The best known packings of equal circles in a 1x0.80000 rectangle (complete up to N = 420)

Last update: 25-Jun-2013


Overview    Download    Results    History of updates    References

Overview

1-6   7-12   13-18   19-24   25-30   31-36   37-42   43-48   49-54   55-60   61-66   67-72   73-78   79-84   85-90   91-96   97-102   103-108   109-114   115-120   121-126   127-132   133-138   139-144   145-150   151-156   157-162   163-168   169-174   175-180   181-186   187-192   193-198   199-204   205-210   211-216   217-222   223-228   229-234   235-240   241-246   247-252   253-258   259-264   265-270   271-276   277-282   283-288   289-294   295-300   301-306   307-312   313-318   319-324   325-330   331-336   337-342   343-348   349-354   355-360   361-366   367-372   373-378   379-384   385-390   391-396   397-402   403-408   409-414   415-420  


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
= 0.8/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 0.400000000000000000000000000000 2.0000000000000000000000000000 0.628318530717958647692528676656 2 1 D2
2 0.267544467966324133600221291114 2.9901571356754651051702999198 0.562188337892633435359300631135 5 2 C2 [1]
3 0.223833696070628178173747954583 3.5740820709477499522033726251 0.590244668065107919146618692211 7 3 D1 [1]
4 0.208578643762690495119831127579 3.8354837559985132088208302238 0.683375737327299495949075064067 10 4 C2 [1]
5 0.188357266740185233085547600489 4.2472478702056009096021369775 0.696617931805065540446616879813 12 4 D2 [1]
6 0.172613694011068949675426115515 4.6346264969493066554781667400 0.702039631505738778917420641655 14 5 D1 [1]
7 0.155983960168048643595298614200 5.1287324615820977135594680322 0.668833180339910418762796893399 15 6 D1 [3]
8 0.151662612389395853535683841774 5.2748662798052623720382222557 0.722614942078649109353318469437 18 6 D2 [1]
9 0.144331195321985151697250280456 5.5428072788789586165453384448 0.736245768767158135905306992717 21 7 D1 [1]
10 0.134224258574381790065222795652 5.9601744758878407466293533232 0.707492618507616034790102994068 20 1 8 [1]
11 0.133347460724735977412358558869 5.9993643347390705830469917441 0.768107615365377910823284439555 27 6 D1 [1]
12 0.126322452478319825311877594386 6.3329992753053977525363008804 0.751972968454567389055505759013 27 8 C2 [1]
13 0.119072443431197967377035895532 6.7185990053378997630620754243 0.723811984017211796270966600357 30 8 C2 [1]
14 0.115156251600485524394653801874 6.9470826714250832899752567115 0.729059479518667398207975091522 34 9 D1 [1]
15 0.112617236481240399971284622636 7.1037083220672239354997665858 0.747069277249179055666419222081 31 9 D1 [1]
16 0.111129887168889730288663097818 7.1987835170227374873831763309 0.775964075149046070806714541383 36 10 C2 [1]
17 0.106170692314551135413282932008 7.5350361061021043118519077572 0.752520101986101792490116012692 34 1 10 [1]
18 0.102981868809774241274631325689 7.7683577628382501922354288036 0.749642030257827819125577503010 41 10 [1]
19 0.101542459539147280449314249484 7.8784776696449727530260232650 0.769323225570554311485525565089 45 10 D1 [1]
20 0.100000000000000000000000000000 8.0000000000000000000000000000 0.785398163397448309615660845820 49 14 D2 [1]
21 0.095500509827938734152534464858 8.3769186305009598740908563702 0.752125928374141410540920639922 43 11 D1 [1]
22 0.093609530103256944275449496239 8.5461384019079239396236097651 0.757046745730220555472543857544 43 1 12 [1]
23 0.092417255226161923847724098059 8.6563921211710270370579531933 0.771425256439958308680909752728 53 12 D1 [1]
24 0.090425531914893617021276595745 8.8470588235294117647058823529 0.770643059829773296103189173072 52 12 C2 [1]
25 0.089919399267489652200258118461 8.8968565906471741975523341526 0.793791945716312991004236027612 51 13 D1 [1]
26 0.087062640429280983805211727293 9.1887863273549798790523229849 0.773921482946352099139921045921 49 2 12 [2]
27 0.085322779317815833632827537474 9.3761596421995116572549545429 0.771886841391011871971806089725 57 1 13 [2]
28 0.084340088756575576984319988601 9.4854061905125512612973071706 0.782142750846249464345523725475 57 1 14 [2]
29 0.083398168183174634737965093958 9.5925368317789734857054608022 0.792083391012325273982684309307 67 14 D1 [1]
30 0.080986667745386336578458896655 9.8781691143919740729777998031 0.772695175030774319822252002165 66 15 D1 [1]
31 0.080214937307274927186321623011 9.9732048276181306828099559620 0.783307123974857560134467990240 70 1 13 [1]
32 0.080021775095394196752104916597 9.9972788537409675559005811763 0.804685593173252340400564548600 83 14 C2 [2]
33 0.077957865600844411603253164885 10.2619536057607853145642040119 0.787578235091706935869053830124 65 2 15 [2]
34 0.076233568624217023440015742984 10.4940646809214830543966311997 0.775945651049225317012203861321 66 3 15 [1]
35 0.075856940904150392372391468868 10.5461674365546326310313992972 0.790894544351602944073948814785 73 4 17 [1]
36 0.075505534593545460651581324040 10.5952497960114761801141001768 0.805972010537677504542200907558 73 16 C2 [1]
37 0.073568574016070527641664260205 10.8742083246746869735044726681 0.786405036232603193447055420818 75 3 18 [1]
38 0.072890905684027670903328698176 10.9753060754642427302698591500 0.792848438573403845221861638769 73 2 15 [2]
39 0.071969970506444457905924439148 11.1157472258289314556582886275 0.793281163660601709449965579052 85 2 16 [1]
40 0.071585465729344188896337168687 11.1754528918579822738961645809 0.804951263864965331222681263606 100 17 D1 [2]
41 0.070165716059027627383043308627 11.4015796450647180046310457936 0.792672270303642767064797297560 83 18 [2]
42 0.069110251506051524732967296468 11.5757066797817878687031802534 0.787760359333533515452399876975 81 4 19 [2]
43 0.068192720630483868045748100180 11.7314574430159895882316676649 0.785243547246417110649555692377 85 2 18 [2]
44 0.067704183267549417852677702048 11.8161088634450688564014820638 0.792033533040683336745619108576 92 20 C2 [2]
45 0.067054307873368633575764955974 11.9306279547436641984041778930 0.794558298293462679108214245104 106 17 [2]
46 0.066674416812048080185571801327 11.9986051359873288491457943212 0.803038133232674601326491901384 88 4 18 C2 [2]
47 0.065589888669514147182363779599 12.1970019499642176746525127114 0.794020163499486923956503353539 93 3 19 [1]
48 0.065277552294993889335577521480 12.2553614814590971697430642112 0.803209517249857903422625973531 107 2 19 [1]
49 0.065078237692110609335543824476 12.2928958799537907277172507014 0.814943562240665982419217140638 99 19 D1 [1]
50 0.064197067266620434474813002025 12.4616284522387168840660869638 0.809208185269689952708267443247 99 4 21 [2]
51 0.063350777978652333386702529866 12.6281006409989232088548793314 0.803774024463320620939195766750 122 3 20 [1]
52 0.062930698071933833900005509666 12.7123967238619944370351189269 0.808701647550231821854342383755 116 1 18 [1]
53 0.062670013122123255384678366458 12.7652757697858299009862718412 0.817438948276751886155250572278 123 1 17 [1]
54 0.062007341694591756722645255595 12.9016980592440962808661677656 0.815342108074403617486282133899 107 1 21 [2]
55 0.060865465652809370527816114909 13.1437423737688000017777915620 0.800137220230281510580485950394 109 8 23 D1 [1]
56 0.060251757547649745566639287656 13.2776209783975967613738456134 0.798339012266576296515534098482 105 5 22 [1]
57 0.059607365793457050070870641312 13.4211601091725140214994046344 0.795306625831041127674550884087 110 4 23 [1]
58 0.059168171244214064726926683352 13.5207829340885768021419257420 0.797377858573125807254304796926 112 3 19 [1]
59 0.058749107103760024718972964133 13.6172282344151386299912378722 0.799676692362754782413028049632 130 1 19 [1]
60 0.058537287973749982127615445398 13.6665026292086838438726176826 0.807376922313816306934900563098 160 21 D1 [1]
61 0.057876159181225037025340338168 13.8226173145836382365572968288 0.802396684664319645206305264504 145 3 22 [1]
62 0.057670909554008571696346797750 13.8718117363972430723771493788 0.809776514550776503459037387612 151 2 22 [1]
63 0.057325065828633922879914388191 13.9555007645608191758408626601 0.812998153836271427936550150507 136 2 22 [1]
64 0.057183306192520156714167615762 13.9900969927591163495795405107 0.821823172710249079025078368632 129 22 C2 [1]
65 0.056929613282050999659491157953 14.0524404414359030657303489320 0.827274637046678761863397920433 156 21 D1 [1]
66 0.056279684084868147915577768199 14.2147208714537730258544215720 0.820931883360533353378235928019 136 2 22 [2]
67 0.055800526792228472010922347178 14.3367822131639571589656452652 0.819240259274373159432672740716 124 1 19 [1]
68 0.055652909818278224269435994077 14.3748099176164549889410039401 0.827074347931220697077386382260 176 2 22 D1 [2]
69 0.055555646379566914274390034963 14.3999764584547422964080772993 0.836306334348300442976743022701 179 23 D1 [2]
70 0.054241213386980203204459544102 14.7489325928690987821868911174 0.808754514930150489371223059573 152 24 [1]
71 0.053841738003889231018002569093 14.8583613690593048851202283523 0.808269842826995083525019564826 125 7 25 [1]
72 0.053591198129264284035739014174 14.9278244921930174952517440003 0.812043539151841942291938930291 146 3 26 [1]
73 0.053086301783543024245574709691 15.0698009302279811198446973240 0.807881548117050806202814551756 142 3 22 [1]
74 0.052699597411196553954433754160 15.1803816214738679982073881702 0.807060701303894676696998509431 125 8 25 [1]
75 0.052365749298295452253889400723 15.2771613262495730373445973656 0.807636231211595147113665025091 159 1 23 [1]
76 0.052314721021861216967473472095 15.2920628147036643266726035167 0.816810487725218831558131606551 93 14 20 [1]
77 0.052171460843930845058510324566 15.3340540414072907804818884424 0.823031781466578727129665764383 184 1 23 [1]
78 0.051611169782250225818066420626 15.5005205922523139596593665517 0.815909319145490547076620045910 151 5 23 [1]
79 0.051265729976640373405183608807 15.6049665217783144992444690201 0.815344729872387087620963266042 128 9 24 [1]
80 0.051120944138678835638499148385 15.6491632437341584036896591505 0.821008408177741967392822485499 169 3 26 [1]
81 0.050997603160693280761027412856 15.6870117499287685455953772680 0.827264589173586827357007944115 163 25 D1 [1]
82 0.050638400255208459687691205093 15.7982873860181879860109483558 0.825721689460585638869493317004 175 2 27 [1]
83 0.050506127621684673772686360735 15.8396621889602578195450816372 0.831430824798504215735732757839 179 1 27 [1]
84 0.050241723652679180606822256891 15.9230205860451077904815548573 0.832661016956739597592324685225 152 10 28 [1]
85 0.050064198790273211019305555533 15.9794827307898325248549411135 0.836629842846397061656324817729 203 4 23 [1]
86 0.050013999247632022259596864289 15.9955214946718550775665644727 0.844775876123802661150013013828 110 17 26 [1]
87 0.049137061066949403359751080795 16.2809900028411840342610372068 0.824892762263135125871725161638 201 1 27 [1]
88 0.048661947946253821208584506536 16.4399501820926795164204452250 0.818316932265282502553959656943 182 2 28 [1]
89 0.048383147704156215305412037093 16.5346827968218012051687187321 0.818159788699350237430270466173 227 2 25 [1]
90 0.048170334424546069622230987009 16.6077319071371332713867337756 0.820090381009069420325466085042 205 25 [1]
91 0.047791502446871553973610993892 16.7393774842993712781880111315 0.816211380322002374611004014420 225 1 26 [1]
92 0.047658199371930979687178123848 16.7861986089044763560329152349 0.820583863666984630559157460235 187 3 28 [1]
93 0.047494500917392956193445065967 16.8440553021377698909549268107 0.823814612438516913398879815312 221 4 27 [1]
94 0.047326151141032882602005497965 16.9039733997379990611813545409 0.826780284833353677555989907088 198 6 28 [1]
95 0.047201245140337711660831003112 16.9487054339659359363027158572 0.831171037064589416644543552724 228 1 26 [1]
96 0.047131905389808397317205105331 16.9736401145579104070832545418 0.837454292974569113696053538972 200 26 C2 [1]
97 0.046465778713880410716871523930 17.2169717616509321654283607308 0.822428325605572609030375836741 235 6 29 [1]
98 0.046250823276994035771814734184 17.2969894007904918615769970854 0.823237029273932274241911235999 237 5 29 [1]
99 0.046141127793533851421622246592 17.3381111007891489476363276215 0.827697209858903673550724872329 206 3 28 [1]
100 0.046020039172959540684213038866 17.3837313999955063419013081021 0.831675396133430650375609059304 201 28 C2 [2]
101 0.045738842021252803765655554646 17.4906045856665022913818991739 0.829758270891230598452809721802 161 10 29 [1]
102 0.045635212533860954892340102847 17.5303226517550791604345729138 0.834180843759733217691146808683 246 4 30 [1]
103 0.045510032736024620800419872092 17.5785415194118220780525286592 0.837744154603203686520185140624 230 1 29 [1]
104 0.045457958233436040732030093043 17.5986786712204302902272557167 0.843942922723918105863015351859 159 12 26 [1]
105 0.045454681838679038324818529680 17.5999471922219235079380964181 0.851934938354260439489487102601 281 2 28 D1 [2]
106 0.044957742918623742735415143232 17.7944876247023522252718541192 0.841346224320627282707002545249 198 1 29 [1]
107 0.044242210462879179973380658410 18.0822791544565575032394105721 0.822464758641412373821908011600 226 4 29 [1]
108 0.044139805122014646223272946034 18.1242304488789298845104990013 0.826312769112061209946470711688 208 5 28 [1]
109 0.044081917164168436306425366468 18.1480310173594793039419855022 0.831777813336513239750513836180 208 6 28 [1]
110 0.044069520685278056908085645381 18.1531359443001483406596644333 0.838936760803140586104521859737 144 18 27 [1]
111 0.044032267581103002691045547771 18.1684942417848584022223495520 0.845132819644799784942469167155 249 28 D1 [1]
112 0.043333593629283880991341683389 18.4614275668883771550616588620 0.825899724032181309731622005207 260 2 29 [1]
113 0.043100863588385861103708704752 18.5611130124912704671846380873 0.824347401167816393377313107184 220 5 33 [1]
114 0.042826108819098200996837070686 18.6801935095079642093377771477 0.821073374901372601172974144266 219 7 29 [1]
115 0.042629163053549346013953222938 18.7664955794479576046758819228 0.820675254190955069848302187498 220 3 31 [1]
116 0.042608626337325656800702557691 18.7755407477004394718946174170 0.827014151892566566063761399425 302 30 D2 [1]
117 0.042403519198549880050340529148 18.8663586211815759625442916268 0.826132201121303349206691988569 279 1 22 [1]
118 0.042234329521309774149027574885 18.9419367861954954237172575410 0.826557557467714748167304123065 245 7 32 [1]
119 0.042093590218440867786102227112 19.0052688746308541162071881419 0.828016106756024358361198793388 269 3 31 [1]
120 0.042053258151036513656099006521 19.0234962800446387367249707732 0.833374927018419116277423613997 215 5 30 [1]
121 0.041928035226567055074104936007 19.0803121509755910587939464842 0.835322691526341640621816553287 243 31 D1 [2]
122 0.041717178046133191103851754376 19.1767525386140742881707540525 0.833776330864597132819865549244 273 2 33 [1]
123 0.041658103776770641786228328925 19.2039465907254125780468155251 0.838231528963274006827718298884 306 3 32 [1]
124 0.041560535361535079408157027930 19.2490301927249093579567182185 0.841092649173055285183747333573 208 7 32 [1]
125 0.041494338549433757497637167098 19.2797385852272371605595899904 0.845176845735045565990827010283 198 8 31 [1]
126 0.041477091827687327486996327525 19.2877553547757078041633078622 0.851230207831279589634732478302 205 4 28 [1]
127 0.041470837350693090281419460028 19.2906642620909082772069730627 0.857727265168794516045843376322 256 32 D2 [1]
128 0.040772418647263162243643191953 19.6211072715868865238978435252 0.835608415322839899094512807422 281 6 34 [1]
129 0.040604698315454488257823688300 19.7021535238328146797752752311 0.835222474992048629492633334048 319 5 34 [1]
130 0.040404836095477188064314160316 19.7996100790902580294747210045 0.833431549722458818190790651261 290 4 33 [1]
131 0.040283518045120274664476021254 19.8592386867489005261780148239 0.834806773387674394248605602806 288 2 33 [1]
132 0.040252186562917299766975108601 19.8746967136689021623304519191 0.839871357801507248396923455358 361 1 32 [2]
133 0.040054517074150308700204913505 19.9727785637513068475399484951 0.837943095005566035581620770787 223 3 24 [1]
134 0.040004280651692916801353660189 19.9978599031787686195725152326 0.842127044860857568479726131270 190 19 29 [1]
135 0.039623004562204550901171957807 20.1902911916755834065519273356 0.832316421127616042573059325303 228 2 30 [1]
136 0.039361073481863738287944388902 20.3246489293187888330275448457 0.827432666692358555281149630441 272 6 35 [1]
137 0.039330789040860073359687373261 20.3402987712474799335672882755 0.832234606980734634346943782226 242 7 32 [1]
138 0.039318454782297415075332384910 20.3466795536478921530113125180 0.837783601737328401079402566779 382 33 D1 [2]
139 0.039040925852364094780192810642 20.4913173172494469336724511745 0.831983862667527499806002242881 275 6 34 [1]
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292 0.027389932112204019315974201794 29.2078124444692322822642492433 0.860249935647496623594881484260 289 89 50 [1]
293 0.027371720593893290216531365184 29.2272455893212027958574257217 0.862048503641229597852132947590 558 2 47 [1]
294 0.027366585808025824903514268125 29.2327294903328142474678350510 0.864666144065934884097834263343 589 47 D1 [2]
295 0.027200724859412082555107105433 29.4109809255021165181373355442 0.857122422883291553815803061399 525 9 49 [1]
296 0.027191946143109374734621530333 29.4204760405766497711551030652 0.859472884264890213053518378638 557 5 50 [1]
297 0.027190762998291987240635814300 29.4217562063356859461179434762 0.862301465028607262910757397051 810 1 50 [1]
298 0.027140435876967612003213228072 29.4763136313116434607944009730 0.862005002381589748608039132008 682 2 50 C2 [2]
299 0.027047899137704676016700656621 29.5771585041443319014242899343 0.859009863406220761575814938671 526 17 54 [1]
300 0.027028620625486151932012062137 29.5982547938704049662458356643 0.860654622223987390093759079526 588 13 56 [1]
301 0.026983773571805253256427716610 29.6474471174744166646374208089 0.860660257509116779427062915791 546 14 54 [1]
302 0.026968004125984256791687854571 29.6647833581864017719834204370 0.862510598901066865821907135089 432 28 48 [1]
303 0.026959408141133162739597324451 29.6742419496741316331002142901 0.864815015244546006596615752975 356 72 49 [1]
304 0.026954790215774993574444371425 29.6793257746004948171266410948 0.867371966447916315059858691739 543 4 51 [1]
305 0.026951916124068547171458291137 29.6824907111366962200002496145 0.870039595681603504420497759506 446 32 52 [1]
306 0.026949833325738645382745208191 29.6847847009114469483601588777 0.872757278480014093053245993558 613 51 D1 [2]
307 0.026747113627037078007753495598 29.9097693738186101787876585517 0.862486103785823329983889019871 653 18 55 [1]
308 0.026652247962902213425979820160 30.0162298172197590055427734408 0.859168394192954975392365896455 659 15 54 [1]
309 0.026621401881189168310026711740 30.0510094686367542961108897902 0.859963875943559391636954211297 584 14 56 [1]
310 0.026572922821775014129012528244 30.1058338732856681480104479581 0.859607570962048463545106196758 726 8 55 [1]
311 0.026521088530336774560299360958 30.1646743905289209124915291957 0.859019386649909194393868576727 715 6 49 [1]
312 0.026465894313054987195697588039 30.2275823570178506823492316693 0.858198255709506076038878671194 745 6 49 [1]
313 0.026427389291553521847933763638 30.2716243051555839199392239174 0.858445537982564546160666680769 578 2 44 [1]
314 0.026384240968562470451709516951 30.3211299863892775950867613181 0.858378326179138556159431228459 547 13 47 [1]
315 0.026426835591463938900052989532 30.2722585619901403828516594352 0.863894611637717557638312495680 895 51 D1 [2]
316 0.026315793045451815698185190533 30.3999958738946225300583332549 0.859369414836272873222788975363 77 256 54 [1]
317 0.026254181683535617755558095863 30.4713363243651097586215969269 0.858056964280448455159716519072 601 5 50 [1]
318 0.026229937117265293552936490668 30.4995012539857427882197072613 0.859174748689486453365361361048 650 21 53 [1]
319 0.026218346598160801502864035605 30.5129843716430018893509851588 0.861115030072689940993214034923 791 49 D1 [1]
320 0.026103966691100260184367465212 30.6466832978586165436317740639 0.856293953266188866965982044893 371 83 55 [1]
321 0.026088972270526150346516738842 30.6642972250691103743251291461 0.857983350815214628242130655818 613 13 56 [1]
322 0.026069551735647441837230702474 30.6871406195328606135671525911 0.859375334451089732768145057142 499 32 51 [1]
323 0.026022867708217859798512041474 30.7421921738227556994654540158 0.858959556505864113781205388954 726 11 55 [1]
324 0.025978266880277257748995024339 30.7949719543208363326961132302 0.858667932273594260681721337791 736 15 50 [1]
325 0.025933154226097639036384863056 30.8485421027159852157358212300 0.858329289021169308651299760121 738 6 52 [1]
326 0.025906580145529061700752565928 30.8801854782080684602589235700 0.859206708672722336569612442528 583 36 54 [1]
327 0.025878156767388208170948306995 30.9141028548124527510025892893 0.859952210666291254201338206151 723 9 54 [1]
328 0.025857628123087029887205921968 30.9386458878538266902151828100 0.861214037266753370431698105000 669 10 52 [1]
329 0.025840307777657113124620329688 30.9593835678583520892946718530 0.862682817215614685589204023607 691 10 44 [1]
330 0.025830778061764582460971449308 30.9708053736167426942599646041 0.864666834892113303786338287527 792 1 37 [1]
331 0.025722538045509997956351508645 31.1011300123101241681339406786 0.860033792639352039908546227222 419 23 48 [1]
332 0.025721790771915031648677968397 31.1020335673323191945206105089 0.862581961575794077254568635847 343 85 52 [1]
333 0.025658863198515251275443763278 31.1783103487722697131865975878 0.860952005063440387011394344227 793 3 53 [1]
334 0.025598009079072475911792747347 31.2524305124196550255810973535 0.859446267940190213446309600319 772 8 57 [1]
335 0.025574584424040945425602623053 31.2810557049745780792226904024 0.860442519794964029259400881216 599 11 55 [1]
336 0.025555377487651609062482286891 31.3045659523738601986692989797 0.861715220578793021784569700379 592 11 55 [1]
337 0.025533047237699746360724811738 31.3319437571397145017355219395 0.862770096276177093239689491142 539 11 45 [1]
338 0.025530236313569610349391048700 31.3353934595110238211770344556 0.865139727616815805424132120884 349 85 42 [1]
339 0.025508358818866925928984373788 31.3622685677563231534608415028 0.866212843519368914598441976965 681 11 57 [1]
340 0.025497754141553846966030270355 31.3753123337335452918433391617 0.868045841976425365667037868911 538 10 50 [1]
341 0.025496900829296905380350446702 31.3763623805121318963812563903 0.870540647728899065169862219002 394 97 56 [1]
342 0.025490940304954718900522402783 31.3836990879658759767503664820 0.872685386613347581039746080486 685 54 C2 [2]
343 0.025287740233667987504655463283 31.6358833414020709910226437747 0.861338875343599931849002250366 780 9 55 [1]
344 0.025266577458661793654518353487 31.6623809183838228898785816914 0.862404796706682356636143059115 521 22 40 [1]
345 0.025230323236626272671445333805 31.7078775605481990272149490511 0.862431498352428499817334917064 643 12 55 [1]
346 0.025194650445464694683103622292 31.7527723487034334806290184062 0.862487201006669800448818590385 772 12 59 [1]
347 0.025137691040002553542978632374 31.8247208435703144121204445216 0.861073311625188433303941721026 714 14 53 [1]
348 0.025104063517577058209810477461 31.8673508549667952771787305685 0.861245924538571891509104112267 688 24 58 [1]
349 0.025080377659040966655109306147 31.8974463174248196163434992544 0.862091684931860648078496587133 643 8 53 [1]
350 0.025061625675758979711446197227 31.9213131003630462210270299951 0.863269521584317442834328478450 587 14 46 [1]
351 0.025045751604207096606690328455 31.9415449231565019091438083515 0.864639636275561303355978500033 752 8 55 [1]
352 0.025004953761503547774941578073 31.9936604414619002691170572833 0.864280393074950591332199709242 409 121 53 [1]
353 0.024957060312864803794213430170 32.0550573653747984245643359153 0.863418695635560951636845544079 729 20 54 [1]
354 0.024955679678082896527640650062 32.0568307623611981530033278596 0.865768843626812499618274917543 87 281 69 [1]
355 0.024955669565855307185669594619 32.0568437520334490447881654578 0.868213814712465409125550029708 254 179 56 [1]
356 0.024953443071286955920099595130 32.0597040542485976894387655197 0.870504137389032301178568921771 477 51 50 [1]
357 0.024951037439775476092505275361 32.0627950613663487025083207150 0.872781068835926244529251917620 716 52 D2 [1]
358 0.024750070461744888240143519367 32.3231403012175397501432592113 0.861183681268607228428137066684 593 48 50 [1]
359 0.024634713192037511999714728166 32.4745002616299108628749567069 0.855557800047294742891131451664 507 35 46 [1]
360 0.024630259554278278930423208243 32.4803722931551449105980939218 0.857630788169670171913310070859 492 42 49 [1]
361 0.024589269734074286963535091623 32.5345164232921301917681040959 0.857153000267826716101262940175 748 6 52 [1]
362 0.024575342320276234007986663167 32.5529544847865120973362701891 0.858553984839304424285314381666 965 8 57 [1]
363 0.024543792715300309390783781552 32.5947993971318656343340845895 0.858716602438801873484064308178 866 12 58 [1]
364 0.024499454933968850982864007027 32.6537876926718215204678769944 0.857973973608620641533276524765 980 7 58 [1]
365 0.024481336007844846719187057914 32.6779551468778685434599369792 0.859058975000605302557290760690 968 5 58 [1]
366 0.024431377186122072967406223658 32.7447770915848995682307868992 0.857900396223194591021238154472 903 5 58 [1]
367 0.024418791989480464928018431385 32.7616534161328440050234352497 0.859358349030394569923478810507 727 10 53 [1]
368 0.024405985919593688223374153726 32.7788437900286402466328893611 0.860796350658113489003787663295 868 13 56 [1]
369 0.024404393256922105273734665216 32.7809829803118168010088621969 0.863022823500665457717935636824 471 54 56 [1]
370 0.024386646356920839103945811718 32.8048386929169942867287881910 0.864103512605917102480777661724 1019 1 56 [1]
371 0.024337406630271112784861993231 32.8712098274658360656191582539 0.862943560072373867439133797142 859 3 56 [1]
372 0.024275460384233554861540316749 32.9550907516293544887216353229 0.860870400628082191868331906072 639 28 63 [1]
373 0.024260927219296362496240764208 32.9748320321288333883377117196 0.862151339888222499337047711721 686 20 65 [1]
374 0.024232692030376433501532050403 33.0132532942347101443968926720 0.862451762061371885266298254989 686 8 57 [1]
375 0.024210217975473170331077951313 33.0438991012167707392221792981 0.863154527034387255269982127489 657 33 62 [1]
376 0.024207112010290863386589693971 33.0481388965319827261818050968 0.865234225317146003949745425313 489 64 54 [1]
377 0.024195623640882113905956083791 33.0638305453007920605849962076 0.866712134348823615312441951208 570 37 53 [1]
378 0.024192744196866064129572559685 33.0677658346684067987691245824 0.868804281353698771813158540457 675 12 56 [1]
379 0.024192204097858655878176323468 33.0685040835452875018031922516 0.871063811365347848920964844242 362 142 65 [1]
380 0.024182731204195553928763984866 33.0814577247256902096231154675 0.872678305299123532376489464437 924 57 D1 [1]
381 0.024071515792360218468854669987 33.2343009431048292582246979957 0.866945384909340667007793548520 864 11 56 [1]
382 0.024013317381431524695125874979 33.3148472280054865738323217336 0.865022831585309074952567821764 662 6 54 C2 [1]
383 0.023995843461698667148159633207 33.3391073031849139123826773931 0.866025539847483890050395275732 691 8 52 [1]
384 0.023988598996017552480422318754 33.3491755868198614515041632748 0.867762502043514362917009674383 769 54 C2 [2]
385 0.023933289720763227947965393595 33.4262447550602816087669180158 0.866014994202041812016257761106 889 10 57 [1]
386 0.023897543756463322319921333035 33.4762437576302060974020182770 0.865672699147885114387878804242 866 6 57 [1]
387 0.023871291302878894427772175886 33.5130592580686843884456025847 0.866009539087011415215373673965 674 9 50 [1]
388 0.023848187223381524775676854412 33.5455266476461757049161711382 0.866567418687838131824112209362 753 5 48 [1]
389 0.023840209433828469161245150244 33.5567521846023065372754254529 0.868219667734134525540524441154 807 12 58 [1]
390 0.023848376539816648640120148446 33.5452603519715547523411273818 0.871048090342808505251465111363 1114 57 D1 [2]
391 0.023749767409904687477827387630 33.6845404080195393587020235862 0.866074725111844395876116408711 500 52 43 [1]
392 0.023613038690450029927681408090 33.8795870572790380621163817259 0.858320944447910577072701752779 841 7 52 [1]
393 0.023602738785749041946853624368 33.8943716346607737066219212255 0.859760000517192757800017603094 592 37 54 [1]
394 0.023596075694255367167226912626 33.9039427727707155479637013643 0.861461095226298512836619443252 630 59 63 [1]
395 0.023586305314978049103719342292 33.9179871250108308969555877336 0.862932475201260113897352049547 662 16 51 [1]
396 0.023567897657812978479593110717 33.9444786979033965056392709933 0.863767300164139840430216621177 553 73 55 [1]
397 0.023543028744323624697409754030 33.9803348451029323696718093945 0.864121992145718249815881262679 626 35 51 [1]
398 0.023442510423282873529346494946 34.1260379351457075475296232491 0.858916989708694876555778860299 734 5 60 [1]
399 0.023438599473962574068301176688 34.1317321834311153551076557118 0.860787787661073501977080707044 244 216 56 [1]
400 0.023375155615757946303843449761 34.2243710865691177745585376985 0.858279814384107138475669072682 992 9 58 [1]
401 0.023320648325443582458694133107 34.3043636195643024020300728643 0.856417430905694574627422128225 1043 6 60 [1]
402 0.023311905356434582621461306461 34.3172292340824450522088791261 0.857909508485621888876235154733 972 9 60 [1]
403 0.023299246322880320677237307726 34.3358745992733755698877747576 0.859109808629327802379659485813 725 32 64 [1]
404 0.023277504668425958538701464377 34.3679449921938978331795807737 0.859635012260493261193296156315 665 65 66 [1]
405 0.023245345069492052087467509870 34.4154925473636463250063869830 0.859383288092978320717640033462 723 58 67 [1]
406 0.023232681281922364745681321272 34.4342519183306597691289695212 0.860566802226601751720228940303 831 8 63 [1]
407 0.023219373669097719015319357923 34.4539870627391982263207372218 0.861698419345016833915658635031 635 27 56 [1]
408 0.023200767068844031107365551688 34.4816185441690949403918396580 0.862431750104988864448439575522 710 36 57 [1]
409 0.023166295061026001515430360106 34.5329280272306591053068365746 0.861978355491442697711492992263 518 102 64 [1]
410 0.023131426654880218485992371445 34.5849831026837130309275213204 0.861486707407336368156424877299 850 12 62 [1]
411 0.023104154286463190140769621116 34.6258075530911327521950356198 0.861552723611534043037032058270 775 11 59 [1]
412 0.023084996066391255201351904084 34.6545434835354253341977169737 0.862217258263106129048473505443 779 10 61 [1]
413 0.023077306638227564048646632416 34.6660904819282422579552119269 0.863734324939753497443165551225 516 91 51 [1]
414 0.023058206565791930013990494366 34.6948058478599262179654605275 0.864393072705368090252876610023 788 13 61 [1]
415 0.023039084103939653067099464458 34.7236025699129703092103179688 0.865044407375861084406734656534 734 11 60 [1]
416 0.023024148941025266059732083639 34.7461268622411880306833556411 0.866004977800985111245801718323 760 14 61 [1]
417 0.023017666903705030055789713905 34.7559117675487910673021481869 0.867598000762104162199230578495 602 38 55 [1]
418 0.023015962110374617980404777391 34.7584861394690072384944794687 0.869549751559630753618263060752 763 5 59 [1]
419 0.023015916469319962433876816207 34.7585550662904853333975816664 0.871626557211529921275623516223 603 87 67 [1]
420 0.023002082502461642363309917591 34.7794596386820805803297484307 0.872656825213688597431147544415 1027 1 60 [1]





Updates

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

29-Aug-2010: First complete presentation from N=1 to N=420.
30-Aug-2010: First nice improvements for N=26, 27, 28, 32, 33, 38, 40, 41, 100, 105 and 121 by David W. Cantrell [2].
10-Sep-2010: More improvements for N=42, 43, 44, 45, 46, 50, 54, 66, 68, 69, 132, 138, 144, 150, 156, 182, 189, 216, 248, 260, 264, 294, 298, 306, 315, 342, 384 and 390 by David W. Cantrell [2].
25-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–September 2010.
[3]   , private communication, June 2013.


©  E. Specht     25-Jun-2013