The best known packings of equal circles in a 1x0.50000 rectangle (complete up to N = 300)

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   297-300  

Download

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

Results

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

Legend:
Please note that all packings (including their coordinates, of course) are normalized such that their width (i.e. the horizontal dimension) is equal 1.
N
the number of circles; colors correspond to active researchers in the past, see "References" at the bottom of the page
radius
of the circles in the rectangle
ratio
= 0.5/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 0.250000000000000000000000000000 2.0000000000000000000000000000 0.392699081698724154807830422915 2 1 D2
2 0.250000000000000000000000000000 2.0000000000000000000000000000 0.785398163397448309615660845822 7 2 D2
3 0.177124344467704704749192123181 2.8228756555322952952508078768 0.591367747560302949672047427901 7 3 D1 [1]
4 0.151923788646684059708830487741 3.2911238223712713478714889306 0.580084717656191386345172372358 9 4 C2 [1]
5 0.141101056459326447763018016141 3.5435595774162694208947927935 0.625475644901659478664848120488 11 5 D1 [1]
6 0.135621722338523523745960615902 3.6867250421135107520700798301 0.693409248097838517925326343365 13 6 C2 [1]
7 0.127166547515124908877372380214 3.9318516525781365734994863995 0.711252276864909443355102456272 17 6 [1]
8 0.125000000000000000000000000000 4.0000000000000000000000000000 0.785398163397448309615660845821 22 8 D2 [1]
9 0.110653654474367741340655060442 4.5186035867963313937887823539 0.692394964906412041401250749201 17 1 7 [1]
10 0.105783167543767350018062597211 4.7266499161421599396459730947 0.703093370410220558014772446967 22 7 D1 [1]
11 0.100233512361812980711521148405 4.9883515824043923067689537591 0.694381995665899017426672706410 24 8 D2 [1]
12 0.096427664499301081081613737870 5.1852339533083273846814570336 0.701074887162327895714146698301 26 10 C2 [1]
13 0.094142364928295586141985453540 5.3111051584568720185160791068 0.723924756137015586693921285372 27 10 D1 [1]
14 0.093365838666289124470030673945 5.3552777669262355400365766702 0.766803187178218992801525852000 30 10 D2 [1]
15 0.091090581831205205682456817276 5.4890416764108682582596182232 0.782020395077194379410398119651 35 11 D1 [1]
16 0.085955228382080002217953105914 5.8169818103146393493737195852 0.742753057385047496076244374298 36 12 [1]
17 0.084652380550897770379015970918 5.9065084377558867324868735460 0.765432927769881082667686725273 41 10 D1 [1]
18 0.083333333333333333333333333333 6.0000000000000000000000000000 0.785398163397448309615660845820 45 14 D2 [1]
19 0.078882401662675827253930313354 6.3385494034289921080468402933 0.742837327277352828388001277557 39 11 [1]
20 0.076558486389562273600443942017 6.5309546149565506221768049234 0.736540345451483812634659053635 39 1 11 [1]
21 0.075053719656362888139207965053 6.6618950038622250655537796199 0.743264866433156259903072044755 45 11 D1 [1]
22 0.073151002198728661815716595557 6.8351763471627435978786734290 0.739678695564936988781589462789 45 2 13 [1]
23 0.071986654782464227954291509799 6.9457318375321806201443920638 0.748879063510356349823388349776 52 1 14 [1]
24 0.071244627914878274325627103835 7.0180730061133940539222952516 0.765412156197806915867227639841 55 1 14 [1]
25 0.070266433941895194884915371942 7.1157730932163230145491284125 0.775560539150189204271231651531 56 1 15 [1]
26 0.069925404472025400649546220148 7.1504770515847603095767246459 0.798772658173624024850866941906 64 15 D1 [1]
27 0.068211221141434775435384587398 7.3301722448753517349200611133 0.789323951734375030235177249966 67 1 16 [1]
28 0.067325459096181718382699377570 7.4266110727250412843872798945 0.797437301818801107264993422928 63 16 C2 [1]
29 0.065263704794652490405193588681 7.6612261221334845244042952908 0.776106486814313277603973846421 55 3 15 [1]
30 0.064000660504103315482893710283 7.8124193728898028892742833677 0.772093746896022497102989802677 68 16 [1]
31 0.063239039575592430245527524692 7.9065084377558867324868735460 0.778954505033258240899845422416 72 16 [1]
32 0.062513529119464986837398831487 7.9982686474872813740198889771 0.785738224057801468068136465618 77 12 C2 [1]
33 0.061362641380780810960072632138 8.1482802687272086503593438242 0.780731855225650963143991475744 62 2 17 [3]
34 0.059913441660665942403647655027 8.3453726933576805383817084746 0.766844524865402961987611579479 56 8 18 C2 [1]
35 0.059073014802394054344042589115 8.4641016151377545870548926830 0.767407755937647211720588284907 87 17 D1 [1]
36 0.058528001390752912998822122969 8.5429194252137425483488998738 0.774835940932423871351782455143 76 20 D2 [1]
37 0.058005263287499033189237805679 8.6199074301548283239360443785 0.782197452598420263396003468222 75 4 18 [1]
38 0.057599687688973850409428335453 8.6806026223595935061023872449 0.792143243443994755142792773403 90 2 18 D1 [1]
39 0.056869687311079409555242253989 8.7920300539895794048062364450 0.792512564229373017714820788255 77 1 19 [1]
40 0.056663836097005681013780908938 8.8239701799226011357810865471 0.806959622831049205220529231512 98 1 19 [1]
41 0.056099074285617284286230158416 8.9128030429584167377195684106 0.810727892311389606953002546479 88 1 19 [1]
42 0.055753293928454377568280133108 8.9680799961635785204543993619 0.820295294549972315075347541306 90 3 18 [3]
43 0.055563768403385707566717104997 8.9986697153091786311327591103 0.834126096657307778808982018695 107 18 [2]
44 0.053876965144146832689219688582 9.2804039474432007240855344601 0.802488455194093560282514848872 109 22 D1 [1]
45 0.053278688524590163934426229508 9.3846153846153846153846153846 0.802600515001020477644060834784 99 21 D1 [1]
46 0.052265194045587846294174911870 9.5665960708742320704392270496 0.789519451645951096890502081732 86 4 20 [1]
47 0.051680888630939204362972417134 9.6747562444325065144402172198 0.788746911006366103895364396224 90 3 20 [1]
48 0.051329154305541199014026688277 9.7410527557829427958074423733 0.794601395758337528598431004864 125 20 C2 [2]
49 0.050502141969400949816447455479 9.9005701639932033634100326910 0.785227580140066157576390542209 104 2 20 [2]
50 0.050142287471139294753868254653 9.9716232588668412804187997430 0.789874616519656831838551349116 103 2 17 [2]
51 0.050010882998682898954840120671 9.9978238739189656834026118539 0.801454902095346760728722938087 119 1 19 [1]
52 0.049246763357710309753212469995 10.1529515019735322161114224424 0.792389290668274744213972336885 103 1 19 [2]
53 0.048809537200804487642610287642 10.2438996285291332231010827191 0.793350532507185761248207100223 131 22 [1]
54 0.048442234418427880520704572183 10.3215717855036700856817415358 0.796199612858575075013494746864 123 3 23 [1]
55 0.048179877701987543711209906807 10.3777764462729990634216119109 0.802183905605113684835321170638 120 1 23 [1]
56 0.047853713201627127034293701652 10.4485099806841937386447408623 0.805747896175842555296394041673 122 2 23 [1]
57 0.047653713411493648625646346508 10.4923617532690428434640095663 0.813295222843526962421495939424 136 1 23 [2]
58 0.047243389501077936665427108193 10.5834912625943501941117521019 0.813373388693226563014110118522 120 2 23 [1]
59 0.047126588165473066409583748155 10.6097220160385208505405710793 0.823310924997848321699943554184 134 1 24 [1]
60 0.047069192209893667609523392109 10.6226594620611087978945350877 0.835227161249785393423821969099 121 24 C2 [1]
61 0.046225992794225535501999773532 10.8164253437615527890364959762 0.818996786818347667015899833339 116 6 24 [1]
62 0.045895850148690101770761968361 10.8942311424700852628705774845 0.820575215804613958504892966452 157 2 25 [1]
63 0.045599739281343847005149976587 10.9649749731039416421165516573 0.823085852530095985023164965229 163 1 23 [2]
64 0.045462299986847309645500334989 10.9981237232752173957920275644 0.831117925188451986755506436566 148 2 23 [2]
65 0.044443613448354213463611989344 11.2502103498183369409461028080 0.806699798123176097431535260635 155 1 26 [1]
66 0.044145386394489884647898198349 11.3262118838857583084278266809 0.808154599134102888065687050335 152 3 24 [1]
67 0.043622899472312524209825004291 11.4618699363931608070669456032 0.801094458858702264951218137924 62 22 23 [1]
68 0.043201265736692864133621358584 11.5737349698836836447482998081 0.797410084158565690753733441318 134 5 24 [1]
69 0.042930061617729661001072932760 11.6468502759731724418663056448 0.799009573150469239437152217856 135 5 24 [1]
70 0.042635291302461060569843903551 11.7273738427849846225327208396 0.799496151606990024839396072618 143 1 25 [1]
71 0.042393241618374906130350178069 11.7943327972183245826854355010 0.801736156603141738672675300991 143 7 25 [1]
72 0.042157793609315461098708808753 11.8602032315447537599255276026 0.804022331813597169923002163248 172 1 26 [1]
73 0.041932470579286307542193317763 11.9239337223070442219814775251 0.806498623258747902848173892280 170 3 26 [1]
74 0.041707829184156647021925114669 11.9881568947715121371123099963 0.808810462484526610526787447879 166 2 26 [1]
75 0.041462593147343994928020639873 12.0590624475215422223038901381 0.810128763724393000938543146191 175 5 27 [1]
76 0.041337977640229239254996582794 12.0954151253255952955659869922 0.816003295444013551751541123725 164 5 28 [1]
77 0.041153613262605951369219562983 12.1496014653547514390450093052 0.819382223249259073781673805889 188 27 D1 [1]
78 0.040922389296232648375756860147 12.2182504149636845751385013728 0.820722682110777389597161024339 190 3 28 [1]
79 0.040776371214460525536308867001 12.2620033393919316583133903132 0.825323304148310047978793839802 159 3 28 [1]
80 0.040726828170627310132321232423 12.2769197224301929689022764774 0.833740756330560801242390460029 191 28 D1 [2]
81 0.040660136398854019834771552820 12.2970566329455840928277494123 0.841400081256199210756908683103 164 28 D2 [1]
82 0.040351006020421063617939452707 12.3912647864828250747577718440 0.838885049554728842245129650526 191 2 29 [1]
83 0.040149313866204279530871168360 12.4535129458557308591222879806 0.840648062364534743342055069000 211 2 29 [1]
84 0.040044221033453281484979358806 12.4861961875171891660050362212 0.846328282301703960966023864792 180 29 D1 [1]
85 0.039356733062302146493957830446 12.7043065085837900924873712491 0.827250191642810942009336067505 140 8 27 [1]
86 0.039034349725912202594995428228 12.8092309340581793514700702702 0.823326733606224277376249433182 168 6 28 [1]
87 0.038822406712210932303739638233 12.8791603185882199626925605857 0.823880134210177421952103653391 183 5 27 [1]
88 0.038755956104951631372920614796 12.9012428088728741582488943252 0.830499645124232599959469903593 237 28 C2 [2]
89 0.038468664389276014689817297669 12.9975918825865445023510626062 0.827530655925570416004833959465 219 29 D1 [2]
90 0.038194738877783002773296068763 13.0908081764852291987335848586 0.824953497919151507475023437500 181 25 D1 [1]
91 0.037807952290728865782753756802 13.2247310342329292561384026301 0.817311431281990592943464796007 230 2 29 [2]
92 0.037473232388110558136722762134 13.3428575048316122468711250208 0.811727030820734102880243134309 226 3 29 [1]
93 0.037325032326601734455544405537 13.3958356854160722665056062727 0.814072720795758829851682083403 224 5 30 [1]
94 0.037116545755325693903764764827 13.4710811532955527044100933762 0.813659736556871684053897838731 239 1 29 [1]
95 0.037041337463301754997468058617 13.4984326766107943460855305469 0.818986593626615485609391750808 218 3 29 [1]
96 0.036793469284642689364368145591 13.5893681601994555098436445624 0.816568421610102880682745210651 210 4 30 [1]
97 0.036608073852591538566964814957 13.6581892293304657397943556220 0.816780499434635355155609994094 238 4 32 [1]
98 0.036476955779110724971034101307 13.7072842105517815654566731321 0.819300306682951444663331396073 239 2 32 [1]
99 0.036361094919354451148096507900 13.7509610507866663856475796634 0.822411108363253028219094708578 246 2 31 [1]
100 0.036212513310819411514713240237 13.8073818767672293415343179023 0.823943067560966553446550115391 248 1 31 [1]
101 0.036081114685429554291876986105 13.8576649961957066550503844524 0.826154236726002432233528277770 252 2 32 [1]
102 0.035941023544174026998796707917 13.9116794875211356938578836683 0.827867668621613214861582716929 255 2 34 [1]
103 0.035904565298857659992845146794 13.9258056973581575427110717097 0.834288849459271344881886623454 174 7 31 [1]
104 0.035799673236013333428892266926 13.9666079269409613453249442702 0.837473999904572871080193556743 258 2 32 [2]
105 0.035617153709178924351362343472 14.0381795828661236222823856016 0.836927018654610693080332202230 213 3 34 [1]
106 0.035524785154266454123315441872 14.0746804753005246944275110753 0.840521165025831414583505763815 222 3 33 [1]
107 0.035490076907234818239428480609 14.0884450971159394824021598329 0.846793521973728029052323549109 259 2 33 [1]
108 0.035474189489398044812362246703 14.0947547272202500355571048708 0.853942418074544477190283170432 217 33 D1 [1]
109 0.035053264279659664521764184611 14.2640067986516931476881572197 0.841517788225853820061957106928 247 3 35 [1]
110 0.034836066725984940353270761905 14.3529407017423035296689809713 0.838746616032047219823059347841 248 4 32 [1]
111 0.034692167026754553375583769515 14.4124752891452603869421304569 0.839393698298279153145986934998 256 1 34 [1]
112 0.034586627008555786210295073319 14.4564545098981078974383869791 0.841810448321954602111698713365 238 34 C2 [1]
113 0.034197885134882832419923765394 14.6207871635309264059483657557 0.830341626358672505658507269366 194 12 33 [1]
114 0.033976666163959316737526691031 14.7159817737025075756944700689 0.826887155470670170339074275208 245 5 31 [1]
115 0.033812840620867094408324013681 14.7872817195794072478800448761 0.826115981029141994682112120675 270 3 32 [1]
116 0.033728283609856567083219774278 14.8243535242891151752079923448 0.829137083695658698609471522448 280 5 32 [1]
117 0.033559646957911044744006845123 14.8988456471868386401798335173 0.827943111291569313344843387381 299 33 [2]
118 0.033421036648604225218737857958 14.9606370758963779761197670480 0.828136085114523097049212265819 265 2 29 [1]
119 0.033391035230150718043282214709 14.9740790171285484912647650289 0.833655456988161249600367522959 245 29 D1 [1]
120 0.033099972685625485952927537917 15.1057526466521180427107389313 0.826069115169024255249837892179 305 2 33 [1]
121 0.032898941383116217301800802759 15.1980574139872080751058834372 0.822865938660027616545603381086 290 6 34 [1]
122 0.032769143187758488004306834410 15.2582567427879570848896929639 0.823132730763227384568375148174 296 2 34 [1]
123 0.032672589642163857972090138470 15.3033477136673556020615909425 0.824996481177404162756800544877 302 5 35 [1]
124 0.032578214153997432801386872080 15.3476798217513307510549287851 0.826905920034870154430865938710 316 3 35 [1]
125 0.032467887201004217734645406482 15.3998317446580021336115496686 0.827938233348228619477252420207 320 3 35 [1]
126 0.032338145197637654872441305476 15.4616165195685272156296623620 0.827905231392529110206146022430 317 1 35 [1]
127 0.032221401821155752656064731828 15.5176364695502701885386291096 0.828461729332254521640291663436 247 3 36 [1]
128 0.032152738848303439166661472212 15.5507747678665587581579054170 0.831430178708538094356839659800 272 1 38 [1]
129 0.032111690007027140829298741067 15.5706535498624589581207383873 0.835787562388498924891025291757 280 1 36 [1]
130 0.032043702323317776757149915505 15.6036900778521038181311982177 0.838703775220033071038457253398 300 5 38 [1]
131 0.031988271722746359424764367637 15.6307287975316848612464845633 0.842233898230128811367628216114 314 1 36 [1]
132 0.031836285162943187589544020442 15.7053499628150776652987179262 0.840617780345121738874174697560 309 4 37 [1]
133 0.031765249959456289370280153386 15.7404711323908077779054653856 0.843210611974200538791313927757 323 2 37 [1]
134 0.031747675676243598892591981865 15.7491844473560608673454057133 0.848610765266058093202330913329 222 6 35 [1]
135 0.031707954593793196249505355526 15.7689137128345248978642021360 0.852805695341224978306945689111 319 1 37 [1]
136 0.031570196507596091110999755836 15.8377221339023094037498494125 0.851673917193855233458847033444 332 2 39 [1]
137 0.031499782194109611131325789225 15.8731256273098574791470568839 0.854113403473558557865426588848 270 7 32 [1]
138 0.031353448896212707770040941609 15.9472089228562265264242477560 0.852372825659176228720549373011 225 8 40 [1]
139 0.031283541947471316585916043173 15.9828449361507019156416218362 0.854725193635759007565715075610 295 5 33 [1]
140 0.031263686459127122158144949397 15.9929956006205395998161816071 0.859781858380831173391999168280 385 36 C2 [2]
141 0.030832247684132922728303277942 16.2167872132563665091274586604 0.842188589993483981665681451850 288 4 40 [1]
142 0.030648977160549346038431441395 16.3137581192624080902690207124 0.838108381360546642575148346006 362 3 40 [1]
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294 0.021583863206326468391301056427 23.1654544517982529022225275667 0.860568724455421507859496354099 645 13 52 [1]
295 0.021507617364507104896785004206 23.2475774292471757982889609263 0.857405938734384260031481165011 576 18 58 [1]
296 0.021465834659063596848891407390 23.2928282520280754726278293726 0.856973000039866459604605270535 621 11 54 [1]
297 0.021414645930970905169837150803 23.3485065133332706154263433659 0.855772081455029736224861774962 633 12 56 [1]
298 0.021393555810296541561152482170 23.3715238567005365822364216830 0.856963019510581104903106650740 638 12 58 [1]
299 0.021358825842630550654695276806 23.4095265200411435484882124449 0.857049302531635097135522453124 542 7 51 [1]
300 0.021293996955586860418567236853 23.4807960686223386468440661418 0.854703531418559057789569276008 718 15 53 [1]



Updates

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

30-Jul-2010: First complete presentation from N=1 to N=299.
02-Aug-2010: Better packings for N=33, 41, 42, 43, 48, 50, 52, 55, 63, 64, 66, 80, 88, 89, 140, 160, 170 and 256 by David W. Cantrell [2].
02-Aug-2010: More better packings for N=32, 39, 41, 66, 86, 87, 90, 91, 92, 93, 94, 95, 97, 98, 99, 101, 114, 115, 118, 119, 121, 125, 126, 127, 128, 131, 132, 136, 138, 141, 143, 148, 150, 152, 153, 156, 159, 161, 166, 171, 172, 174, 175, 176, 178, 179, 181, 182, 183, 184, 185, 190, 193, 194, 195, 196, 197, 211, 217, 218, 219, 222, 223, 224, 225, 240, 243, 247, 251, 252, 254, 257, 262, 263, 264, 265, 271, 276, 285, 288, 290 and 299 and a new packing for N=300 by Eckard Specht [1].
04-Aug-2010: Better packings for N=52, 198, 240 and 264 and a corrected packing for N=64 by David W. Cantrell [2].
04-Aug-2010: Better packings for N=57, 85, 86, 87, 93, 95, 98 and 290 by Eckard Specht [1].
10-Aug-2010: Even more better packings for N=35, 41, 42, 46, 49, 50, 54, 55, 67, 68, 69, 70, 71, 79, 93, 94, 98, 103, 113, 114, 115, 117, 129, 132, 142, 143, 144, 145, 146, 147, 148, 150, 151, 166, 171, 172, 173, 174, 175, 176, 177, 184, 193, 195, 199, 217, 218, 224, 225, 226, 243, 247, 249, 251, 252, 253, 257, 262, 278, 287, 288, 295, 298 and 300 by Eckard Specht [1].
11-Aug-2010: Two better packings for N=49 and 50 by David W. Cantrell [2].
30-Aug-2010: Some improvements for N=57, 91, 104, 117, 144, 165, 175, 176, 180, 185 and 216 by David W. Cantrell [2].
19-Mar-2011: Two better packings for N=33 and 42 by Milos Tatarevic [3].
19-Mar-2011: More better packings for N=71, 102, 292 and 293 by Eckard Specht [1].
24-Jun-2013: Due to Julian Wiseman's discovery that the case N=7 could be slightly improved for a 1x0.60000 rectangle, the same is true here for N=13 [1].

References

[1]   , program crc, 1999–2010.
[2]   , private communication, August 2010.
[3]   , private communication, March 2011.

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©  E. Specht    21-Jun-2018