The best known packings of equal circles in a semicircle (complete up to N = 250)

Last update: 16-Apr-2011

Overview    Download    Results    History of updates    References

Overview

1-18   19-36   37-54   55-72   73-90   91-108   109-126   127-144   145-162   163-180   181-198   199-216   217-234   235-250  

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 isosceles right triangle
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 0.500000000000000000000000000000 2.0000000000000000000000000000 0.500000000000000000000000000000 2 1 D1
2 0.414213562373095048801688724210 2.4142135623730950488016887242 0.686291501015239609586490206323 5 2 D1
3 0.333333333333333333333333333333 3.0000000000000000000000000000 0.666666666666666666666666666665 7 3 D1
4 0.276768653914155215717770973809 3.6131259297527530557132863468 0.612807102315627401679548476730 9 4 [1]
5 0.261203874963741442514768206917 3.8284271247461900976033774484 0.682274642960738735672090837380 11 5 D1 [1]
6 0.240253073352042147999877060493 4.1622776601683793319988935444 0.692658471061220938666994505356 14 6 D1 [1]
7 0.229142727332434474733758841965 4.3640922478382885174453284989 0.735089452850849786047748320111 13 1 7 [1]
8 0.205604646759568224693193969093 4.8637033051562731469989727989 0.676372332306029255659665555408 13 2 8 D1 [1]
9 0.205604646759568224693193969093 4.8637033051562731469989727989 0.760918873844282912617123749834 23 8 D1 [1]
10 0.188262725163240989137465394437 5.3117259358320060660316814761 0.708857073717800238401537800688 19 1 8 D1 [1]
11 0.180813296439582931755295509684 5.5305667209830479862609238345 0.719255859725666861515923434231 23 8 [1]
12 0.174492346043946343810794646535 5.7309103961966758208402076267 0.730741891870087614270421371134 21 2 10 [1]
13 0.166736205797815901445200032195 5.9974976353522081040288910186 0.722825020420142042721852306941 25 1 9 [1]
14 0.163481073179191446749135004898 6.1169160475469901264781529361 0.748329716058964172169441424130 23 3 11 [1]
15 0.158625813390680839940422928770 6.3041441908139607292494317181 0.754864460215653026123293079296 27 2 12 [1]
16 0.151911224282959640349762191330 6.5827920531884828216018737039 0.738464642020725334282575464147 29 2 10 D1 [1]
17 0.147955904479076327030170123453 6.7587704831436335362164371093 0.744292288787532983254375322575 40 11 [1]
18 0.146657116265538436708891253926 6.8186258223528189292195982640 0.774299151047651713209349455118 33 3 13 D1 [1]
19 0.141008083760465126860613679638 7.0917919975334995436155413454 0.755564628060337263225161629375 37 1 14 [5]
20 0.139095851255553663989391355519 7.1892870346129257071315563827 0.773906233460284392114102652573 40 1 13 D1 [1]
21 0.134390382953145787637749518707 7.4410086348860438828296981282 0.758552551272313472352004461552 37 3 13 [1]
22 0.131288876303240632245712664145 7.6167915223090136219250629643 0.758417837802575263372084257492 39 3 14 [1]
23 0.128649676452470314264601082848 7.7730471430252721071897260639 0.761334005560963563755389620516 47 12 [3]
24 0.126229454067600370337946457283 7.9220813191861256801102101656 0.764826003561812720663201454622 45 2 13 [2]
25 0.124584600939271017574727611620 8.0266741833322703775643431926 0.776066139559870445725543183057 23 14 17 D1 [1]
26 0.121917986048602687422534759871 8.2022352272235642498650669607 0.772927756751658535656104273284 47 3 16 [1]
27 0.118570768623816923999271262653 8.4337818806981509093547453372 0.759187467290307308173020257802 51 2 17 [1]
28 0.117105282006001788553894567443 8.5393244682912661458619574967 0.767964236127491551806135100180 53 2 15 [1]
29 0.115456141678356872441258327524 8.6612975755403886690909628653 0.773146997772663263385791229405 67 15 [1]
30 0.113965603325473159881345473754 8.7745773357958768304439908049 0.779289524480345937499120112312 58 2 17 [1]
31 0.111855081493207919511660560499 8.9401392109371637720155312089 0.775716673862835458647216998330 59 2 17 [1]
32 0.110502488840286058184874465159 9.0495699281972054490464756432 0.781491202553442869015260436183 62 2 17 [1]
33 0.109835130273955511846140435159 9.1045551410168762600027182161 0.796207885591587386987940542679 64 2 17 [3]
34 0.107103121134104680260306094216 9.3367960654283076306436749492 0.780033341853335641558595452601 67 2 17 [3]
35 0.105783385012527529973759390042 9.4532803982551111071035114986 0.783308718129604383694837854092 67 3 18 [2]
36 0.104344255391410342613764288256 9.5836612782261549099683016571 0.783916101589526377333778240091 74 2 18 [3]
37 0.103223971063542207533249113251 9.6876722499314032336613014354 0.788483926957397924476620322518 69 3 18 [5]
38 0.101505935589673642686583068779 9.8516406374735347136284143919 0.783062576955058042181484376264 69 4 17 [2]
39 0.100515207677333030487447211782 9.9487433106652960864454718618 0.788057944004556341496601180205 73 3 19 [1]
40 0.099066823853548926392800107127 10.0941966351753230165490186141 0.785138847074407244330950649645 74 4 20 [3]
41 0.097539055719853182722657761532 10.2523034759753076609221562210 0.780137126039091158259165467742 77 3 19 [2]
42 0.096615980540062609007691011372 10.3502546308614214766857912509 0.784110406440291567246742511399 78 4 21 [2]
43 0.095593885556814456701722189745 10.4609201119423951779831561186 0.785884422203043204792129743565 80 4 20 D1 [1]
44 0.094497292784693290216814076218 10.5823137418173783416641847443 0.785816974239972097971821314928 80 5 21 [3]
45 0.093666768138434146333617184258 10.6761450178579555378587818491 0.789611710814926393948376010168 88 2 19 [2]
46 0.092716338800976109854126225032 10.7855855066342641616954497507 0.790861392220479705557265817802 90 4 20 [5]
47 0.092056532078109381522021451718 10.8628901982914842487203753949 0.796594079235310267134950272541 90 5 19 [5]
48 0.091151466676148447974902892314 10.9707505152154554049323084877 0.797624628212448095064094404767 90 4 22 [1]
49 0.090569956715334948592803091939 11.0411888916215407216105594507 0.803885871822929322732238529377 96 5 20 [5]
50 0.089093785529075211414743560472 11.2241274075581413327558012649 0.793770261990085154914999654727 96 4 23 [4]
51 0.088364179559579711980137803642 11.3168028604367694608557717878 0.796439279382239788112197944537 99 6 23 [3]
52 0.087817433817507436917753348600 11.3872605532756783220589277487 0.802037774958399075758633069094 97 6 21 [5]
53 0.086833227715247571123047179187 11.5163287869397483855680806617 0.799241000157492125586937863842 107 5 20 [5]
54 0.086024724524272915212284435041 11.6245649786165594740654964284 0.799227348783520529356534385242 105 3 20 [5]
55 0.085034616681704255693300451207 11.7599165965918492476647673340 0.795397463762481324977312026362 107 2 21 [3]
56 0.084232218436600018938706310867 11.8719418597846961500975961326 0.794647461748123215536422021918 101 6 23 [2]
57 0.083551186608543770120847620186 11.9687109254979998717297417963 0.795811289341310235860974791565 113 3 24 [2]
58 0.082906284736989599064700932150 12.0618117573641364621565876436 0.797320437671332168190006945595 117 3 24 [2]
59 0.082127709357950922726128311941 12.1761584222632200478325980834 0.795905356037319028551382680639 125 4 24 [3]
60 0.081505849981968187989940587985 12.2690579905765456985798387847 0.797184429753972440458193326898 129 4 24 [4]
61 0.080994720910621467867473140675 12.3464836813686979480913414018 0.800337667477514420249493973720 124 6 25 [3]
62 0.080610377983216918709021033355 12.4053505890792375068581201018 0.805756096786040765371032767542 129 5 24 D1 [3]
63 0.079902784728944282835789240146 12.5152083671716671630813291512 0.804441330937441472976000433829 128 7 24 [2]
64 0.079358884120611330517355245231 12.6009836337942781314900319183 0.806122558575183000290816362914 132 6 22 [5]
65 0.078767398641700702337515900681 12.6956077926202356319292770946 0.806559401541477174185020652339 131 6 25 [4]
66 0.078252927327473012271589300631 12.7790746512932088776372958731 0.808304723862077977227810110234 132 4 26 [5]
67 0.077610460456189716692324479792 12.8848610628265685080230786231 0.807133398677719555502556721226 146 6 24 [3]
68 0.077098474373952776476094769684 12.9704252661300852680296137524 0.808407766107583203726973497039 129 8 23 [5]
69 0.076587056192720970122201745956 13.0570366549098744433344070369 0.809449650324845930915565562017 137 5 27 [2]
70 0.076061046753782301041080774334 13.1473341832529122304150469523 0.809939596659347996118303977363 136 8 27 [4]
71 0.075572173580384116622662502751 13.2323837283352254409578359106 0.810983785592246404304340133056 154 5 26 [5]
72 0.075107368269508650166404543106 13.3142729274135698029741277507 0.812320814645509652222072691793 144 4 27 [2]
73 0.074391154696433685038856423279 13.4424583686148917970020562882 0.807970408972034217050716874969 154 3 22 [3]
74 0.073814151887090250484604115224 13.5475376257069115942559155433 0.806382294783943519635432486279 166 4 23 [2]
75 0.073302177703898282140658418963 13.6421595009014174657681750413 0.805981388420082364562063437095 160 5 27 [3]
76 0.072942793329582360457972160478 13.7093735289466131620396754782 0.808738967005769064693452457866 155 4 27 [3]
77 0.072651240855613832753602478339 13.7643898194029127528450661936 0.812843230870503533235719774766 166 6 27 [2]
78 0.072020041392488325131784951807 13.8850239553500143571143046622 0.809154272499414143651239898178 159 5 29 [3]
79 0.071569712383540241464531929381 13.9723909276178980542019801436 0.809311349444702395802827112565 169 5 26 [5]
80 0.071321221934945490647412579161 14.0210721699655394380977710796 0.813874671726999967513305880791 171 6 26 [2]
81 0.070782426789738040589279165590 14.1278004351353903371411888098 0.811644614643629324273350239981 174 5 27 [5]
82 0.070417262106960783171239785349 14.2010633483739135212831392605 0.813208891633028059348817466555 175 5 26 [5]
83 0.070065755199955661478379241156 14.2723074509961581490595866824 0.814928868588858893455399407534 172 6 26 [2]
84 0.069810787636327915246489502250 14.3244337137319904136647698179 0.818755739827951717976801904095 180 5 29 [3]
85 0.069236879710708157632342075042 14.4431696543560478019418289414 0.814936737052762058731188715627 175 7 26 [4]
86 0.068889696390622911874886862883 14.5159588791004952901758449150 0.816275926232258993740173226764 169 4 30 [5]
87 0.068438323749736608578027119208 14.6116962720590975236814371412 0.814981923435234575416348469237 192 5 28 [5]
88 0.068154499745286623419316957712 14.6725455213858749280821482530 0.817526307053328297311493974032 195 6 27 [5]
89 0.067866307455630784618687818612 14.7348520567996693281112328168 0.819838752403872778852152691718 192 7 28 [3]
90 0.067565909065404280434944012397 14.8003632872310337602783032220 0.821727372210206458398036304074 197 7 29 [3]
91 0.067075539860670328771868654670 14.9085643153555732021232046207 0.818841304663268094956749988661 201 7 28 [2]
92 0.066734287888443416751957952446 14.9848006420875151958517908465 0.819437593115886809334196967899 191 7 27 [5]
93 0.066309736827657244370592660887 15.0807414995335684556308190711 0.817838502856488398058024242184 197 9 29 [2]
94 0.065989864733662591610222934810 15.1538422458666201379808017218 0.818676502542612134677583417361 186 10 30 [2]
95 0.065624461214805828445766928983 15.2382203448001112694254649651 0.818248282849375349103423877050 213 7 27 [2]
96 0.065147868190103551087812916524 15.3496964333808774815832598495 0.814894988105300379137192587200 205 5 32 [2]
97 0.064842120563689588264432958596 15.4220742829928647804274739111 0.815673116244034896539980681741 215 5 33 [5]
98 0.064570058230011335736311251179 15.4870543315572358100978507788 0.817181314286102707716649411486 201 9 31 [5]
99 0.064355159479183188001003140181 15.5387696665326056233909944535 0.820034137215038144020926378589 212 8 30 [2]
100 0.064027617963500979241270325618 15.6182602415421920704795047460 0.819907172416006656860830200621 223 7 31 [2]
101 0.063858713932114389981668286256 15.6595699854378442554638029305 0.823742939702851339960931536905 214 10 32 [2]
102 0.063495074407009239306180667504 15.7492531403287829810908636767 0.822451392686134499313227516980 227 8 32 [3]
103 0.063331902707158091316996652340 15.7898303580728351055372397199 0.826251559504841305386522782877 217 8 29 [2]
104 0.062968389993709074665175170172 15.8809840953517484755033199278 0.824723772787166952991473480265 218 7 31 [5]
105 0.062745134956874283465793036261 15.9374906227760240354000420809 0.826759911758837109908133575327 238 8 29 [5]
106 0.062344645156484646343280711621 16.0398699437619161109698945353 0.824013213293852708261629101705 236 7 29 [2]
107 0.061996918594682384859981849864 16.1298339121933755110444744412 0.822534233860434333169826821994 234 4 31 [2]
108 0.061788945136324301359260655904 16.1841248105742943513177416980 0.824660728068894019837749854877 237 9 32 [5]
109 0.061537390875819579361104707617 16.2502827267728484704791384751 0.825533403725141814023255492543 239 9 34 [3]
110 0.061391600386263953508502223241 16.2888732938740056181817646380 0.829164291557079369237493830466 248 9 34 [2]
111 0.061049338306594488898369470864 16.3801939175478498684660038692 0.827398819103411607140089729465 239 10 34 [3]
112 0.060828833302910104790631923203 16.4395722505524221855777031465 0.828832919262482500441203525882 242 8 31 [2]
113 0.060514942949160794310900317560 16.5248441337887395828497765140 0.827625180351681975816728574845 248 7 32 [5]
114 0.060258226781778034522487238732 16.5952443908023853496480311420 0.827880288033595681006210299516 247 7 35 [2]
115 0.060005068920134665373887989529 16.6652587522393560930383745530 0.828139908105325570590995905819 257 9 31 [2]
116 0.059787346818217172425287959466 16.7259470978113472287906569657 0.829290226778333667911817747745 265 7 30 [2]
117 0.059404880382065989701249133920 16.8336337615435138767642676515 0.825771916290571072650898618312 257 8 33 [2]
118 0.059125781995969617040722861510 16.9130955438046680222914673972 0.825022510805842279128814676004 247 10 31 [2]
119 0.058839042929252045505213347427 16.9955177755423064613645189894 0.823963847533626901331196509263 258 8 33 [5]
120 0.058589552976949916116335253676 17.0678892258047484740478767115 0.823856572329316986676431213383 263 7 33 [2]
121 0.058398608171074243638896829099 17.1236957749160167825176433538 0.825316179589115578574051900368 262 7 33 [5]
122 0.058202497466008939928676787362 17.1813932999011558183730921382 0.826557493552509605936028604834 268 8 33 [5]
123 0.057936521442589162227686619789 17.2602699489116991247620089420 0.825733567149428069741923097634 263 12 33 [5]
124 0.057716970061430165940733340774 17.3259268276845690253374931187 0.826149661001862471105247682998 268 8 35 [5]
125 0.057576519673373672599891802255 17.3681911597454742330847742560 0.828763904424596390631750612897 276 9 34 [2]
126 0.057363112651374015289551967877 17.4328057488359997208294779248 0.829212726649664853764521273638 275 7 36 [2]
127 0.057171156228861998441429337793 17.4913376947791207817967968582 0.830209440554416305110255315208 262 10 35 D1 [5]
128 0.057023696401740350559430156158 17.5365692352676074839812796413 0.832435699537370983389199253404 279 6 38 [2]
129 0.056858450373147929471207551799 17.5875352465156478634363424957 0.834083911739617287391924743484 283 8 39 D1 [2]
130 0.056618980749969982541393005584 17.6619216162864282222360009153 0.833484335103022576136262898162 295 11 33 [2]
131 0.056444896325568687438174775459 17.7163935997348102049166235665 0.834738896155499748302125777381 282 10 35 D1 [3]
132 0.056158328518030259864979400004 17.8067977874188119493210808958 0.832592075551898841485023833229 295 10 32 [2]
133 0.055939287202258638427906066672 17.8765238174079450255959707935 0.832368224817342705249540970817 305 8 33 [2]
134 0.055730684474279227538697124271 17.9434365365011629873158110177 0.832383663448406954423780342396 310 7 33 [2]
135 0.055528166810711735565664773323 18.0088783303232271930746874114 0.832511873526721646003152123706 291 9 34 [5]
136 0.055358822867300920493028004721 18.0639679134267705502916859647 0.833571001236870211838632296728 300 7 35 [5]
137 0.055238484395898973200967454026 18.1033207361902602000173244516 0.836053503389536971980045255520 310 9 40 [2]
138 0.055119941185259247683891942509 18.1422544817125182232728069228 0.838545384889537064198954731115 301 9 39 [3]
139 0.054860827582506637196968634678 18.2279423053921995136097745375 0.836699492044430848715038723701 306 9 40 [3]
140 0.054657786986961786222348533413 18.2956547479418194850400639337 0.836492629927384969596768641463 302 9 37 [2]
141 0.054474270635525176887770061360 18.3572903011546521720904922279 0.836819817478828237653944984747 311 10 36 [3]
142 0.054235525271534080671143823875 18.4380992899658970753180769215 0.835383785220096183835698756908 326 10 37 [5]
143 0.054005084581005638864411108291 18.5167749992139501509365932104 0.834133059932049878074338670082 312 9 33 [2]
144 0.053791339351778583437457835021 18.5903532436757499257974466370 0.833330358506362519582943376651 331 10 33 [2]
145 0.053625624712269202846110339348 18.6478014077327166669733053948 0.833955211476529853115991917496 300 10 35 D1 [2]
146 0.053348124250109897128550357978 18.7448015100163526394051762477 0.831038529413507792760479194953 326 11 35 [5]
147 0.053133901021816626742358470423 18.8203760832355011886147903710 0.830024162712084555603601124208 339 9 37 [2]
148 0.053013007219523555227568453336 18.8632951128213193702942939720 0.831872164599347950071247046457 322 8 38 [2]
149 0.052831261764795599078621238822 18.9281869596829406494223003407 0.831760381458785312621970814524 329 6 37 [5]
150 0.052736861205212853159147445572 18.9620689807218469575863255372 0.834352958933365340114116359905 328 7 37 [2]
151 0.052567292135960856989219918725 19.0232359204195746197396446532 0.834522701157250551815805948419 345 7 37 [3]
152 0.052458588737549760657134096155 19.0626554023974696092148350800 0.836578673829956307847027172456 339 6 38 [3]
153 0.052276717600272380098699339472 19.1289745398014363477716078508 0.836253692135940178709756786514 347 5 40 [3]
154 0.052096320393393956927719442935 19.1952136436646379369180171683 0.835920192347595766828088452089 353 7 36 [3]
155 0.051945924431120772599258750615 19.2507884102819160058526290304 0.836497510151150088340861435436 348 12 36 [2]
156 0.051804192590392057461166005662 19.3034569210806791900198544989 0.837306403422038580747392244083 348 9 40 [3]
157 0.051662700565055199100219587580 19.3563245641944414351289954897 0.838076873717810203556722999494 317 6 41 [3]
158 0.051496533944469923890069304549 19.4187826520193859770997080801 0.837998190620886019046469676430 364 9 40 [3]
159 0.051335338235602895511422347968 19.4797586685902886977465405564 0.838030790660873328461175574306 301 8 38 [3]
160 0.051222039860754298080917216304 19.5228460779475476881049376020 0.839583157598944701547102490677 361 8 40 [2]
161 0.051034822131128688105572270067 19.5944642940187702230810237176 0.838666088525813494762244294948 366 12 38 [3]
162 0.050876215101825949925229181889 19.6555502015736612761332975453 0.838638121240273164151327372103 377 12 38 [3]
163 0.050695270124969824783703309196 19.7257061168602507263724973577 0.837823394652232519130461137155 374 13 37 [2]
164 0.050483162710326520798413005956 19.8085846114282027544792246613 0.835924307253835104966815745477 378 10 39 [3]
165 0.050373923838376980968173783043 19.8515407139707032278279527106 0.837385626948619541562813586158 386 7 38 [3]
166 0.050230766889190433625359125229 19.9081173139564014002611002859 0.837679140835695077479296315156 384 6 40 [3]
167 0.050155747221105973834495200259 19.9378945665311774871508433088 0.840210059088698191928025295652 369 7 41 [2]
168 0.050038681955291150266225434219 19.9845391789792896453930499863 0.841300216452454138185328632779 392 7 40 [2]
169 0.049938580982068043467491457178 20.0245978226549973003200595930 0.842925312229868008588567480736 284 10 37 [3]
170 0.049821392725851391486219286306 20.0716990290220981840835255368 0.843938198868796101239150147122 377 10 42 [2]
171 0.049648179120251106520322901365 20.1417255923512362918901335077 0.843010057965135974590268432444 400 9 43 [3]
172 0.049460977153927605801878259634 20.2179588342522632022132420898 0.841557561791343906070091296497 383 8 44 [3]
173 0.049314832883948881592696062737 20.2778746579810993779869921330 0.841455648860644864246041885622 402 9 41 [2]
174 0.049103119141474123843295117455 20.3653050454663849742662120805 0.839068475678787268959842868725 406 11 40 [2]
175 0.048926671215871176011511191699 20.4387499731560115154691189528 0.837836704693084971278937823746 406 11 43 [2]
176 0.048739673766468695518495009095 20.5171664626111501746558090685 0.836195641199352458143780531435 399 9 46 [3]
177 0.048568609843831599884174760464 20.5894301528377766270273478027 0.835054091205466867992528106813 405 11 44 [3]
178 0.048449849313697606607096734789 20.6398990743049180561639264835 0.835670091873121581336573326987 395 10 41 [3]
179 0.048367167313452924026834891491 20.6751822681552480272199764605 0.837499068866062522074152083554 395 8 44 [3]
180 0.048219418339379643225718916707 20.7385330316878577564123220481 0.837040429795716673186377537511 403 10 42 [3]
181 0.048142630866476295943282025530 20.7716109818240394364476283642 0.839012072241978220605163597733 416 10 41 [3]
182 0.048041101916841943934176071620 20.8155092223108721142075912604 0.840092880311919670710940183333 423 6 42 [3]
183 0.047895781453815042540193936796 20.8786655034385497623382502487 0.839606152472210354328916723721 422 6 43 [3]
184 0.047809800090653111600335784513 20.9162137909775848664923631837 0.841165930372622854376663154294 429 7 43 [3]
185 0.047711997360871162250729321919 20.9590890198217690552843342555 0.842280836100597398097408310208 432 8 41 [3]
186 0.047629163892296303139347879425 20.9955396710573639326812048196 0.843895858145470557675612071733 444 4 39 D1 [2]
187 0.047459686748637229637816702444 21.0705141248896732797201655993 0.842405777988260822808644867149 431 6 45 [3]
188 0.047372985601548134190924734503 21.1090769834722073371500638350 0.843819111566487050888527080439 429 6 45 [3]
189 0.047239667869985712361972831662 21.1686501004246465717194012108 0.843539591336359850232262115139 422 9 44 [3]
190 0.047133350669894809336948663176 21.2163995512146721551199737013 0.844190043241083919391309766971 434 10 43 [2]
191 0.046976238840096570870994560849 21.2873577087327381556052838803 0.842984999944606690153035634048 437 6 45 [3]
192 0.046818965317731558348937252303 21.3588658615929306688040158676 0.841733957154412999842065347731 435 13 41 [3]
193 0.046677137770317288543369684687 21.4237643473485517665393559645 0.840999503505663794591358510160 438 11 44 [3]
194 0.046566760008289661053424740026 21.4745453585773048780073747306 0.841363697415822381669456158746 360 18 43 [3]
195 0.046459661361732405104955572294 21.5240484043577976838051518493 0.841815052200271873866535973072 420 15 42 [3]
196 0.046361388872078310142219261373 21.5696730475273082503960488255 0.842556324234042093988072664394 482 7 42 D1 [2]
197 0.046234828663281372642005406144 21.6287164657360735530763355465 0.842237796320056010946354757955 450 9 45 [3]
198 0.046108443313943600471025127013 21.6880017655593063226698323619 0.841891463754719524196155268171 462 14 41 [3]
199 0.046011204980442141959239953984 21.7338363649695174535677161036 0.842578331533400980075489073729 462 12 42 [3]
200 0.045925579937177115105495919437 21.7743575882531687224984484815 0.843663557026418060736014976813 468 13 41 [2]
201 0.045805770099319394159316873640 21.8313107242106713885084524794 0.843463766905464469587879211706 459 10 42 [3]
202 0.045706004515499749895611354674 21.8789634009877529178787751460 0.843971694903436944987029050159 440 8 44 [3]
203 0.045598196784532409376636311314 21.9306917930407055645216965800 0.844153393300382353629164632442 446 9 44 [3]
204 0.045482723793930015186593055605 21.9863701332121391729521199382 0.844020690795690442260512451428 466 10 47 [3]
205 0.045405558181820127178382504680 22.0237354201360464498160465412 0.845282532659087604289656038539 342 9 37 [3]
206 0.045315606543795988629519948894 22.0674526122371129759014282701 0.846043728930035840289623948751 465 13 50 D1 [3]
207 0.045158063790004260950869118175 22.1444392445663278432681165775 0.844249800258506914626475266556 460 11 49 [3]
208 0.045035630332158932107182880221 22.2046409170812129231643121329 0.843734527756587436518836872198 468 13 48 [3]
209 0.044916766538605184111365340363 22.2634013323402808307625403291 0.843321653006529054913845623456 455 12 47 [3]
210 0.044747780865852762778279764134 22.3474769172990340533719599223 0.840992834815719025477930258221 491 10 47 [3]
211 0.044632716935228222702022197045 22.4050891065228550157286377492 0.840657515670527857336299931564 497 8 50 [3]
212 0.044560724820715835484536649148 22.4412866716905376110025437764 0.841919075336165579278136247997 476 13 45 [3]
213 0.044484054534350586244309783047 22.4799652474978425133603467600 0.842982051929222781712864494052 492 11 50 [3]
214 0.044364254453869719695626841222 22.5406695617934345123230512462 0.842384067350015283694691854878 472 9 48 [3]
215 0.044291867298695488258218343734 22.5775082648062712097851013087 0.843560888786257830276606665871 483 12 46 [3]
216 0.044205544144813163164034613395 22.6215968911975158220877448229 0.844184217516058943866659541903 483 12 46 [3]
217 0.044104656297529097831634587809 22.6733429970300802996044288166 0.844225786891457128973010411025 496 11 47 [3]
218 0.044040449149775568573498578880 22.7063987608104589338548740052 0.845648666332889873969315880163 501 9 47 [3]
219 0.043960522487962605587766247053 22.7476823159648028296629839591 0.846447061387623688002736176138 500 10 46 [3]
220 0.043873558198413590324981740001 22.7927717983940193321601988899 0.846951207955417120859865435896 473 9 49 [3]
221 0.043786214316032567296756495927 22.8382383729813428137771216016 0.847416793345254622546110337638 425 9 43 [3]
222 0.043675823952184178371503973754 22.8959618734334406364706274899 0.846964453468570221058052427922 440 7 49 [3]
223 0.043590637025177062458420285865 22.9407062673211314959077604991 0.847464061772288872076095512358 527 15 45 [3]
224 0.043491195240204935987098638545 22.9931597528403055705381663388 0.847384860412887766884660488474 515 11 46 [3]
225 0.043422671977047805614872168786 23.0294441698239177719705363530 0.848487798731831752617978708618 523 13 49 [2]
226 0.043297949555254623393623801509 23.0957819081906238696513097561 0.847370020931597065958121529347 507 9 46 [3]
227 0.043178999946733505641794515837 23.1594062213951314488833173399 0.846449420525605485705340239811 538 12 45 [2]
228 0.043121154538565252112337247009 23.1904736944288802696557037835 0.847901889744904948215000829101 514 13 44 [2]
229 0.043023577180191332566906602128 23.2430696269582678114606088980 0.847770912567985211895984765371 484 14 47 [3]
230 0.042978286377706935054374608525 23.2675633274830915637909704879 0.849681225983527170259073720792 545 12 44 [2]
231 0.042853638751926357175186344436 23.3352412799496049918436506727 0.848432671677639228864463620804 533 15 47 [3]
232 0.042795962401677000409819255378 23.3666903109722811321186659574 0.849813400618988674177908885281 553 10 46 [3]
233 0.042668510795014132383259678858 23.4364870338256853752699029784 0.848400445074334697144482899157 536 11 51 [3]
234 0.042575012883230164198897048953 23.4879553117854523041679666732 0.848311645899376366884796203168 523 14 49 [3]
235 0.042469229728586713961237539156 23.5464595517936624123188441536 0.847708672657552560976798410976 540 11 50 [3]
236 0.042373194552835731045912102960 23.5998255631419520143894434830 0.847470155041084739818909170674 564 12 46 [3]
237 0.042276476563515127593717996208 23.6538160529443567824036656746 0.847180423076460385968781012131 567 14 47 [2]
238 0.042203298991905187595315693728 23.6948301172333756363422685067 0.847812380200869221604052029374 551 12 48 [3]
239 0.042105450229978638564665030000 23.7498944801214949942870965057 0.847431352875081513479365144896 557 10 47 [3]
240 0.041981279111216265119145591736 23.8201412908552131301368810263 0.845965341990644699029774216143 561 13 49 [3]
241 0.041892963553749135370798948264 23.8703570998740383904700014893 0.845919830542193131753086815064 566 10 51 [3]
242 0.041793102943229431602492189592 23.9273930284231755295640390874 0.845385111553712327951272339060 551 11 50 [3]
243 0.041698207016647027472114587202 23.9818465000371249171900105870 0.845027867643931578072529035396 561 11 51 [3]
244 0.041629312396728498086630531291 24.0215353659933739119822618919 0.845703829504713548720420108567 402 17 42 [3]
245 0.041524607912420698535850018212 24.0821057747033759243264645036 0.844903600517333124789964716010 557 12 52 [3]
246 0.041451599501467709544541098636 24.1245214183976714736118101969 0.845371669805198408692142563732 580 10 47 [3]
247 0.041369993907840121252558217311 24.1721089499722583933513657709 0.845469339591756000943235629462 595 10 48 [3]
248 0.041312575863900977620508259540 24.2057044153909140631087746585 0.846537546557244509253554428563 583 10 48 [3]
249 0.041218400962635375161307083547 24.2610090795735502322467153884 0.846080375802457249238723637859 597 11 48 [3]
250 0.041124237386118139971190331763 24.3165603439872926878234458163 0.845601450294898472718013382952 598 13 46 [2]



Updates

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

23-Jul-2009: 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. He had adopted the code for this case, great work. Many thanks to him!
17-Aug-2009: David W. Cantrell sent me the coordinates for ten better packings ranging for N=24, 35, 37, 41, 42, 45, 52, 60, 72, 78, 83, 94 and 129, thank you! The original reference is here: sci.math Forum.
18-Aug-2009: Calculation form provided.
31-Aug-2009: More improved packings for N=65, 68, 73, 77, 79, 87, 88, 91, 95, 97, 101 and 145 by David W. Cantrell [2].
17-Sep-2009: Once again, further improved packings for N=56, 57, 58, 63, 96 and 100 by David W. Cantrell [2].
21-Oct-2010: Claudia Orquídea López Soto, PhD student of John E. Beasley at Brunel University, UK, sent me some nice improvements for N=50, 60, 65, 70, 80, 85, 95, 100, 125, 150 and 175. Many thanks and welcome to the list of contributors!
03-Dec-2010: Some improvements for N=94, 119, 168, 248 and 249 by E. Specht [3].
03-Dec-2010: More improvements for N=72, 80, 95, 99, 103, 114, 125, 150, 170, 175, 186, 200, 230, 232 and 250 by David W. Cantrell [2].
28-Feb-2011: A bundle of better packings for N=19, 37, 41, 46, 47, 49, 52, 53, 54, 64, 66, 68, 71, 74, 79, 81, 82, 83, 86, 87, 88, 91, 92, 93, 94, 97, 98, 104, 105, 106, 107, 108, 112, 113, 115, 116, 118, 119, 120, 121, 122, 123, 124, 127, 130, 131, 132, 133, 134, 135, 136, 140, 141, 142, 144, 146, 147, 148 and 149 by Claudia Orquídea López Soto [5].
17-Mar-2011: Slightly improvements for N=41, 74, 77, 83, 91, 93, 94, 100, 101, 106, 107, 112, 115, 116, 117, 118, 120, 126, 128, 129, 130, 131, 132, 133, 134, 140, 143, 144, 147, 148 and 160 by David W. Cantrell [2].
19-Mar-2011: More improvements for N=73, 78, 89, 141, 145, 150, 154, 155, 157, 158, 159, 163, 164, 165, 166, 168, 169, 172, 173, 174, 178, 179, 181, 182, 183, 184, 185, 190, 191, 194, 196, 198, 199, 201, 204, 205, 208, 209, 212, 214, 216, 217, 218, 221, 223, 225, 226, 227, 228, 231, 232, 238, 239, 240, 241, 242, 243, 244, 246, 247, 248 and 249 by E. Specht [3].
21-Mar-2011: Even more improvements for N=131, 145, 148, 150, 154, 157, 158, 159, 161, 162, 163, 164, 165, 166, 168, 169, 179, 181, 182, 184, 185, 187, 189, 190, 192, 194, 195, 198, 199, 201, 203, 204, 205, 208, 209, 212, 214, 215, 216, 217, 218, 219, 221, 223, 224, 225, 226, 227, 228, 229, 231, 234, 236, 237, 239, 240, 241, 242, 244, 246, 247, 248 and 249 by E. Specht [3].
21-Mar-2011: Further improvements for N=145, 150, 155, 173, 174, 196 and 225 by David W. Cantrell [2].
21-Mar-2011: Slightly improvements for N=154, 156, 159, 163, 166, 187, 201, 207, 215, 216, 217, 227 and 237 by E. Specht [3].
16-Apr-2011: Improvements for N=163, 167, 168, 190, 227, 228 and 237 by David W. Cantrell [2].

References

[1]   , April 2009.
[2]   , August 2009–April 2011.
[3]   , May 2009--March 2011.
[4]   , private communication, September 2010.
[5]   , private communication, February 2011.
[6]   C.O. López, J.E. Beasley, A heuristic for the circle packing problem with a variety of containers, European Journal of Operational Research 214 (2011) 3, 512–525.

Datenschutzerklärung der Otto-von-Guericke-Universität Magdeburg nach DSGVO

©  E. Specht    21-Jun-2018