.7777426848753188408129767326 0.851005975075142305826061972623 730 14 56 [1]
301 0.023236480482963347803747558402 25.8214663980593507790887893217 0.850953471496105338022594167245 744 13 55 [1]
302 0.023215425055959923683019565366 25.8448853964001169748588529616 0.852233976723685224668674755412 544 7 54 [1]
303 0.023200617088860396991193494332 25.8613810874920873796119206752 0.853965495942935477817267940158 489 13 50 [1]
304 0.023173681497999945436747130176 25.8914406867887734667796700430 0.854795590680273986943801801962 611 14 56 [1]
305 0.023157299062031371677388693317 25.9097573681966195894622771631 0.856395290423114232449603784204 634 7 58 [1]
306 0.023139834197111943590490737592 25.9293128416142808035502237899 0.857907637979985849690872695304 523 18 54 [1]
307 0.023134143833945586175676545278 25.9356907394860135250426421050 0.860287991288593997351770142695 435 25 48 [1]





Updates

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

08-Aug-2010: First complete presentation from N=1 to N=307 by Eckard Specht [1].
09-Aug-2010: Some nice improvements for such small values of N=22, 37, 43, 62, 86 and 204 by David W. Cantrell [2].
11-Aug-2010: Better packings for N=52, 111 and 150 by David W. Cantrell [2].
24-Jun-2013: The old packing for N=7 wasn't the best: Julian D.A. Wiseman [3] found the right (which has now D1 symmetry)!

References

[1]   , program crc, 1999–2010.
[2]   , private communication, August 2010.
[3]   , private communication, June 2013.


©  E. Specht     24-Jun-2013