The best known packings of equal circles in a circular quadrant (complete up to N = 600)

Last update: 04-Jul-2011

Overview    Download    Results    History of updates    References

Overview

1-12   13-24   25-36   37-48   49-60   61-72   73-84   85-96   97-108   109-120   121-132   133-144   145-156   157-168   169-180   181-192   193-204   205-216   217-228   229-240   241-252   253-264   265-276   277-288   289-300   301-312   313-324   325-336   337-348   349-360   361-372   373-384   385-396   397-408   409-420   421-432   433-444   445-456   457-468   469-480   481-492   493-504   505-516   517-528   529-540   541-552   553-564   565-576   577-588   589-600  

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:
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 circular quadrant
ratio
= 1/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 0.414213562373095048801688724211 2.4142135623730950488016887242 0.686291501015239609586490206326 3 1 D1 [-]
2 0.276768653914155215717770973809 3.6131259297527530557132863468 0.612807102315627401679548476729 5 2 D1 [-]
3 0.240253073352042147999877060493 4.1622776601683793319988935444 0.692658471061220938666994505355 8 3 D1 [-]
4 0.205604646759568224693193969093 4.8637033051562731469989727989 0.676372332306029255659665555408 7 1 4 D1 [1]
5 0.180960531653868302007734251938 5.5260668769074292471840516069 0.654934280329013433966393380911 9 1 5 D1 [1]
6 0.167632184592678242543441991309 5.9654415554498328490451219672 0.674413183471530066358686982685 11 1 4 D1 [1]
7 0.163243161142003991913329124676 6.1258308954830123575921725923 0.746153230469759897379439667492 9 3 7 D1 [1]
8 0.150154807084741322297665096450 6.6597934452783805797025686526 0.721486914900988249479759677467 16 1 7 D1 [1]
9 0.139429107850471597338690683436 7.1721035543914774826620608538 0.699857140175223854936269498606 17 1 7 [3]
10 0.135072985268351994987422353792 7.4034048926458649548526577564 0.729788453972177402373005711137 19 1 7 [2]
11 0.127840596125867863162619218933 7.8222413717113083030406255990 0.719101592783959497365179991504 21 1 8 [1]
12 0.125441937332820688361738236682 7.9718156564085488072559389623 0.755312622806943012206842214130 17 4 10 D1 [1]
13 0.118280294043166541218397318535 8.4544936930495687601038227293 0.727491853864772795469510131294 25 1 8 D1 [1]
14 0.115456141678356872441258327524 8.6612975755403886690909628653 0.746486756470157633613867393908 13 8 11 D1 [1]
15 0.110212024387959910148530104616 9.0734201241043966664888887115 0.728801419181536200430090727506 27 2 10 [1]
16 0.107771974554386810881887575168 9.2788501290319491439860485932 0.743347103958489487613193326311 33 10 [2]
17 0.104571563125836898331644050466 9.5628292253472698979081068634 0.743594403391500608336021956248 33 1 10 [3]
18 0.102431610396020738261052586657 9.7626113280246542499913981828 0.755440906199197232431357820980 31 3 9 [3]
19 0.100690695170108760185749609484 9.9314042703805066859664918851 0.770534823131822034080312767981 15 12 14 D1 [1]
20 0.097064861746472257626375696150 10.3023893714693739368078703213 0.753726990868941872875322013364 37 2 12 [1]
21 0.095341841141922509471481302590 10.4885744603089339528745666809 0.763565600475853365526889907199 25 9 14 [1]
22 0.092357812035589777023034182141 10.8274544184162062167681167064 0.750636959072117204744558312931 41 2 12 [3]
23 0.090855472209869065449883091814 11.0064916914424028839460532792 0.759433948404002683987378066114 46 1 12 [1]
24 0.089194524043168924059610389804 11.2114505988732418203069013267 0.763743659451594169732007874488 45 2 12 [1]
25 0.087830975905185831831546124292 11.3855048255356648799061692615 0.771428032845733415126285385673 47 2 14 [1]
26 0.086028881020132430920462047286 11.6240033363444603657209581415 0.769700710435914608852759922229 45 4 15 [1]
27 0.084931362539244210178247227126 11.7742135543619742187587223752 0.779040325019433751152111117631 49 3 14 [1]
28 0.082763227492774399007205548930 12.0826607455261998509962972622 0.767172204402321550836691887403 51 3 15 [1]
29 0.081560146728344274824249725274 12.2608901542409064828013378670 0.771638673984489488696048598282 58 1 14 [4]
30 0.080566190424326355573787305069 12.4121544624760800047588701634 0.778909324738657890046016413694 57 2 16 [1]
31 0.078902983099998773519415823186 12.6737920508358516569176234937 0.771984412017757815941752166433 61 2 15 [3]
32 0.077900800624123658619302848501 12.8368385432270960413220790852 0.776772446448571520034754038959 62 3 16 [1]
33 0.076699896995223654046948938047 13.0378271572160412807196868912 0.776539394278285243947270191546 61 3 15 [1]
34 0.075703248146737660204245987796 13.2094728361149426970022209607 0.779413522075449301864183954871 65 2 15 [4]
35 0.074555758644084592766278171429 13.4127801552367686456988144673 0.778198560579299226425426824167 63 4 17 [2]
36 0.073839675073728961639686362588 13.5428548270492709504022466736 0.785130856559117299664607746209 73 1 17 [3]
37 0.072791736622433279819024691717 13.7378230881747581573526812911 0.784198264235434764070001446823 67 4 16 [4]
38 0.071770791677304644383725433205 13.9332446616472246952327121554 0.782959073774033364648399646019 69 5 17 [2]
39 0.070892474060411576015783723099 14.1058696745135764607310032881 0.784015889031355994152698136593 73 4 19 [3]
40 0.070088729311365557825494148499 14.2676291869631660656011076005 0.785988796237099760268526068865 76 3 19 [1]
41 0.069105932745766031249481120868 14.4705376263265588309865376557 0.783203310268623406258561330627 84 1 19 [1]
42 0.068490562566375397099520563816 14.6005517041984081056678277039 0.788080802990641913998519198214 81 3 19 [3]
43 0.067695356551689090397906481699 14.7720619395282386177292843231 0.788217743369574189626113184894 85 3 17 [2]
44 0.067092515149803725825711474525 14.9047922524175250835306516529 0.792247383686289074099780444298 89 3 20 [4]
45 0.066425531708864791464822896781 15.0544523208768748110496107752 0.794223227304972335339919010405 87 4 19 [2]
46 0.065644625979000921535821451997 15.2335394571353074712855276881 0.792895913265817776098895771967 89 2 18 [2]
47 0.064812778834776259705369595528 15.4290560901461478210893059540 0.789730904453696841390942749409 91 2 20 [4]
48 0.064258263442779429307939858065 15.5622008193619799904542773593 0.792791888770875501114746456382 90 5 19 [2]
49 0.063682934382299404892909647152 15.7027940012442140019551466905 0.794881161781889330499456970367 94 3 22 [4]
50 0.063100032760197409718776519382 15.8478523109545130437755204866 0.796322826867597267408784098160 98 3 23 [4]
51 0.062382503172326925548522372140 16.0301358417371610476780683326 0.793881647217257301662781329314 102 2 22 [2]
52 0.061878492264348670578387579783 16.1607040412028685362475543890 0.796421143421084119134768435457 100 6 22 [4]
53 0.061441520765396898744355981343 16.2756388113880732228026354432 0.800312820480516041743544388666 106 3 20 [2]
54 0.060866975785767450648172816380 16.4292703406143336094135977438 0.800234368121923451912726723248 107 3 22 [1]
55 0.060346853664467115054859642554 16.5708722042092294775724312704 0.801183404384133775975509122987 109 3 21 [2]
56 0.059814591562794433327835044209 16.7183286531377887289877232178 0.801423921496557858217604111399 106 6 19 [4]
57 0.059296796748175769239897558746 16.8643173803610969341844748934 0.801673103847538813734015776962 107 5 23 [4]
58 0.058761842243376827266422977046 17.0178463067623128450019381735 0.801085352089837252756838213066 113 3 23 [4]
59 0.058229778872367047562758347311 17.1733435943812279653348482399 0.800206886815844254254500445004 122 1 24 [2]
60 0.057804691718488802552161420565 17.2996338233242490594559228521 0.801931772320686708768375357260 115 5 23 [3]
61 0.057337512219285693771525584590 17.4405892633696467959364309383 0.802172035029203643582985231063 119 5 23 [2]
62 0.056905282892276613010398287620 17.5730608684088195558122363077 0.803076382820407282986418833778 120 5 21 [2]
63 0.056439910614597870967413316364 17.7179586060746794744666290176 0.802736804566316951078676771641 137 6 23 [2]
64 0.055980958572200128520988752432 17.8632168063051912693934839458 0.802270137001571085329698308153 119 8 24 [4]
65 0.055601707670656490568141957029 17.9850591266596071133919643698 0.803802972932216613776241535867 123 7 24 [4]
66 0.055310994377172886502427586954 18.0795881770063568285059150041 0.807656810133795776050061301162 132 3 23 [2]
67 0.055118371461092659214755488175 18.1427711576327500934733255655 0.814193345836162318181373478224 143 4 24 [2]
68 0.054554936898038103346297664058 18.3301467632338486601138602369 0.809537590066106053329050602206 131 6 24 [2]
69 0.054233098590001789332564367072 18.4389243100403680370327176085 0.811779199217707723374854237446 128 6 24 [4]
70 0.053838632742106852500240476249 18.5740229472414955185966387841 0.811607545151048806608009618658 129 8 24 [4]
71 0.053412118127950514142737189158 18.7223430758627953582862921732 0.810210639067615741675305906983 138 8 23 [4]
72 0.052970965638075540919473044087 18.8782663852591691430173218247 0.808105881781491753007683300460 144 6 25 [4]
73 0.052649642462630304848924598240 18.9934813082497304012451979780 0.809419576621298788825523900162 143 7 24 [4]
74 0.052361336091511592848657362660 19.0980611772836736126378929953 0.811546017117317421016711227769 144 7 26 [3]
75 0.052039382895073807651284596791 19.2162155730455976280841524041 0.812429211630030117082969496881 146 5 27 [3]
76 0.051659307512809368731043404430 19.3575959134187274229040055215 0.811279952021712755889856761284 141 7 28 [2]
77 0.051316388023587913838037419370 19.4869521904063756546298592055 0.811078477374530619963319201691 147 6 28 [3]
78 0.050969656550876277737480334185 19.6195161527492744067785953168 0.810546637341256936581032599330 153 6 29 [2]
79 0.050729212034025278045646845412 19.7125080383522699508522928740 0.813211133335418047754548139307 152 5 29 [3]
80 0.050469995687303621559202946822 19.8137524361144912731025677245 0.815110548696462771051519381953 158 6 28 [3]
81 0.050128110030006909440350418768 19.9488869498849159854424003461 0.814156082518475298148213236384 167 5 27 [4]
82 0.049949115342851746776562775774 20.0203745979479235612568655147 0.818331832518990549522400232870 166 4 26 [2]
83 0.049667229534383780971846297946 20.1340000111686711562767054356 0.818988784954226624755170089667 172 4 26 [3]
84 0.049370544769394135430585458801 20.2549922159238872449127242969 0.818983432117788210068104617487 165 5 28 [4]
85 0.049029773123640570574387596041 20.3957704939456949724057490773 0.817332341868926859217415179828 161 6 31 [2]
86 0.048718942595282814644415806633 20.5258970480370085241798536286 0.816496166455246995037935000763 170 5 29 [2]
87 0.048394069366077102053419807327 20.6636890242789174073979187298 0.815011110533421407757225667294 164 10 29 [2]
88 0.048135265567885533375817493726 20.7747892984965949279189767383 0.815585334534385043170649649276 176 7 24 [2]
89 0.047860179787749405180935826811 20.8941964788850696113084456881 0.815452464116388024756907262991 184 6 29 [2]
90 0.047690815763978770675769818572 20.9683978766265405020743178542 0.818789006964155765801287983644 179 5 27 [2]
91 0.047425323601201752506735799656 21.0857812675981426409324735713 0.818694719999048242546398808393 187 6 29 [2]
92 0.047143089137604011316317166039 21.2120168256505507452273966356 0.817869274064476455018250347123 188 7 28 [2]
93 0.046875798407782704872814368992 21.3329699752691954281864520515 0.817410657208555087783341761293 190 5 30 [2]
94 0.046622128741765958668086661336 21.4490420533749716179023006056 0.817282206043588609123310148589 185 8 32 [3]
95 0.046445632686155081367553277443 21.5305496376215520441112573377 0.819734782334550166044193921916 181 9 29 [3]
96 0.046162935180602263933367986218 21.6624007136400965164973213000 0.818310368443578690906239652703 194 6 32 [3]
97 0.045988795089836153717255600968 21.7444270511233075130939097693 0.820608078240195913679059489031 203 5 28 [2]
98 0.045753164215951218411157236927 21.8564118380989177812161362255 0.820593998022545235661841312316 200 5 29 [2]
99 0.045543244595615184739876255041 21.9571532261071724462668179588 0.821378102805228565944852087925 205 6 30 [4]
100 0.045421461297472835606397648395 22.0160243954026664690495626841 0.825243658559129076240356879119 215 7 30 [2]
101 0.045119237135243855605965803316 22.1634952958651194053329283991 0.822441206105212739084038346091 209 7 31 [3]
102 0.044947642441240489331457360867 22.2481079248436207742093875360 0.824278548898446144540947597906 207 10 30 [4]
103 0.044706931172835039134122944581 22.3678962918309863196243102281 0.823468394295755059523307914649 211 7 32 [4]
104 0.044442427278154635366223205852 22.5010212367843151886060161095 0.821653806427610304229114741468 202 9 32 [3]
105 0.044187434847504235562011350418 22.6308678802268449682656727750 0.820062347329021299508777045362 195 7 29 [3]
106 0.044021539019473353049777899203 22.7161526442235547382018822891 0.821667860600638335349634255074 192 7 28 [4]
107 0.043802414903252930474652138218 22.8297915128358887687249726431 0.821182863980673776294110165302 218 9 32 [2]
108 0.043650810083632774908072229540 22.9090822847055960229079782092 0.823129871453586752355863030021 229 7 31 [4]
109 0.043437766544493837733801993550 23.0214414678914883648406678430 0.822662049195041380459198189457 227 7 32 [3]
110 0.043253319427263113291638813855 23.1196128584223151063407852741 0.823173842249816928123684321706 251 4 32 [3]
111 0.043146290244897913890471382024 23.1769636352050189117760693217 0.826551448682255961473510932328 220 7 30 [4]
112 0.042900591891557121480073940343 23.3097017059291664420805622650 0.824526431521379613995151407024 215 8 33 [3]
113 0.042796186715069609522420065930 23.3665678360047659767842854359 0.827844146002697442028944722836 237 8 30 [2]
114 0.042568121129894523227427201137 23.4917579976938445348681985023 0.826292491057393896861877381437 220 8 33 [3]
115 0.042379410845075479510806140895 23.5963638960356330077793356254 0.826166653244822537474429037859 223 7 33 [3]
116 0.042207300576844288904254584632 23.6925836604821541786518123486 0.826595687000613232699300133418 217 7 31 [3]
117 0.041999098154912066645377558893 23.8100350705517118563335871049 0.825516547046539059273698016613 222 9 34 [3]
118 0.041827105447510659508904074709 23.9079417354115525443494958023 0.825767186055307019165334190939 205 9 29 [4]
119 0.041664363838892453146555708345 24.0013265021108886717590166302 0.826297545911413919917877771473 229 13 31 [4]
120 0.041528702890877599133163133735 24.0797311350570730245380074365 0.827823918623417020519716880206 240 6 34 [4]
121 0.041343986760658147036947762851 24.1873142469069712617301971770 0.827313416772490452904096737161 248 9 33 [2]
122 0.041138228676572956727695854654 24.3082901760782754807989949329 0.825868683019252666620603218126 251 7 30 [3]
123 0.041010465996361532835978001803 24.3840194375923564011819309843 0.827474294049452961827028583082 235 8 34 [3]
124 0.040839089927107800350836839437 24.4863438873114818991308205469 0.827244307972901309679401696773 277 4 33 [3]
125 0.040665412126370336744932079773 24.5909225484408429621956694106 0.826837871703773816296132940500 294 7 31 [3]
126 0.040539377876827879351713862469 24.6673741032319928322457263502 0.828294343954681816359714541925 295 9 32 [3]
127 0.040374700053353078211844978714 24.7679858594256233059860320294 0.828099133434300360755901786453 276 4 32 [3]
128 0.040224180670223725750051286983 24.8606679698079734289259969439 0.828408171822489609667194118123 275 4 34 [2]
129 0.040096986299053851013745128894 24.9395301816884056246161633945 0.829608448097520317966773612517 268 3 33 [2]
130 0.039955883725213528940846256897 25.0276031154071510256615311949 0.830165775016647342358694170608 279 6 34 [2]
131 0.039893166794404111069602866521 25.0669495644119149143686402331 0.833927532608340912218060976772 289 6 32 [2]
132 0.039706823934972434809033287886 25.1845879599358648118611753874 0.832461625777531111220021316978 210 14 32 [3]
133 0.039571148531012335795435869060 25.2709369609600595014677173097 0.833045923505749942290673099027 264 10 33 [3]
134 0.039418708598368303843710418315 25.3686646660310417329542392778 0.832855338933808333670694612497 272 9 32 [4]
135 0.039228586335224107904497680509 25.4916144939457229537073279715 0.830996272364471066583865771395 247 10 34 [3]
136 0.039040377175033596131444600850 25.6145066303176527051152714216 0.829138171183073000399821901151 232 16 36 [3]
137 0.038869939420518748988257412818 25.7268216752639908612668974017 0.827957960424028986102883579711 271 6 33 [4]
138 0.038753806920383718788845877120 25.8039165559763425789308412299 0.829025368053954319788259622712 261 11 36 [3]
139 0.038630068470524239119782337252 25.8865707360323828839219029588 0.829708897660789355458683575743 300 4 34 [2]
140 0.038508217172066278583491662471 25.9684834416431094106342649973 0.830414362271771321944425869175 283 10 38 [3]
141 0.038360609675454577588528214836 26.0684073704871294186685216911 0.829946515315334754813538454147 304 7 37 [3]
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595 0.019065810984877064980631034797 52.4499063162430659253117394558 0.865142253456320321324160187367 1447 11 74 [3]
596 0.019052974788866087634248111647 52.4852423876803788791910718594 0.865429782359517500206512981636 1095 25 63 [3]
597 0.019046101391334905818654541463 52.5041833734516237451928420982 0.866256499963114531348756344875 1492 17 69 [3]
598 0.019027181397767250138410239739 52.5563917794647859126027575856 0.865984447608947887459113571316 1350 26 71 [3]
599 0.019018384780189938935674833170 52.5807008091258566728440013965 0.866630707315075307376544200586 737 113 55 [3]
600 0.018995948824214276328477016848 52.6428034342402844830581387227 0.866030572157202593558241518794 1382 10 71 [3]



Updates

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

25-Oct-2010: First complete presentation from N=1 to N=250. Packing of circles in this container was initiated by David Alan Paterson, CSIRO Materials Science and Engineering, Highett & Clayton, Australia in 2009. He had adopted the code for this case, great work. Many thanks to him!
26-Oct-2010: Early small improvements for N=16, 45, 46, 50, 52 and 82 by David W. Cantrell [2].
27-Oct-2010: More improvements for N=10, 35, 38, 43, 48, 53, 55, 61, 62 and 100 by David W. Cantrell [2].
27-Oct-2010: Further improvements for N=9, 17, 18, 22, 36, 39, 42, 47, 50, 51, 55, 57, 60, 64–69 and
09-Nov-2010: Extension to N=370 by E. Specht [3].
09-Nov-2010: Some improvements for N=55, 65, 67, 82, 97, 98 and 100 by David W. Cantrell [2].
16-Nov-2010: Again, a few improvements for N=51, 63, 71, 86, 90, 91 and 92 by David W. Cantrell [2].
22-Nov-2010: Many improvements for N=31, 66, 81, 85, 93, 96, 113, 121, 128, 148, 149, 158, 167, 168, 172, 184, 185, 186, 187, 188, 197, 200 and many more by E. Specht [3].
22-Nov-2010: Extension to N=499 by E. Specht [3].
16-Nov-2010: Some more improvements for N=70, 76, 130, 172, 191 and 284 by David W. Cantrell [2].
26-Nov-2010: Much more improvements for N=43, 56, 68, 73, 78, 98, 105, 110, 117, 131, 132, 134, 135, 136, 138, 146, 187 and many more by E. Specht [3].
27-Nov-2010: Probably the last refinements for N=43, 68, 73, 81, 85 and 93 by David W. Cantrell [2].
02-Dec-2010: Extension to N=600 by E. Specht [3].
16-Apr-2011: A bundle of better packings for N=29, 34, 37, 44, 47, 49, 50, 52, 55, 56, 57, 58, 62, 64, 65, 66, 69, 70, 71, 72, 73, 78, 81, 84, 87, 89, 91, 92, 97, 99, 100, 102, 103, 106, 107, 108, 111, 118, 119, 120, 121, 129, 134, 137, 139 and 146 by Claudia Lopez Soto [4].
17-Apr-2011: Some improvements for N=59, 88, 150, 153, 176 and 215 by David W. Cantrell [2].
26-Apr-2011: Even more improvements for N=55, 62, 66, 78, 87, 89, 91, 92, 97, 98, 100, 107, 113, 121, 128, 129 and 139 by David W. Cantrell [2].
04-Jul-2011: An correction for N=128 and a slight improvement for N=131 by David W. Cantrell [2].

References

[1]   , April 2009.
[2]   , October 2010–April 2011.
[3]   , October–December 2010.
[4]   , private communication, April 2011.

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©  E. Specht    21-Jun-2018