The best known packings of equal circles in a 1x0.40000 rectangle (complete up to N = 299)

Last update: 05-Aug-2010


Overview    Download    Results    History of updates    References

Overview

1-10   11-20   21-30   31-40   41-50   51-60   61-70   71-80   81-90   91-100   101-110   111-120   121-130   131-140   141-150   151-160   161-170   171-180   181-190   191-200   201-210   211-220   221-230   231-240   241-250   251-260   261-270   271-280   281-290   291-299  


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
= 0.4/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 0.200000000000000000000000000000 2.0000000000000000000000000000 0.314159265358979323846264338333 2 1 D2
2 0.200000000000000000000000000000 2.0000000000000000000000000000 0.628318530717958647692528676661 4 2 D2
3 0.168629150101523960958649020633 2.3720691218521717454061960774 0.670002523442130933307637917607 7 3 D1 [1]
4 0.136669234721606424128467239518 2.9267742723137007054895296790 0.586801746666601560813927656865 9 4 C2 [1]
5 0.122291236000336485745322130030 3.2708803433706352873054732000 0.587287225892779898799343094103 11 5 D1 [1]
6 0.114835192865495968749289508460 3.4832527382830412900002273573 0.621428461359890309554359981934 13 6 C2 [1]
7 0.110533615595889603202655493362 3.6188086116932805672893123093 0.671702050023025336601165188768 15 7 D1 [1]
8 0.107846155007274420463776511862 3.7089871212656516886038413553 0.730784286300017387294073945469 17 8 C2 [1]
9 0.101542459539147280449314249485 3.9392388348224863765130116325 0.728832529487893558249445272197 22 8 [1]
10 0.100000000000000000000000000000 4.0000000000000000000000000000 0.785398163397448309615660845821 27 10 D2 [1]
11 0.090250667337725817063198995036 4.4321001916048481002837558659 0.703693290665031692128218632133 21 1 8 [1]
12 0.086774673184824642143883403054 4.6096399481450647030607556377 0.709671068964420482220141771548 26 8 C2 [1]
13 0.082135439251618401626952783726 4.8700050020383664774938537005 0.688802003647898839772925867675 25 1 9 [1]
14 0.080292014132332485963920887997 4.9818154933907121964337569724 0.708863511514220726716584507947 30 9 D1 [1]
15 0.077614097156614570939266394471 5.1537029309618203342490070044 0.709679663463021453352722127961 32 12 D1 [1]
16 0.076056320952540401801665669087 5.2592604400310447215066896343 0.726909745240630745149682528204 31 1 12 [1]
17 0.075395404095677962031020119732 5.3053631689856541408945564353 0.758976884580350389100237412424 36 12 D2 [1]
18 0.073966982608467729686829578568 5.4078182709890352710665629251 0.773460592697101150543835511684 37 13 D1 [1]
19 0.072238100898120671618301231814 5.5372441277786456875935016359 0.778710664587450848878714812890 41 1 12 [1]
20 0.071489902589948347471580051352 5.5951957620409593847806360102 0.802803556244141808613052169312 41 12 C2 [1]
21 0.068605355934390964205598509778 5.8304485787163630416613413895 0.776292195206150758141701223337 49 15 D1 [1]
22 0.066673730362367988706179279434 5.9993643347390705830469917441 0.768107615365377910823284439555 52 12 D2 [1]
23 0.064970614526482707350360766966 6.1566294688032507913724049205 0.762520750288327935339910335713 47 12 [2]
24 0.063237221919466704808226692686 6.3253885584886126715596216786 0.753783607044408357094633461248 51 12 C2 [1]
25 0.061236269596560969334222027705 6.5320765395295080406179875693 0.736287356453942609205579415141 51 13 [2]
26 0.060459691595432752335947766327 6.6159781739643675400497630078 0.746440309650878862580904332054 64 15 D1 [1]
27 0.059324748157101969026110778925 6.7425486399155430447836541511 0.746320639178402513655770152992 57 13 D1 [1]
28 0.058471825688645937006223433070 6.8409014989534015277607017219 0.751867341679898184953357471196 60 2 16 [1]
29 0.057746433976699528327157134337 6.9268346537450003342553013764 0.759518260908626514120228284948 67 1 17 [1]
30 0.057234765046146988483348257685 6.9887593611590752971153079003 0.771846509975236813608128011979 70 1 17 [1]
31 0.056604833061588746227185712143 7.0665343993644678038686834586 0.780114954130651016608306435624 70 1 18 [1]
32 0.056335630166834727287170807361 7.1003022210885547084180617321 0.797638619232051685538107989760 79 18 C2 [1]
33 0.055681720774830597504472114127 7.1836860361687867456466160028 0.803579996881155945578943512731 77 1 19 [1]
34 0.055555601006057610311065853495 7.1999941096197526769974818042 0.824184606430305634816390261101 67 1 17 D1 [2]
35 0.054079179187037156440614954680 7.3965619673436255789173053267 0.803929764648643602723536782842 71 20 [2]
36 0.053226366147400207862541526713 7.5150724904322183635122374218 0.801024900137468780441408265251 81 20 C2 [1]
37 0.051888656244421807380930532927 7.7088140058165651454700235277 0.782413712686224184810082826678 69 4 18 [1]
38 0.051079948261548001484999472136 7.8308614948443922871184931488 0.778707531958686279025768646192 86 20 [1]
39 0.050536970007648632655719154782 7.9149976727821452360522874802 0.782299203448973319910211114575 83 2 18 [1]
40 0.050010694212584780649886074089 7.9982892918799612201215570071 0.785734167923567026674447154647 95 16 C2 [2]
41 0.049244178420356284119927645812 8.1227875625324724380267951028 0.780878616684905152078270409859 79 2 17 [2]
42 0.048333796137424027303068356163 8.2757828262176758253292977940 0.770621255568053306284900103948 93 3 20 [1]
43 0.047800651505993843500001577440 8.3680867811988503428465226778 0.771659964814615537793675928891 102 20 [1]
44 0.047426966620235086373940376980 8.4340203159722399455910767584 0.777308210165977195379700386452 87 2 20 D1 [1]
45 0.046924229919799661102137494421 8.5243807023292278045095950026 0.778209813857846180812981373469 104 2 22 [1]
46 0.046632751263577313850701096377 8.5776624617132874966698714536 0.785651223075213767306963407001 112 1 22 [1]
47 0.046339123808318821317012632026 8.6320147453498421937709656543 0.792653487691974980836959108135 108 3 22 [2]
48 0.046079750151179080327542668363 8.6806026223595935061023872449 0.800481593374984173617980065754 116 2 22 C2 [1]
49 0.045606402153225626990791181727 8.7706984351912749688227059874 0.800456225006147447617753378004 114 2 23 [2]
50 0.045469007353211252107927729412 8.7972010669318024935934999920 0.811878109753656037060276780502 124 1 23 [1]
51 0.045104182523331597585651209782 8.8683571594072691487287401377 0.814880057448888395570228391870 96 3 24 [1]
52 0.044998055581666161978898926969 8.8892729881193908985154024278 0.826952797328472011728808184816 115 1 24 [1]
53 0.044948291444021147825214434396 8.8991146748739543275239727361 0.840992508760917019614881675373 122 24 C2 [1]
54 0.043958668604815488426695866770 9.0994566645310470513824927055 0.819544805130413318940753684833 131 2 25 [1]
55 0.043680623572147088327992887106 9.1573784275153904279134320662 0.824195483047665033246794207919 121 25 D1 [1]
56 0.042822441517597989655596692692 9.3408966379373534223073322238 0.806530430869813552226835134604 105 4 23 [1]
57 0.042343660635921679162697190482 9.4465144012765240346637900375 0.802678328657423190323435633286 113 1 27 [1]
58 0.042127671943700618799652783019 9.4949467071088006985171154753 0.808449309814534940492833717128 138 22 [1]
59 0.041702880667412736525888555679 9.5916635397459741174362198443 0.805886728274768113747937992915 131 1 23 [2]
60 0.041106014192067563815155980701 9.7309361625528273701671920203 0.796254440766646051607613259714 153 1 24 [2]
61 0.040693643573987878237154941552 9.8295449821968588959251453188 0.793364694684721864850072396202 145 2 24 [1]
62 0.040312851371889284633053765120 9.9223941345643835373896723030 0.791349997979713385522705376185 147 3 25 [1]
63 0.040199241787773280542873925740 9.9504364314070519290933676669 0.799587791320313516313109341215 165 25 D1 [2]
64 0.040007318109490393371439577895 9.9981708072832161684949002338 0.804542024881897504944952557416 107 2 20 [1]
65 0.039613423837161346037717125804 10.0975871624800093474644367708 0.801102338114106071598306980690 131 22 [2]
66 0.039206395224708574180262036224 10.2024171747346162143488853722 0.796796915156774686385349861751 157 2 26 [1]
67 0.038982472898959211265093314879 10.2610216913837591812967144297 0.799656469776784030975993258402 166 2 27 [1]
68 0.038817146488575927579339158732 10.3047244886411709534907206939 0.804722245760677307482227595187 165 27 [1]
69 0.038589972841912029520368569234 10.3653869267709252475848995683 0.807026727191379616409413737972 168 1 27 [1]
70 0.038422105620617615725000622347 10.4106735833175351176158292386 0.811615335463267495095687126872 121 6 28 C2 [1]
71 0.038244419254201498489138741143 10.4590423335048119324740798387 0.815613435192757122284187059780 153 3 28 [1]
72 0.038122897320735944223330277416 10.4923819570877828576777861621 0.821853060108471542698892231635 173 1 28 [1]
73 0.037860461129469029184906659290 10.5651116776455851360369220269 0.821834824161825950460062791229 150 2 28 [1]
74 0.037785399186422303999311879853 10.5860996208221836778827359447 0.829792739480888281537441489340 148 2 28 [1]
75 0.037739779879100519289905205893 10.5988959469663314436003040893 0.838976643012146332063548007848 193 1 29 [1]
76 0.037420236049282145483244758295 10.6894034413145674569245985959 0.835827236554449354745815389962 187 2 30 [1]
77 0.037240424492423385096489641354 10.7410161256722662454240709252 0.838706197137125600172613853742 185 1 30 [1]
78 0.037096963913258499375164039169 10.7825535516948199761485450842 0.843065309330245291158955619276 169 30 C2 [1]
79 0.036480662848382205534384579859 10.9647130498271264851502417173 0.825738272279298334157417518297 136 11 32 [1]
80 0.036168741757889461464348885143 11.0592732995127841150236127195 0.821952403698552378498399038987 134 7 24 [1]
81 0.035984612662402776097500019159 11.1158623201723542385349316745 0.823774918547102216112796114700 209 2 28 D1 [2]
82 0.035780164466717547467877509574 11.1793784618311951303222569745 0.824495707142013066131110698606 166 4 28 C2 [1]
83 0.035243299392891578052017672001 11.3496751691948081363566776660 0.809694319111758757082072772388 200 1 33 [1]
84 0.035138695601805936167558201639 11.3834618260400706102572602907 0.814592559828708597920637414910 220 1 29 [2]
85 0.034816304259847710867448148820 11.4888701860669467298203396497 0.809234044157387327985685574364 189 4 29 [1]
86 0.034551946195428241352574439418 11.5767719056278863697937679489 0.806368143459192479232692709480 173 6 28 [1]
87 0.034448121126407134848198001673 11.6116637691850551766434417240 0.810849425012329843000730253187 214 2 30 [1]
88 0.034197878050793188755886506023 11.6966321537813181041471107560 0.808296823524876460746210371323 204 3 30 [1]
89 0.033988701748872304677884155539 11.7686166113500176246182817510 0.807512103831598413134331061915 222 3 31 [1]
90 0.033870220701084425437077272253 11.8097842801240787167424080850 0.810902135089519463714293374528 223 3 30 [1]
91 0.033671343669196838221394282439 11.8795378031179469919551941929 0.810311809864292441366667871744 219 4 31 [1]
92 0.033551128938007240936364792215 11.9221025539582894858615997513 0.813377181894720814487714670698 218 4 32 [1]
93 0.033409591967047324962421295178 11.9726095546012508026013886279 0.815295740081802405957632463378 198 4 31 [1]
94 0.033342487302360741984346304388 11.9967054758892829722763651831 0.820755352956751522917020960434 115 24 32 [1]
95 0.033171885464463632809341851110 12.0584041093627878392278054514 0.821020121725449334545991094145 232 2 32 [1]
96 0.033004858658586366078690027843 12.1194277526753824364975300361 0.821328454330917993041571157482 232 5 33 [1]
97 0.032936386659298485538590705557 12.1446230315939468232192395660 0.826444170334470760602839442212 165 11 30 [1]
98 0.032825163769342199422031047202 12.1857731711787833872245129241 0.829334553203884198447019940327 242 1 33 [1]
99 0.032688829737536424174276296472 12.2365959017701372705289954657 0.830852291768545312393879933041 247 3 34 [1]
100 0.032594949358809154888303176104 12.2718398975482826322612033213 0.834431139133619855373519397784 204 3 34 [1]
101 0.032566460319533686655779883996 12.2825752653282992330930277122 0.841302868380519995298001773265 243 34 D1 [2]
102 0.032527056506916112453323956285 12.2974545795420935564287291110 0.847577816418407205148038668554 266 2 34 C2 [1]
103 0.032332681091494836424674090410 12.3713835814630491936978441222 0.845688730776271083004925509685 243 2 35 [1]
104 0.032287761387534007636744205764 12.3885950220889618856273431942 0.851528309278661098777077313085 205 2 36 D1 [1]
105 0.032260139385853187915707991527 12.3992024713758424539741171831 0.858245745957965359449860294057 225 35 D1 [1]
106 0.031731493691567183558833800165 12.6057727974621682590392715402 0.838256208602009425581692538894 216 3 35 [1]
107 0.031541800171578138485828045990 12.6815843681754801900169076622 0.836077644250088785881708715211 188 9 31 [1]
108 0.031414377532443719231537845558 12.7330232657608245841260649391 0.837086916926279545858502729257 231 1 32 [1]
109 0.031350372910099047271274883502 12.7590188846253265826091621363 0.841398631841355868304141093292 171 5 27 [1]
110 0.031027261800950910493829370020 12.8918885129509226601413037168 0.831705295935878969215872629965 219 1 37 [2]
111 0.030766977122688111970568084952 13.0009522354093389291411886633 0.825244269668085616142132858842 285 3 34 [1]
112 0.030693828753465406148658413572 13.0319356119701765848080456367 0.828724227688187295532376674466 299 1 34 [2]
113 0.030444669101860517789985291365 13.1385891783450331900102942392 0.822604045740244596850744351198 273 3 34 [1]
114 0.030307368798832845048964926246 13.1981104217600123677864375955 0.822415336006123064775429521295 287 3 36 [1]
115 0.030164462966474390441257256673 13.2606372089094026161192646097 0.821824184059430419084669110210 300 1 35 [1]
116 0.030136306469873515273715488090 13.2730266862619556824336630662 0.827423627255196031599244542932 314 35 D1 [2]
117 0.029930015192882771075766506242 13.3645104228052072669648767883 0.823170157874771764791848492825 292 6 37 [1]
118 0.029774325988679966003711354776 13.4343931127803795375982627402 0.821591176821486657262331341068 301 3 36 [1]
119 0.029672043740627260756173162947 13.4807026943114126130032669002 0.822871012944627682302730923943 304 2 36 [1]
120 0.029514485726977941540684541548 13.5526671106580366049986571106 0.820996995888163523467660636520 306 3 37 [1]
121 0.029423596115510824950081282160 13.5945313560478626831731993911 0.822747843842712107030242781389 308 3 37 [1]
122 0.029319431505049224621387318012 13.6428293274074666234462709788 0.823684327232276982511619395594 307 3 37 [1]
123 0.029269866457453052420700781894 13.6659318409068823616718985004 0.827630476727663883543877931834 283 2 35 [1]
124 0.029138155669407739935167666343 13.7277048190102682413909336026 0.826867047888570634140365615302 309 4 38 [1]
125 0.029051749467083980612284125158 13.7685339897759111460013414395 0.828599133788459192281648461991 279 4 38 [1]
126 0.028972186238373444887551348153 13.8063450479343867810904906036 0.830659359753897745666045741117 318 4 39 [1]
127 0.028923908506835989736886455677 13.8293896174323202034686477964 0.834463913879620927751631093727 320 4 38 [1]
128 0.028830034139415396427413301246 13.8744199214503960158559828127 0.835584094174133695193187413579 325 1 38 [1]
129 0.028722392508076534436235102009 13.9264164671352784357998844615 0.835835508637060644526087453878 320 3 39 [1]
130 0.028657163269701260592625072264 13.9581156807280122393821072864 0.838493363704214738763588715190 332 2 39 [1]
131 0.028635013329730917539052803629 13.9689126522840277439197900375 0.843637655802932928744991342785 182 20 37 [1]
132 0.028582875088661643649006521915 13.9943934526962061938506521107 0.846984836639863042070424997122 302 1 39 [1]
133 0.028466972445732656026128114241 14.0513713132835112818039305494 0.846494390122715931930998161409 264 5 40 [1]
134 0.028409750533377982620875190527 14.0796730872398515568274383281 0.849433764813764339774079134723 284 3 40 [1]
135 0.028388475305432615042213079041 14.0902248428767582516283748769 0.854491576407928605281418139752 267 3 40 [1]
136 0.028378658499944252543435618432 14.0950989632151134246610879867 0.860225898384740345764419721540 273 40 C2 [1]
137 0.028119048886690898301913132417 14.2252322122219810283612962314 0.850769088644281565158834229135 195 24 29 [1]
138 0.027970157541658982881310387662 14.3009562747094544378418485468 0.847927641168080786137628455447 371 2 41 [1]
139 0.027864545488624242998161411493 14.3551596835950835187018495642 0.847634468244750168624385832480 271 10 37 [1]
140 0.027826454909139112134717997039 14.3748099176164549889410039399 0.851400064046844835226721275878 376 2 38 D1 [2]
141 0.027777784092733368914057834888 14.3999967263277657863644622139 0.854484501469705669288978416848 67 89 37 [1]
142 0.027359612987747249582205539260 14.6200898448065150612380950015 0.834830150247109633571759980650 273 7 35 [1]
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293 0.019268752742089564465157073910 20.7589980189149874060509194880 0.854406826232471635973988013943 562 8 55 [1]
294 0.019266875537644070720638626100 20.7610206033910725261763831088 0.857155854044122782536607344252 736 4 59 [1]
295 0.019226433709666146176692333639 20.8046903570524793606371895694 0.856464501457009492657960280013 508 9 55 [1]
296 0.019205610234789917395660124293 20.8272476172312283947930180014 0.857507276935444942348441202877 694 9 62 [1]
297 0.019185804819413448413992501737 20.8487474862276257590072083532 0.858630625344128383172657367593 538 9 58 [1]
298 0.019180899578615387983548637766 20.8540792552793714552614790839 0.861081162845858489059595236566 393 68 58 [1]
299 0.019167850760922819367289711008 20.8682759997001536934717683861 0.862795573616588293548154021806 779 3 60 [1]





Updates

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

12-Jul-2010: First complete presentation from N=1 to N=200
14-Jul-2010: Many improvements for N=23, 34, 35, 39, 40, 41, 45, 49, 51, 52, 57, 59, 60, 61, 62, 64, 66, 70, 71, 72, 73, 79, 81, 82, 83, 84, 85, 88, 91, 92, 93, 98, 99, 100, 106, 108, 110, 114, 115, 116, 119, 120, 125, 126, 127, 128, 131, 132, 134, 136, 137, 139, 140, 141, 142, 143, 145, 146, 149, 150, 151, 154, 155, 156, 157, 158, 159, 160, 161, 174, 177, 178, 179, 181, 183, 184, 185, 187, 188, 190, 191, 193, 194, 195, 196, 197, 198, 199 and 200 and new packings for N=201-220 by Eckard Specht [1].
14-Jul-2010: Some better packings for N=23, 35, 41, 47 and 200 by David W. Cantrell [2].
16-Jul-2010: A few better packings for N=25, 34 and 49 by David W. Cantrell [2].
19-Jul-2010: Again, many improvements for N=40, 51, 56, 58, 59, 64, 65, 70, 71, 72, 74, 79, 80, 86, 93, 94, 98, 101, 104, 107, 108, 109, 110, 111, 113, 114, 115, 116, 117, 118, 119, 120, 121, 123, 124, 125, 126, 131, 132, 133, 134, 135, 137, 139, 140, 142, 145, 147, 148, 149, 150, 151, 153, 154, 155, 156, 157, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 174, 175, 176, 177, 178, 179, 180, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 197, 198, 199, 200, 203, 205, 206, 207, 209, 210, 211, 212, 215, 218, 219 and 220 and new packings for N=221–299 by Eckard Specht [1].
21-Jul-2010: Some nice improvements for N=40, 59, 65, 101, 110 and 215 by David W. Cantrell [2].
23-Jul-2010: More better packings for N=60, 63, 81, 84, 112, 116, 140, 144, 180, 185, 215, 216, 220, 225, 264 and 270 by David W. Cantrell [2].
03-Aug-2010: Further improvements for N=58, 71, 79, 92, 98, 109, 124, 132, 141, 145, 148, 164, 173, 176, 181, 182, 183, 184, 197, 214, 217, 219, 221, 222, 223, 224, 226, 228, 230, 231, 232, 234, 240, 243, 244, 246, 247, 248, 250, 254, 256, 257, 265, 268, 276, 280, 281, 285, 286, 287, 288, 289, 290, 291, 292, 293, 295 and 297 by Eckard Specht [1].
04-Aug-2010: Two better packings for N=205 and 253 by David W. Cantrell [2].
04-Aug-2010: Some small improvements N=51, 56, 58, 64, 70, 71, 74, 79, 82, 86, 88, 90, 93, 94 and 97 by Eckard Specht [1].
05-Aug-2010: Better packings for N=106, 107, 108, 109, 111, 113, 114, 115, 117, 118 and 119 by Eckard Specht [1].

References

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


©  E. Specht     05-Aug-2010