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]
143 0.030524298789264589533492081681 16.3803926652641153828629773615 0.837157741896084015328406139388 275 7 31 [1]
144 0.030504838873534247385607364780 16.3908421897548842500395054962 0.841937456881066416655487509693 370 4 37 [2]
145 0.030253161384122585174416403796 16.5271983860307962618322145900 0.833852814202376873880622303560 328 6 37 [1]
146 0.030118710794236567075190203235 16.6009761644804070634507050088 0.832157402424695738473087788104 277 12 38 [1]
147 0.029996034415041702998703677546 16.6688700606794812591385578372 0.831045667089338324028150566056 283 4 37 [1]
148 0.029939498172237215628250668980 16.7003467166877274125073108721 0.833548006376398697350155209215 261 5 34 [1]
149 0.029919015690121834039814954783 16.7117797316133543015899887123 0.838032265218344238903384442300 92 77 31 [1]
150 0.029755549335495874515855989992 16.8035882773483848638322066043 0.834462975880592609459164064941 372 6 39 [1]
151 0.029583427716873667128551761728 16.9013545281236010954471418502 0.830335872390082551952113794627 327 13 41 [1]
152 0.029531729632610148868772065103 16.9309419468570328039457135660 0.832916036404120503036626971709 354 1 28 [1]
153 0.029452878299917718960141566102 16.9762695145960259207867637132 0.833924599289888976904728517620 395 2 39 [1]
154 0.029363586774117616058513576296 17.0278925339162738891691234297 0.834293378324210248849271808242 372 5 39 [1]
155 0.029249317150331426391025879193 17.0944161680825584450513559620 0.833188044661220999638591909413 401 3 41 [1]
156 0.029202766669026161242808650602 17.1216654115968985813040308575 0.835896416856976459357125926188 280 29 33 [1]
157 0.029119214409784814953283095528 17.1707928985881902393001058735 0.836447773001234587300795755761 400 4 41 [1]
158 0.029058237273437755400315142620 17.2068248770565271406988543682 0.838253714598074049266034218180 248 13 37 [1]
159 0.029023709820116717074324623872 17.2272946187411020833301661615 0.841555648488510663092337686466 412 2 40 [1]
160 0.028924095514503622084926890786 17.2866252550328261838732677513 0.841045371338874604273600461713 410 1 40 [2]
161 0.028851810426012807835773947769 17.3299350237377038364657519759 0.842077152858825353924304928936 200 45 40 [1]
162 0.028767689234845603615444808841 17.3806104452199844341826628796 0.842373779345573595502993550844 346 7 42 [1]
163 0.028736775955143162594938170194 17.3993074512073972842148676220 0.845753019223176756581116257244 282 4 40 [1]
164 0.028709087374056238416836270953 17.4160882749564112552321117268 0.849302672110844077075778118630 301 3 41 [1]
165 0.028668598518972321055370654619 17.4406851339143670456630968213 0.852072870860392321433961934593 430 1 41 [2]
166 0.028563251083422741014652857837 17.5050101453676989423969868229 0.850948410626955988129458683395 405 3 42 [1]
167 0.028521023700264348432896607831 17.5309275450504158017386292587 0.853545267128190685717536738065 397 6 42 [1]
168 0.028481894514575358161479073921 17.5550120004878682688690552102 0.856301879764605831916022811599 287 9 42 [1]
169 0.028468419676542691635782768725 17.5633212409042937584518768204 0.860584048662543282414044205842 250 20 41 [1]
170 0.028462230114497167626611467785 17.5671406628578347797942746729 0.865299874763444728644778048198 415 42 C2 [2]
171 0.028155157990097527246117191432 17.7587353683419355390731957250 0.851710332925249407634745479438 358 12 41 [1]
172 0.028055713370242029128209255968 17.8216819298680564563841416175 0.850650079956962006609674106382 340 13 40 [1]
173 0.027974980513621630868689035472 17.8731134327882389889190832980 0.850678696365816421371397001069 257 16 30 [1]
174 0.027877534651427880692080167858 17.9355888622091666546476133509 0.849645663697377912369990674468 331 10 40 [1]
175 0.027854494020919565807362294623 17.9504247905018453153372348457 0.853116741871191664619540041000 451 5 40 [2]
176 0.027788671525522512066090112544 17.9929436187971375092462035689 0.853941473803440948363129202835 460 41 [2]
177 0.027469518079713647735583102232 18.2019938809648087632091720381 0.839180174806112089003547672645 377 13 40 [1]
178 0.027440138405255882778077978281 18.2214824362631503444674369989 0.842117060461904518779259728714 275 13 37 [1]
179 0.027420855016873229220200646629 18.2342964758877334748770386618 0.845658238880964472199987127873 399 36 D1 [1]
180 0.027320875054589898960751665082 18.3010243632734650383872027425 0.844192683334723072501042862819 470 3 42 [2]
181 0.027160747740299551798438346006 18.4089188111021267215977191482 0.838961221247833073920423215687 422 7 42 [1]
182 0.027070140901292798062709387233 18.4705355551408056179385867269 0.837977367091122178116679390215 432 10 44 [1]
183 0.026991401655121140447351188802 18.5244177530563276613385068302 0.837687112290307021509373123392 462 6 44 [1]
184 0.026942110285249565751056626824 18.5583087110196825381376629191 0.839191179512605682812623055413 471 5 44 [1]
185 0.026894728096383453585744927931 18.5910040885386466938777427936 0.840786853900850290200174233572 482 4 42 [2]
186 0.026801101672654646396249086073 18.6559495242747254884309853919 0.839456323918890867432211663466 487 3 43 [1]
187 0.026729066848563772908624074399 18.7062273005189625774923482658 0.839438858346503149739964251965 396 14 46 [1]
188 0.026668676074390265577253248509 18.7485872416496272966600328433 0.840118658659688009470229095395 402 14 44 [1]
189 0.026603324440616178478757357325 18.7946435460010931299621538007 0.840453121189481071803443017516 328 11 45 [1]
190 0.026549309863213483240776269847 18.8328812528869574057665604009 0.841472527957415063753016389914 461 3 44 [1]
191 0.026539268843604086931301889821 18.8400065934935899481527810097 0.845261607464905888613850278425 171 71 40 [1]
192 0.026432957846190840434234036648 18.9157794186114219569386038564 0.842893342861832451743252206722 485 6 44 [1]
193 0.026396682215842580358984795563 18.9417744211775757405880176843 0.844959446211852181731161050186 450 6 45 [1]
194 0.026343816719120851538742440017 18.9797858575705293447577982313 0.845938889327519253945017324595 517 3 45 [1]
195 0.026268710378111639270662074695 19.0340520262701665359250706642 0.845457895602292669841933939870 515 3 45 [1]
196 0.026247895915702708712779968149 19.0491459431945099274331694048 0.848447413677487703247112002336 307 14 40 [1]
197 0.026202902783482076315423261043 19.0818553246395521496393299048 0.849855140441242763252824625118 508 2 45 [1]
198 0.026179488125024927902621827711 19.0989219350721703208939510793 0.852643253576696405267105941211 519 45 D1 [2]
199 0.026098878874533704510192820051 19.1579110506498048766106293594 0.851680391382775337999813864237 454 14 43 [1]
200 0.026064142423986239629381457131 19.1834433631647642691847260306 0.853683218504362457951662830452 414 4 48 [1]
201 0.026027930375516013092727922674 19.2101328375436504502630811264 0.855569311520231635254449246681 511 1 46 [1]
202 0.026009679124707085858275318327 19.2236127790227349849328472674 0.858620447443356784678108379011 332 7 44 [1]
203 0.025994436777372496251725088401 19.2348849210396214719474365332 0.861860010376684901563640421961 394 1 46 [1]
204 0.025984795945105457435179389557 19.2420214134558517801772198480 0.865463301780434428338316999856 500 2 46 C2 [1]
205 0.025888784816969074081007973899 19.3133823597726295744165501145 0.863290697022397505535073042000 534 5 47 [1]
206 0.025819747808011556669016179442 19.3650226066443617233972085838 0.862881346737591753653417541863 514 4 48 D1 [1]
207 0.025727497645450246695495097588 19.4344590713983407702831867939 0.860885331634938833338637420498 444 16 43 [1]
208 0.025683654829247473440333872666 19.4676343115552574132344752097 0.862098426949051811220951656855 462 2 47 [1]
209 0.025660189630547876283852777491 19.4854366705365766764092999010 0.864661013277847913110574176641 437 47 D1 [1]
210 0.025430942147710306904662007948 19.6610883346694199533115965023 0.853343850029346992941640969237 400 14 40 [1]
211 0.025324204178219183871859708381 19.7439570649979002489095282893 0.850225128501076739996082178736 397 20 52 [1]
212 0.025263657277136025527715930223 19.7912754481714417738295677010 0.850174689663201852389316531854 403 14 45 [1]
213 0.025191584740009428865580669440 19.8478978262092794738302681118 0.849318236423463060423224767291 391 10 45 [1]
214 0.025152777696328487642633393911 19.8785202189809928082078444794 0.850678676387596954809155633715 381 7 44 [1]
215 0.025121237583400945744594094755 19.9034780169580065384503463598 0.852511782381892619192026255775 298 25 35 [1]
216 0.025009389414484396042022159896 19.9924912885086606037076077011 0.848867286774895044423825406923 555 1 46 [2]
217 0.024957350991830491677875932769 20.0341775120151726003019628353 0.849251995226075820567939677543 387 4 39 [1]
218 0.024954417219028271464737577037 20.0365328355069504085170782129 0.852965028973228023730444550078 259 38 36 [1]
219 0.024951548307616929837312621404 20.0388366219087729890313143370 0.856680700182480532508192478952 440 40 D2 [1]
220 0.024756927609435777493786222180 20.1963671699484863865561869786 0.847219694251594984782917323779 417 4 45 [1]
221 0.024654375527023819224242549709 20.2803757674554275958267636074 0.844034415777567839973395757392 583 5 47 [1]
222 0.024583954303142997308698963937 20.3384693054069111131584552263 0.843016981007456352966460355836 583 5 49 [1]
223 0.024546145899888458355453388975 20.3697966287355963534401211832 0.844211675138051678901788104853 580 4 49 [1]
224 0.024511374327676122114566726924 20.3986930033312445260258858337 0.845596567634978667249393006780 586 5 49 [1]
225 0.024456307413175520448382456503 20.4446236119288982907349293700 0.845559466456781114016275656177 593 2 48 [1]
226 0.024434837930513668593500524012 20.4625871234288563850084176021 0.847826980668358899271184178864 381 22 41 [1]
227 0.024382320286205430405171065019 20.5066619637049299043109956640 0.847921776946988992984431296056 586 5 48 [1]
228 0.024303644990277716800630557247 20.5730457386131576948386882409 0.846169838153929051075527058787 598 7 50 [1]
229 0.024270585218318338859973367772 20.6010689689765963182710787541 0.847570528729306749550315656000 602 5 49 [1]
230 0.024218583902030023269277282863 20.6453028807390160723155442943 0.847627807278537049410889405172 607 7 50 [1]
231 0.024182835029953092954747519875 20.6758223087034738677893793638 0.848801766413551336802477289372 606 5 50 [1]
232 0.024168042122966510371217051544 20.6884776787465900196513629881 0.851433613689858284670163575428 158 123 44 [1]
233 0.024108474399411274020936580295 20.7395952027644660290543524720 0.850893580131117113525885755851 594 5 49 [1]
234 0.024084929815576748642220957374 20.7598695046488668146658457318 0.852877183144722676695103976157 570 4 51 [1]
235 0.024034321211533409925537408993 20.8035831592391195764034579913 0.852926195354949097015529444690 614 4 51 [1]
236 0.023998028408993557240587846661 20.8350449244663295282037743293 0.853970753870807933847860840796 606 4 48 [1]
237 0.023959665258176061074171647376 20.8684050721192210984583712713 0.854849588125676421809767984199 611 4 50 [1]
238 0.023947748728349824886379776742 20.8787893038183576478372863061 0.857602840399127879432805808042 393 8 48 [1]
239 0.023932825228432520546903511811 20.8918084358044459952441373919 0.860133193708411219501111874422 522 4 50 [1]
240 0.023906597128458200936964534529 20.9147289893802761196012674593 0.861839978314580788883901926398 631 4 50 [2]
241 0.023837570897961177375643411182 20.9752915739734579672920117087 0.860440623719258296453361717775 633 6 52 [1]
242 0.023816374464428687945713293248 20.9939594604032903169644903966 0.862475038159362464419992281349 480 6 53 [1]
243 0.023789483750642631837469824590 21.0176902214825642491547171627 0.864084425208600013840522360323 555 3 52 [1]
244 0.023773925474852827831909549391 21.0314447451634086238852489440 0.866505829143070235534523520683 524 2 52 [1]
245 0.023766787833391966426231571578 21.0377609084180822308270194547 0.869534726796852126215110759462 270 64 48 [1]
246 0.023764395432472400734695193056 21.0398788145388552037335084388 0.872908085088057451052319086453 493 51 D1 [1]
247 0.023552493213875367425720524415 21.2291749947490653262533638699 0.860895814329373753078915281449 603 8 52 [1]
248 0.023493272150631340734062439717 21.2826887967823320700014527591 0.860039837620290722173686819674 350 29 43 [1]
249 0.023435397799192867521272085097 21.3352469748655331455519592549 0.859258575164927208490010874387 547 3 53 [1]
250 0.023355890353995782344336889832 21.4078758044203092360784454324 0.856865648704522807569138560941 541 4 52 C2 [1]
251 0.023322991809768651564780342356 21.4380729572858202119650331835 0.857871241911462781108396289627 509 12 44 [1]
252 0.023302577855523479490307661727 21.4568535335451527986603365818 0.859781991619316672596093398964 480 10 54 [1]
253 0.023164683161765541621232760977 21.5845818614637819353053473251 0.853008027880186513757101780084 586 15 48 [1]
254 0.023083379387426491015983171478 21.6606066039164885767808423533 0.850378681129596969459046934589 438 9 38 [1]
255 0.023021477795255211112176199464 21.7188490003474648092822133949 0.849153972495287427170744637785 559 9 49 [1]
256 0.022989168271594785595932718181 21.7493731871020148446912225388 0.850092827614683207781038476886 652 6 52 D1 [2]
257 0.022949936262126711223912094941 21.7865528814181679760679905850 0.850503214039743094389603031167 502 15 54 [1]
258 0.022921626939540217620628036102 21.8134603324117029143993302993 0.851707465743239052386597088547 549 5 50 [1]
259 0.022842320989425122366770557694 21.8891941949102085598557044243 0.849102447428288195319043913286 685 5 52 [1]
260 0.022749079140619140019153761688 21.9789116258000751864729459991 0.845436238674083149431853110984 683 5 53 [1]
261 0.022738827242976757328087022617 21.9888209122320864342659022342 0.847923164578122260040595532288 692 7 54 [1]
262 0.022738847982564955100319041160 21.9888008567265909964819347276 0.851173465006494273517274057624 288 70 45 [1]
263 0.022742944515625178490254324584 21.9848401624724955197272443089 0.854730104485577342208760755451 428 10 37 [1]
264 0.022686356854871270942962255755 22.0396779967180476129353839511 0.853715789279887982163599202798 527 1 45 [2]
265 0.022619624326634222264045283090 22.1046995644069348544237644409 0.851915495715389456136886974227 645 13 53 [1]
266 0.022545816626216991389068308962 22.1770631904538833736918972554 0.849558805818353870459253633600 695 6 52 [1]
267 0.022505251361713945564281194540 22.2170369023561622708192247028 0.849686788527305516108581750664 593 13 52 [1]
268 0.022475576482647568823107159347 22.2463704272959855679482911260 0.850621473734303670062320245182 689 9 53 [1]
269 0.022433443041362350209228618065 22.2881525175653879813169973039 0.850597329933115644943196688896 714 6 53 [1]
270 0.022404903893470077438210156233 22.3165429486946068669293850993 0.851588529739621455549984637620 658 5 53 [1]
271 0.022401124925992715560822018887 22.3203076475786531633707212247 0.854454251954406789872236259957 160 151 49 [1]
272 0.022330800265160510296636066671 22.3905992648224547751174850904 0.852231037102768223226324163248 723 4 53 [1]
273 0.022309942945733705340077142644 22.4115319889517780790719263125 0.853767138288501692020765020896 602 18 52 [1]
274 0.022276160952751874110322264175 22.4455192732943849832922750904 0.854301414582833788533881482381 692 9 52 [1]
275 0.022241928046184515186987003304 22.4800655303698974346184570324 0.854786048364534386397855698201 696 13 53 [1]
276 0.022210809994061184149439728019 22.5115608180742626769231286227 0.855495529063495618428399117106 466 7 52 [1]
277 0.022195759728175783452382722984 22.5268252190209578976110236528 0.857431959537241971694888666061 464 16 49 [1]
278 0.022181481611371754803750404940 22.5413256319030362679878207584 0.859420615552270540345163399779 568 8 56 [1]
279 0.022153411110297019040185595947 22.5698876579597007095281626660 0.860330432635262903229822670710 691 9 56 [1]
280 0.022143157008859841031584934347 22.5803393707564726536446029809 0.862614946053261735334957632075 622 2 54 [1]
281 0.022090720123257037265442629563 22.6339384687419786541377759106 0.861600485724955226537232146150 742 7 55 [1]
282 0.022060035660516849909989614503 22.6654212030537297459348250083 0.862266268434812780262755465468 749 5 56 [1]
283 0.022037939881602512214869180100 22.6881461101273305009934657588 0.863591366347640205560013579149 729 3 55 [1]
284 0.022026935871964960496770940892 22.6994804409623233554311163794 0.865777675722038955306009593483 482 8 54 [1]
285 0.022019462748245911473512947087 22.7071843539793182076679615990 0.868236751784762776163327975686 550 15 58 [1]
286 0.022012220400374815776763766029 22.7146553553264526103451189236 0.870710149096804878842352933415 666 1 55 [1]
287 0.021953790770444916435502275178 22.7751100130333882759705743935 0.869122127098316440029558568732 483 10 55 [1]
288 0.021930049967546014927976525729 22.7997656521505082806708670671 0.870265162262647426413288089086 552 11 56 [1]
289 0.021866843179344365245297082582 22.8656690816855040590952350912 0.868260195803609209981902627671 619 6 49 [1]
290 0.021804543254270266341907428040 22.9310008546993305564392862895 0.866307060367706487637674201770 625 12 50 [1]
291 0.021776541570683319599002782964 22.9604870165942814223880542051 0.867063041134784510603445358994 512 16 55 [1]
292 0.021758449031095686655696674980 22.9795790722690854338429922883 0.868597529879460139427385793531 611 9 53 [1]
293 0.021749636546474226692427229566 22.9888899031305244680599858281 0.870866323402620008294491613201 610 9 55 [1]
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.