The best known packings of equal circles in a 1x0.20000 rectangle (complete up to N = 347)

Last update: 18-Aug-2010


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   337-347  



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
= 0.2/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 0.100000000000000000000000000000 2.0000000000000000000000000000 0.157079632679489661923132169164 2 1 D2
2 0.100000000000000000000000000000 2.0000000000000000000000000000 0.314159265358979323846264338328 4 2 D2
3 0.100000000000000000000000000000 2.0000000000000000000000000000 0.471238898038468985769396507492 6 3 D2
4 0.100000000000000000000000000000 2.0000000000000000000000000000 0.628318530717958647692528676656 8 4 D2
5 0.100000000000000000000000000000 2.0000000000000000000000000000 0.785398163397448309615660845820 16 5 D2
6 0.084524052577349764562923561227 2.3661903789690600941748305755 0.673335869308202289813958373700 13 6 C2 [1]
7 0.075077640500378546463487396284 2.6639089703277447388644303619 0.619782269328363353231892192330 15 7 D1 [1]
8 0.068958825895264200037012280989 2.9002813984066853513413480322 0.597571093485630614143172665685 17 8 C2 [1]
9 0.064797003745975065702681339008 3.0865624710683202099544536317 0.593570399322825616131283377466 19 9 D1 [1]
10 0.061850317545375803602987430032 3.2336131476329479615843419943 0.600902131304683676485706937541 21 10 C2 [1]
11 0.059693491089449820242245977452 3.3504490414256880287612406436 0.615696265875270834741186988630 23 11 D1 [1]
12 0.058070323909482229503460718766 3.4440999556288380507529505865 0.635637659775061893264141808994 25 12 C2 [1]
13 0.056819747776422020642924518493 3.5199029884288257963736184002 0.659267751452390399782190482367 27 13 D1 [1]
14 0.055836716635766173763732845739 3.5818725034396117346635326963 0.685626599107221933343095035512 29 14 C2 [1]
15 0.055050511413105583784558276553 3.6330271030395379562185920112 0.714058596289082273473333907963 31 15 D1 [1]
16 0.054412188832468264872991509398 3.6756470248934025388101606792 0.744101624749586933356144970288 33 16 C2 [1]
17 0.053887046240469780235174544026 3.7114671141465857807661797477 0.775420995947011705127707060490 35 17 D1 [1]
18 0.052886076962755087244445810439 3.7817136661667984101537610419 0.790815344455234446008737060520 45 18 C2 [1]
19 0.050516276701207074309823924356 3.9591199720232161922626407977 0.761616150296221531119415780737 47 18 [1]
20 0.050000000000000000000000000000 4.0000000000000000000000000000 0.785398163397448309615660845820 52 20 D2
21 0.047148169098807494744715475145 4.2419462690240190730639787583 0.733278306188245785277584143706 41 1 13 [1]
22 0.046068938270063788377973868859 4.3413199329137279517167784758 0.733430497133873458931388368901 46 13 D1 [1]
23 0.044304881270818043187669315550 4.5141752841516464729904138803 0.709170835737437981420973886692 45 1 14 [1]
24 0.043493177134262624676914080897 4.5984224004285485777538732265 0.713137683498819676332202783454 50 14 C2 [1]
25 0.042269960955124823410608895093 4.7314924234807445891673402155 0.701654926810895543438906302271 51 15 [1]
26 0.041669750984812399697557708378 4.7996447128492580983381671328 0.709144983946183126908603623144 54 15 D1 [1]
27 0.040801395066564593485098624771 4.9017931782409432261412147437 0.706047088296368968222594274657 61 20 D1 [1]
28 0.040316555991033377499214845493 4.9607411913974272402473049989 0.714899115780357248309824234727 58 16 C2 [1]
29 0.039722870247268030405054363364 5.0348828963021660547505210156 0.718785186793354217420726040843 61 2 20 D1 [1]
30 0.039233338586232517375172577515 5.0977053497604348621332110022 0.725356762434894640090506126910 68 22 C2 [1]
31 0.038839455689618423398312547549 5.1494027516317361553294119974 0.734560956099863801777410224292 68 1 22 [1]
32 0.038487051145727202116114240103 5.1965529716142937217629314667 0.744559020093295010865735072614 68 2 22 C2 [1]
33 0.038064941614196181081443283241 5.2541785569273204867237574355 0.751076463824206327124517886934 73 1 23 [1]
34 0.037778569938339541578791725201 5.2940066372663358125720887964 0.762236642501044014238256523917 75 1 24 [1]
35 0.037613330478421455521863720079 5.3172637853682968354677625573 0.777806379695233258238035671433 72 24 D2 [1]
36 0.037268501951520713452221634031 5.3664620128859016031946521930 0.785427765958661693321991502257 80 1 25 [1]
37 0.037018969085496882770975369672 5.4026355930682861007053370864 0.796471502622184161744693742147 82 1 26 [1]
38 0.036957211574933433058655583170 5.4116636909818117926959238280 0.815270759309987774418980456856 78 26 D2 [1]
39 0.036683735733267490578414877458 5.4520074360536130999231513302 0.824387876477657997511598266532 79 27 D1 [1]
40 0.035938232559818867487942312885 5.5651039507049152797416629625 0.811508919819025754200348680512 85 1 22 [1]
41 0.035747662088825697908318683246 5.5947714707339607125776612405 0.822998458603318725670617630009 96 1 22 [2]
42 0.034834280165172963023606057020 5.7414707308910724512007266272 0.800539652495362356448972241618 98 29 D1 [1]
43 0.033958773032007974007889262796 5.8894942939042355776841849062 0.778919028020134565038746763495 87 19 [1]
44 0.033756830962913366790012090851 5.9247267677385998852948809121 0.787582198814161828623770682873 109 24 C2 [1]
45 0.033333333333333333333333333333 6.0000000000000000000000000001 0.785398163397448309615660845804 108 32 D2 [1]
46 0.032596219936162102470717868142 6.1356807750005663413387359035 0.767736498476290127870584626704 105 2 25 [1]
47 0.032262319368807558355748046949 6.1991823251668518575216003233 0.768438130937215197167619711964 107 2 25 [1]
48 0.031741816059838887383237003268 6.3008367140356727368293699631 0.759669439505799269609623393297 114 1 27 [1]
49 0.031376373296674426833354747845 6.3742229896658561533829675481 0.757742146178704458612464330046 109 3 27 [1]
50 0.031176464698490098904180455769 6.4150955515391121239944819299 0.763384985264789401827597867539 124 27 D1 [1]
51 0.030768153182597123182619923435 6.5002276481489523084428876052 0.758390547376508656308863781455 122 1 28 [1]
52 0.030497677357300217794848863406 6.5578764460279816578322580471 0.759725584362670243026235614891 120 2 28 [1]
53 0.030272848686719203069348012687 6.6065801097780615817647497987 0.762960979088657593441775907333 129 29 [1]
54 0.029984018689822097418773875547 6.6702199618054816970416988642 0.762593881842471544096267071251 129 1 30 [1]
55 0.029740240386981680892591733039 6.7248952058755662161348277085 0.764137504310215272610818241471 131 1 30 [1]
56 0.029616937785747821994635221242 6.7528925997286399872995345107 0.771592877651146578588950241476 139 30 C2 [1]
57 0.029385881769402513183695232202 6.8059894057099954411822859451 0.773165001092711378303613711905 137 1 31 [1]
58 0.029190054222239494253979220981 6.8516488005569649086067732432 0.776278707383718881896018545854 135 2 31 [1]
59 0.029017737744080922979966436948 6.8923360519651904270013609979 0.780367171734440206302599531038 142 1 32 [1]
60 0.028871754277476068519829366314 6.9271855834554346229032588536 0.785628940136958320313083202249 145 1 32 [1]
61 0.028697122670198869263698727823 6.9693398288914242746191719402 0.789089783303957553763931093832 147 1 33 [1]
62 0.028617382523073494241674128842 6.9887593611590752971153079004 0.797574726974411374061732279019 154 33 D1 [1]
63 0.028443473954043213802453498439 7.0314899060200831396745455302 0.800618649914533882091702970779 152 1 34 [2]
64 0.028345645458148414677783947821 7.0557574811727126212794837453 0.807741790061903844845449270584 150 2 34 [1]
65 0.028201133568197784273773044231 7.0919135046946681316400025979 0.812019324380774299899489130291 151 2 35 [1]
66 0.028139143063145233484305339610 7.1075369833115712760141845005 0.820891100810544297060939251615 160 1 35 [1]
67 0.027993448514499621332879920743 7.1445288313230517970650393167 0.824721819539384575758748460948 161 1 36 [1]
68 0.027956977538776886207519482085 7.1538491499160095388690325351 0.834851486522198981030471718720 169 36 C2 [1]
69 0.027812487772841806267220477793 7.1910143973274825899546659986 0.838394928473584072569442651701 167 1 37 [1]
70 0.027777800548174088041633491128 7.1999940979181145183777687510 0.848425332906614692610115305678 84 15 27 [1]
71 0.027279504769527760019684098383 7.3315113925164643960539091414 0.829948586373963149743966360793 188 1 38 [1]
72 0.027109714466787749343476270481 7.3774292327947986645284075612 0.831193733319072283126732410963 162 38 C2 [1]
73 0.026704745881396873341022844175 7.4893054922991983481488677632 0.817748274538508643358249370203 105 9 28 [1]
74 0.026475820607740561657212822669 7.5540623636620091961056491596 0.814798941827253466618324648798 174 4 32 [1]
75 0.026341555891749357970839864565 7.5925659373311180768829885916 0.817455249878942161669759513488 178 32 [2]
76 0.025974590752105054609119829142 7.6998325751789345111498467238 0.805435739601622572895152488031 190 1 34 [1]
77 0.025636928895985156395012313684 7.8012464289870844996707784585 0.794955100115072027415771090295 186 3 34 [1]
78 0.025532079059729030689171789773 7.8332829665819850115636089763 0.798705804842593712828364909319 194 2 34 D1 [1]
79 0.025288095148916152906272276737 7.9088598339354158411457426239 0.793558964665969400347629280263 182 1 33 [2]
80 0.025130533303249093564651149639 7.9584463085843969460909426260 0.793621224495756130629720371724 202 1 35 [1]
81 0.024961590336989002183904093053 8.0123100050894052100370935005 0.792774000351716813405319853437 190 5 36 [1]
82 0.024777535620545268229346529892 8.0718277662029525423130987071 0.790769568306444814764634792085 197 3 36 [1]
83 0.024596236628240359656645212368 8.1313252520252813960020525804 0.788742594089834325640231707294 195 4 38 [1]
84 0.024486787539065672300089422241 8.1676700008494164574077184961 0.791157208128924643700292215746 186 8 38 D1 [1]
85 0.024324423413740596101729963680 8.2221887276893159827428482816 0.789994216451864807216322922036 193 6 38 [1]
86 0.024197674598329396477495869713 8.2652570265494837054695983631 0.790980164280158686843835410948 216 1 38 [1]
87 0.024054996135079111387690298150 8.3142811113713881161524524663 0.790769140123991900842202157006 203 5 39 [1]
88 0.023964152169326008376353025628 8.3457991163984917517360641728 0.793828499249725463011660780943 220 2 38 C2 [1]
89 0.023819794994046352405071851165 8.3963778886421598096382560401 0.793205875445861057736427857650 214 3 39 [1]
90 0.023720963436127793299365231197 8.4313607471521810827676755729 0.795475914643597360222071001670 228 1 39 [1]
91 0.023626150640801775532889659048 8.4651961735402197040653545776 0.797897688197152964639201626385 223 2 39 [1]
92 0.023514548846143155349208012886 8.5053726230773039434396085285 0.799062971215602720187228383917 230 1 40 [1]
93 0.023413606816332240363302717861 8.5420414534547204315774962387 0.800828402428167122973496171086 225 3 40 [2]
94 0.023318676271278656602769822706 8.5768161825865597197294351450 0.802889017036193453539317963768 231 3 41 [1]
95 0.023239142833345730138846468768 8.6061694028155374803461174211 0.805904707827667759978545228155 234 2 42 [2]
96 0.023170725042556317811729756593 8.6315814301309811169707086323 0.809599735052279932751778808392 242 1 42 [2]
97 0.023101069888288189047557560049 8.6576076764910470405302565304 0.813122164206328620754437126452 243 1 42 [2]
98 0.023039875075589540163771334181 8.6806026223595935061023872451 0.817158293236963010568354650433 246 2 42 D1 [1]
99 0.022927738166668222058309077749 8.7230584432770192309550056067 0.817480681960991765236715736295 244 3 43 [1]
100 0.022881112120071583326585408723 8.7408338786364157737380452387 0.822383021350763279674608921417 254 1 43 [1]
101 0.022790971947177767038369312299 8.7754045972035096162659799603 0.824075388624749843948904137622 249 2 43 [1]
102 0.022754152336011154859252561543 8.7896045102711293195062946784 0.829547715015178118457036209676 249 4 45 [1]
103 0.022718029191773014002652894110 8.8035805532122009041961291639 0.835022942960149678918780693637 237 44 D1 [1]
104 0.022620050782971421088872971066 8.8417131295992407646108643823 0.835873135611507319675283922901 255 3 45 [1]
105 0.022586342535343294479035018867 8.8549086549554606206362656455 0.841397072208383722224853648314 211 45 D1 [1]
106 0.022509898445356309716209131129 8.8849801115499521862871675116 0.843670402670673216063354971009 262 2 46 [1]
107 0.022481657186936888078514247253 8.8961413447853519471103681495 0.849493963328646776703887974877 240 1 46 [1]
108 0.022470940523537839138250918772 8.9003840222221314063947069872 0.856615903438338376183694089961 218 46 D2 [1]
109 0.022301955382957583086584992721 8.9678235188665737899580134478 0.851593347687777523901553563693 274 2 47 [1]
110 0.022236615073398454529071990528 8.9941746682146306705637199606 0.854377728313294011180032260497 242 47 D1 [1]
111 0.021939239930453824808338400718 9.1160860920427927503175958554 0.839239683208257075518990832847 208 9 40 [1]
112 0.021831637708329681664612111211 9.1610168083584332412319848821 0.838514411272277792296243540579 250 6 39 [1]
113 0.021791488616319364922145546648 9.1778952563260216236346108191 0.842892361031347145305010480494 273 1 40 [1]
114 0.021495847111172447340057809474 9.3041227435996313260759530860 0.827434942982759449911882055940 288 3 43 [1]
115 0.021425201345660810658802759987 9.3348014225549143245248016107 0.829215747799013451564901261667 298 1 43 [1]
116 0.021249279743191863994701106276 9.4120837231708404648683240424 0.822746975000810225651461573768 290 2 44 [1]
117 0.021134661204481532060747479380 9.4631278005814774720279561030 0.820911461803077450865252408307 292 3 43 [1]
118 0.021084204421255848374667299304 9.4857740896484490652198777421 0.823979336708914017817614550568 297 1 42 [1]
119 0.020883659859864774035782546227 9.5768654221557045059482115899 0.815229826352210885779339593874 302 3 44 [1]
120 0.020813957282804213714093530819 9.6089367957545114966954160173 0.816602003093788013407740077376 313 1 44 [1]
121 0.020689523049209363956781918076 9.6667283979580604150962014996 0.813591130728335876267308121178 304 2 44 [1]
122 0.020586014716687988398375963047 9.7153335773082210990322456058 0.812127591347609393917083275909 314 1 45 [1]
123 0.020553007096033781907577990351 9.7309361625528273701671920201 0.816160801785812202897803591247 325 45 D1 [1]
124 0.020394425835144745237345723336 9.8066011574275114270660373156 0.810148306361865088469233564972 318 1 46 [1]
125 0.020322511577886842178394769803 9.8413032874156308890926760746 0.810932393949582383131332817561 316 3 46 [1]
126 0.020226752846638489501213877478 9.8878945877483375423557674736 0.809734713059295945033448216458 316 2 46 [1]
127 0.020155021079308652041673091027 9.9230856277953495938035212769 0.810382607124690624917438611403 324 1 46 [1]
128 0.020102874050706594697191685189 9.9488261974645467256255248641 0.812542619910173608659411616563 325 1 46 [1]
129 0.020001905161608479177554786547 9.9990475099281362798392328237 0.810685331217204632951612733744 323 3 48 [1]
130 0.019941052552694549098510509253 10.0295608504865434423553127785 0.812006275048028469870710267459 338 1 48 [1]
131 0.019859770160478376100973220711 10.0706100012177815079605894192 0.811595459660147689708690151077 287 36 D1 [1]
132 0.019789101289386861357901992012 10.1065731624337312462701932177 0.811981157086116531781968432379 334 2 48 [1]
133 0.019745036704881803256928217024 10.1291277898993011730095937866 0.814493098645371603847517468101 337 1 48 [1]
134 0.019690055618071636154123936171 10.1574116335374366688454546506 0.816053367292093632036359036621 333 5 51 [1]
135 0.019624651023863632144418726831 10.1912640258825201120616967607 0.816690550794239018361121857424 353 1 49 [1]
136 0.019559084400308558537495305988 10.2254275254748047939030954087 0.817251689138880262760241170863 346 3 50 [1]
137 0.019494770697918633233953144100 10.2591614489393854572155365275 0.817855741295820789522172825958 351 1 50 [1]
138 0.019474501536030627368840800306 10.2698392372185367263895644495 0.822113281149980182126309962245 365 50 C2 [1]
139 0.019402722688937781637879584706 10.3078317000339073601130209732 0.821977690730233081614210199675 358 1 51 [2]
140 0.019349767227858497843590697917 10.3360416507777626368829937155 0.823378272609746936455406060288 360 2 51 [1]
141 0.019299864778513010852351117849 10.3627669051165923208879272796 0.824987792176019074769633040947 357 2 51 [2]
142 0.019253496896377514551216946028 10.3877233874138145168262892284 0.826851388437849597977628544281 361 2 52 [2]
143 0.019214940160731447463968464624 10.4085674130138264121560623252 0.829342625238554798781450795446 366 2 52 [1]
144 0.019172069820366453445389934816 10.4318418341842454034052921147 0.831419819329887978101919786290 368 1 52 [1]
145 0.019124794027687288752685154276 10.4576289663803232149524644134 0.833069841044618045768371070157 366 3 52 [2]
146 0.019089088077777755809559749262 10.4771898576352985371771389450 0.835685941920299718837482988863 379 1 53 [2]
147 0.019061390214171943514413227395 10.4924141289181572724501809025 0.838969853939907685325013872022 378 1 53 [2]
148 0.018998643430656269386221250424 10.5270674051011129448490607949 0.839125224271665854217786839704 374 3 54 [1]
149 0.018980269061929650669912513182 10.5372584207015857882391849678 0.843161707595067971913386798950 387 1 54 [2]
150 0.018961762330914637085159860092 10.5475428132503560488245959258 0.847166031358726805012677029465 393 1 54 [1]
151 0.018897984682813392597327561906 10.5831390678334212137835926494 0.847086595901579493554582461579 316 2 53 [1]
152 0.018880546742273365892267891690 10.5929135808446631649590887160 0.851123531431317800401021965234 395 1 55 [1]
153 0.018869889939550259644952602946 10.5988959469663314436003040895 0.855756175872389258704818967967 405 55 D1 [1]
154 0.018807774894311212657423312895 10.6339001356558130911032599324 0.855687985539181630301045489858 392 3 56 [1]
155 0.018791120418197754850544438093 10.6433249082005446852372411847 0.859719794988328322077753565701 403 1 56 [1]
156 0.018784755819323387214698267312 10.6469310500307503955979986481 0.864680337859600223724649770386 313 56 C2 [1]
157 0.018631505479996644649543668273 10.7345056047524517073152481299 0.856082120103580186029586722804 171 42 29 [1]
158 0.018555601980318251752435181389 10.7784161468939715947051876822 0.854529501091329735705783683683 273 10 44 [1]
159 0.018529472433616946494800034439 10.7936154532468828860314833014 0.857517733626380830513989822450 328 6 45 [1]
160 0.018376963609636770120054985487 10.8831907298940937535130770464 0.848764819864678126277992345994 319 1 58 [2]
161 0.018270631498207221833658705895 10.9465291344540936326950802581 0.844214621710439537942879342959 394 1 58 [1]
162 0.018242069458436795322421020343 10.9636683741219817549062335716 0.846804391512915779690140783498 429 1 52 [2]
163 0.018102731366781063787019985970 11.0480565583050459494599157928 0.839065172610210525852003528356 377 4 50 [1]
164 0.018016918469994105566898354588 11.1006774178995012236641793795 0.836228089356072234032617294530 420 4 52 [1]
165 0.017992306331201388048750009579 11.1158623201723542385349316748 0.839030009631307812707477524185 437 2 52 D1 [1]
166 0.017900384796189156346517358394 11.1729441728302032586330682038 0.835512013608342050012678818248 422 52 [2]
167 0.017794178251731241958818268657 11.2396311406255289685944698236 0.830600564064023253478875677572 434 3 53 [1]
168 0.017769630920114218155094687843 11.2551577969811010931863084216 0.833270436338252020869688607677 448 1 53 [1]
169 0.017670959048961068095108886507 11.3180048375336276004456956292 0.828947111409059900262416426252 415 6 53 [1]
170 0.017597614131897344459899261797 11.3651770348504819907170062819 0.826944531383154930574685970116 437 3 53 [1]
171 0.017570284086487097500356357497 11.3828552239411662563086432713 0.829227228933372044232819108663 440 2 54 [1]
172 0.017488659220934484570886930808 11.4359824543092246750013792294 0.826344906565878821999666876161 441 4 56 [1]
173 0.017413818330824168591939370427 11.4851318763318185502883689278 0.824050825777798874392436304464 441 4 56 [1]
174 0.017389317949191233068268440270 11.5013136561403710578860820196 0.826483564981755079671188333661 448 3 54 [1]
175 0.017327936686768345534722978176 11.5420550995387976513260203992 0.825375608790782160074360758571 449 5 56 [1]
176 0.017257892104308169854179797154 11.5889008223705980195567180195 0.823394655566987139679429352889 449 5 56 [1]
177 0.017227550349753667028417565356 11.6093115933258471784301222473 0.825163860699290502490442807948 456 2 55 [1]
178 0.017168993872479793767858620284 11.6489062484075029192627775919 0.824194231518502495008152184956 444 4 55 [1]
179 0.017118136168072994223091779177 11.6835149595911995144343953209 0.823921548578756232261371201022 457 5 57 [1]
180 0.017084667304732212817538035832 11.7064029654591406552932395729 0.825287817389269102672549524456 471 3 56 [1]
181 0.017037719043471376413552420262 11.7386605266646474011252102162 0.825318075050851694258575655513 26 150 57 [1]
182 0.017002321952486092089707934491 11.7630992142667814905693945493 0.826433163198269718802503298594 419 6 56 [1]
183 0.016958986068433382988417319663 11.7931578688109248348217637512 0.826743394320708768166299839586 486 1 58 [1]
184 0.016926341702645105372235547282 11.8159023085741967105496864067 0.828064007436250787232314243720 452 6 58 [1]
185 0.016867959461749889103972291658 11.8567987108057655438524676208 0.826830907856349080187027877220 478 3 58 [1]
186 0.016851144243951518985002890329 11.8686302309581846126544991532 0.829643688210322852059853195983 494 2 58 C2 [1]
187 0.016799439702926256548799225343 11.9051589539121623589993182887 0.828993410051666439112153008925 483 3 59 [1]
188 0.016754126269586444381562096751 11.9373578055847350009140838773 0.828936560523479891131961789883 484 4 59 [1]
189 0.016735599450278746246069842140 11.9505728249649774383586088921 0.831503778410885644562142117129 505 1 59 [1]
190 0.016687805470685709619201234327 11.9847993405319763598610916815 0.831135696185447773712486787944 489 4 60 [1]
191 0.016664264048079061478220829904 12.0017301347943166889463873180 0.833154455589512593427971068185 481 6 61 [1]
192 0.016630409160404547470617339942 12.0261623193361555544842046353 0.834117003627923330862079281155 463 2 59 [1]
193 0.016598725533982748946071190444 12.0491178428450939395900190189 0.835269596595686483859691876001 493 5 60 [1]
194 0.016565012376738234107830373035 12.0736402395240098171062242506 0.836190321266246102622080882840 507 3 61 [1]
195 0.016534560573301617772030844863 12.0958763381314366380973541576 0.837413202002991135962203641079 508 3 61 [1]
196 0.016505011620721321446910667689 12.1175316077274815373761162937 0.838701881455222420805919964993 502 4 61 [1]
197 0.016473065030899144358258880128 12.1410314124816795009048328471 0.839720835077098071247416931105 512 3 62 [1]
198 0.016451009996530175800379926859 12.1573082772537201340620654728 0.841724948662685341437531240462 513 4 63 [2]
199 0.016437757485613613151929367439 12.1671097882445794833409166311 0.844613640572995064088624794543 518 2 62 [2]
200 0.016412404337172521282671549397 12.1859049954685302998130814215 0.846241438983481981175588015578 518 2 62 [2]
201 0.016367412125580736332143483202 12.2194027049284523891803508435 0.845816144182205000204986919335 517 5 63 [1]
202 0.016352971230128453120207638717 12.2301933505223315618456384469 0.848524901227830607742222296252 486 3 63 [1]
203 0.016335003007438524680796380123 12.2436463531059856199278650103 0.850852643469082364521193657399 503 1 63 [1]
204 0.016300788824294591946405099898 12.2693448860536903921860158885 0.851465953098424706745077718599 532 2 64 [1]
205 0.016283013314130158511121657663 12.2827388359648980509710738615 0.853774725350276045854077098535 425 10 68 [1]
206 0.016273117664787137595561954257 12.2902079441589365054245272217 0.856897008629124767843883027018 511 6 66 [1]
207 0.016263528253458056226661978142 12.2974545795420935564287291111 0.860042196071619075811980413668 551 2 64 D1 [1]
208 0.016214100185949151206700931970 12.3349429019389196564309731901 0.858952041708590788673766274067 541 4 65 [1]
209 0.016203997541366615653336750861 12.3426333217730397223368861243 0.862006420047583051771000811189 549 3 65 [1]
210 0.016197533588485788024643407773 12.3475588988543546246824681583 0.865439972255752560041872669407 562 1 65 [1]
211 0.016161527617815283384135782989 12.3750677986364766824020889835 0.865699466224168734958096190081 386 3 67 [1]
212 0.016145071184917156211072513779 12.3876815226953886281900031084 0.868031861317900267060624714658 451 1 67 D1 [1]
213 0.016134147011892969370338165664 12.3960690238271617822141969974 0.870946543986360342651537055321 371 2 41 [1]
214 0.016129033240963740118320822674 12.3999992443471226463338104120 0.874480892049830303853277753481 84 147 52 [1]
215 0.015952300606546573411517567313 12.5373765786436561908862445601 0.859419071205959744466843196026 477 8 64 [1]
216 0.015913440570239784739764443474 12.5679923909117528576778239482 0.859214903200468374024023774770 515 2 67 [1]
217 0.015891603043212421653537834491 12.5852627614822947018157404045 0.860825309534636617955804689667 585 1 61 [2]
218 0.015771122920346430310110032169 12.6814051865627572490309455816 0.851729332237320134456804582653 485 2 61 [1]
219 0.015722929737572856961970440006 12.7202756317140375672334327659 0.850415053499018680970802756911 421 2 55 [1]
220 0.015686317077668465837711019409 12.7499654004014930889716901085 0.850324195243458568050365161497 426 3 52 [1]
221 0.015682287475429829002194754889 12.7532415352893718660860006566 0.853750502281365982444318086341 582 2 60 [2]
222 0.015559743033148525320473202077 12.8536827101783990404162027832 0.844262874891620481367933020501 367 19 49 [1]
223 0.015517403922453149948743592611 12.8887538791593145615540274635 0.843456851208792833386909406189 427 12 56 [1]
224 0.015507959828895195431526141245 12.8966029191892887320506832427 0.846208201229594161899618550207 607 1 62 [2]
225 0.015414147582570061905741172194 12.9750931038285289007467326971 0.839733387215851758667067856875 431 15 57 [1]
226 0.015389196500966192504915604177 12.9961301090309189629066281907 0.840737088365913906662063789423 598 64 [1]
227 0.015356598653442342204768214781 13.0237173291735346729801243452 0.840883445089585094032685283209 603 2 63 [1]
228 0.015346914376732703074329206786 13.0319356119701765848080456365 0.843522874611190640095454829386 622 63 D1 [2]
229 0.015272592914550707564691259814 13.0953532984862932932240242546 0.839036610561781682331349305061 596 5 64 [1]
230 0.015222320161255296675073465158 13.1386015982669462589059476959 0.837161826641038944264208483502 612 3 64 [1]
231 0.015201874655402777754787190543 13.1562721396942702423699647704 0.838544571014172453755529774643 612 2 64 [1]
232 0.015142113599889771663744247424 13.2081957172383066416072545603 0.835566196652504196578102087456 607 4 65 [1]
233 0.015117805531553526486639687665 13.2294333051556144046527847728 0.836475657595386771213396398798 576 6 64 [1]
234 0.015076674297568630323818306803 13.2655250125190669704968446161 0.835500742380818244556290815277 623 64 [1]
235 0.015068153234936757636857744045 13.2730266862619556824336630662 0.838123070711082187180269256849 641 65 [1]
236 0.015016542614505362088761531017 13.3186449860175025871899345597 0.835933607529879650085234456925 617 6 67 [1]
237 0.014973694439890158212358313492 13.3567571318402788297085873797 0.834691817218458604353761809896 622 5 66 [1]
238 0.014950862594059463768230803946 13.3771545783229571951590745645 0.835659460765490945068666943992 627 2 66 [1]
239 0.014905189208923561018049600925 13.4181456670313422382567243215 0.834051302272508554913948109984 623 4 66 [1]
240 0.014879010241468044728855165942 13.4417543071915315411975446796 0.834601583079518836653760838463 637 3 67 [1]
241 0.014862284441798202704729078474 13.4568814628205163967604396989 0.836195945026853067351303397826 636 3 67 [1]
242 0.014836179634504955722389074246 13.4805593439199062220172454255 0.836718572681878468370374915344 637 67 [1]
243 0.014809157149471654085913467510 13.5051575171606167197224206396 0.837118296261802455831201453678 629 4 67 [1]
244 0.014767514027629843301439000956 13.5432409020098017908653944627 0.835842572369863032839809885432 649 67 [1]
245 0.014753543757896975820203890400 13.5560651245534200211588081854 0.837680989527493191859539710813 662 1 68 [1]
246 0.014730251355284468986647723026 13.5775008298314012101750209965 0.838446390519674576968175243659 579 5 66 [1]
247 0.014694705450223536505121284924 13.6103442615760352794461598215 0.837796613888470287618176207542 652 2 68 [1]
248 0.014674608405075370061658536002 13.6289837847276295105880810860 0.838889192810941396040874738932 658 1 68 [1]
249 0.014649091515656425591467869864 13.6527237737744450647356517370 0.839345194660910211556591845294 649 3 69 [1]
250 0.014634379134311154385152716173 13.6664492674710298632993761936 0.841024195063422296468457850543 528 11 62 [1]
251 0.014600230402988223793220518672 13.6984139619513178004035415345 0.840452197768251295268028946105 647 3 69 [1]
252 0.014588637635141936384049837876 13.7092993192338039470929013764 0.842461168131950293817949047372 684 1 69 [1]
253 0.014552269390974786019006708181 13.7435608582149102878890747265 0.841592486845076384832011071610 668 4 70 [1]
254 0.014532613401094451547740273609 13.7621496203110993132052421193 0.842637989014679713897624648301 669 3 69 [1]
255 0.014513418548511107535018076549 13.7803508753985090978574371014 0.843722240688971905529127903972 681 1 70 [1]
256 0.014509007894035429903235140881 13.7845400223552745611916493112 0.846516205069960399325204068404 699 70 C2 [1]
257 0.014463641995490234585823322848 13.8277758853793550367791249032 0.844516866353316474203631629199 652 2 71 [1]
258 0.014445626927518965109263661803 13.8450204344540675993242002684 0.845692291816090049014281684397 524 44 72 [1]
259 0.014435160581007234983049919587 13.8550588944016236215185728832 0.847740399236298212441532702184 524 43 72 [1]
260 0.014413174289362975687017575021 13.8761938199554980081329553128 0.848423133387342664962452960584 619 7 71 [1]
261 0.014387579303850699025408068254 13.9008790691059404632632217697 0.848664127658030921612750354613 594 10 70 [1]
262 0.014371863768082800219595407348 13.9160795862929269003872302923 0.850055637745041645206410617237 551 12 67 [1]
263 0.014366690561041046744812733409 13.9210905357947303079845512587 0.852685938218558960590476309803 336 68 63 [1]
264 0.014350281438593147706317960993 13.9370088911376300453633434738 0.853973985486785965858619045605 689 2 72 [1]
265 0.014320797267990881240711127863 13.9657029044765435529441658780 0.853689902081884290690593799021 691 6 73 [1]
266 0.014313369385927822521587072972 13.9729503660144366460359795408 0.856022681838515205015529925028 438 44 71 [1]
267 0.014303340044255015949052755592 13.9827480421491247766678475536 0.858037098367476432803421727304 527 39 71 [1]
268 0.014292030283053914975780604014 13.9938130579768184435983733659 0.859889263765055670480049945888 672 1 73 [1]
269 0.014262199526132351999241974542 14.0230824588832790531408902512 0.859498598330525844825327455489 597 10 73 [1]
270 0.014246778156412688340224152772 14.0382616900633777990790969154 0.860829149438162762316173905515 521 3 73 [1]
271 0.014244878888492294503185880197 14.0401334097385496492533803402 0.863787052952327236855790903992 310 51 58 [1]
272 0.014238001658633590609014677856 14.0469150654105090657609967063 0.866137534651738444338322605773 725 1 74 [2]
273 0.014209463222242525918136550078 14.0751270383622670593328771890 0.865840445035959475404519608955 666 3 75 [1]
274 0.014195842152846120803722814064 14.0886322802555289452245671356 0.867346765370856160423660736598 514 2 74 [1]
275 0.014191231430249436539894248469 14.0932096684497969986455018983 0.869946882240718867042682298351 685 2 75 [1]
276 0.014188659811412899725474884561 14.0957639874575369730964380536 0.872793918302347383772448391343 553 75 D1 [1]
277 0.014160843031753967781160959437 14.1234529294283066848626299824 0.872524968410982040942736426148 433 12 76 [1]
278 0.014116166064385175597376198196 14.1681529593645316880539225842 0.870158146758657611321257858953 777 1 76 [1]
279 0.014098552326307416566690758670 14.1858536515700862870888182402 0.871110244838859626593903680727 720 1 76 [1]
280 0.014088086615541166581914287747 14.1963919911857659366480711879 0.872935055876175036827774541766 595 76 C2 [1]
281 0.013983608467831717954930832653 14.3024599451626213258398716911 0.863107137252211926137270114186 495 6 57 [1]
282 0.013941395746630152069637111937 14.3457659214891055366533520878 0.860957069065790524832492947229 648 16 64 [1]
283 0.013919134422206143447868374666 14.3687095715468035205890639273 0.861253045331061174849826597818 641 12 65 [1]
284 0.013913227454569556067358998519 14.3748099176164549889410039401 0.863562922104656904301388722654 776 2 70 D1 [2]
285 0.013888890503232114628545060033 14.3999983262491380977267968825 0.863574570227555048773634355710 129 178 60 [1]
286 0.013791290771027291139314549267 14.5019058274196854338615130272 0.854467876817118395973892796753 751 4 72 [1]
287 0.013769254078541286196578363811 14.5251150758914593184332159842 0.854717510721256648237336481340 776 2 71 [1]
288 0.013762144311223339911835162742 14.5326190074097302606138251072 0.856810105308370738202381353084 790 1 71 [2]
289 0.013687800076955091270090952378 14.6115518107779699297124884181 0.850520971132049081770621148327 731 8 74 [1]
290 0.013657595543380449787546093650 14.6438660717944460411769534583 0.849701470102479962781924972470 773 4 73 [1]
291 0.013636245931610259492524716776 14.6667932657608395684684616870 0.849967884330770617471586846031 779 4 72 [1]
292 0.013618614715406392253028436848 14.6857814968320595572424995810 0.850684647987998875593358880390 520 30 66 [1]
293 0.013567581001326075240508572286 14.7410212609345981930061206339 0.847212477992727502417445107308 489 69 64 [1]
294 0.013536582566144829517507407268 14.7747778305731774794479504387 0.846223887537236169923659987235 511 53 65 [1]
295 0.013513784049919584181209925764 14.7997037144596172744116606743 0.846244467200001054469821658300 515 60 64 [1]
296 0.013494426217611419450977107745 14.8209339748719597364793849226 0.846682209170764492151185133762 478 45 62 [1]
297 0.013455789208680202683639836307 14.8634908661456948727634603844 0.844684794800813029366520267575 500 57 65 [1]
298 0.013426950715053645052692164867 14.8954147702180594985359638575 0.843899890833342391774088301843 561 36 68 [1]
299 0.013398721675631212469921340433 14.9267971110817196318468477164 0.843175147076241465241974063069 578 29 68 [1]
300 0.013396433701093527841692147654 14.9293464561149840940108014029 0.845706230142844946798684455272 427 42 60 [1]
301 0.013352345561474024572363782900 14.9786416985092703124861716109 0.842949387674881917287018198649 613 17 64 [1]
302 0.013333559467980713870438025355 14.9997456028362978638865707216 0.843371701170239922522663651053 600 20 65 [1]
303 0.013302442233656677545616471048 15.0348331898016040527291364960 0.842219454206847344225877774858 715 9 72 [1]
304 0.013295482255178390257042863315 15.0427036914815940402950783818 0.844115062795944628193077444102 654 11 66 [1]
305 0.013264867245188762869062269553 15.0774219072973423174817636335 0.842996035535006329430052868329 717 10 74 [1]
306 0.013236924939678425089883264991 15.1092493846881899379277161947 0.842200540667445163638876204338 720 7 74 [1]
307 0.013214968824599887324485606512 15.1343527672722637980110384728 0.842152104676292815318502968845 723 8 74 [1]
308 0.013204128112838473908519884548 15.1467782113942444359410227600 0.843509644071054162417598973119 735 4 73 [1]
309 0.013191827033906840590059225147 15.1609022378736246782910995419 0.844672301838885797282779256846 802 6 76 [1]
310 0.013163055632690517950300429198 15.1940404706107104233388932925 0.843713512273247345215403921228 818 4 77 [1]
311 0.013146350589477126566525979471 15.2133475095429244327606318760 0.844288134793198393144482315699 822 5 77 [1]
312 0.013127887079755592926432455927 15.2347440821926599948846580240 0.844625396786184972920574717415 826 4 76 [1]
313 0.013107836875915356707965691291 15.2580476773772364868595002750 0.844746246382425042141212384748 821 8 79 [1]
314 0.013077028645298377887078567743 15.2939941805439556970105102486 0.843466183149608741126875107468 842 3 77 [1]
315 0.013067342446435727658765583117 15.3053308903335863495225637133 0.844899349620822798493788084605 813 2 76 [1]
316 0.013056280709957701444835647229 15.3182981005811985086548766538 0.846147191571937217809390673087 864 1 78 [2]
317 0.013037610856048166177261148709 15.3402338977788799539279714393 0.846399050659474552174071295580 757 3 75 [1]
318 0.013016578896678743575622719033 15.3650203780527287889212935851 0.846331893255295679676422763773 842 4 77 [1]
319 0.013000246502312497746176236635 15.3843236714337521356509449392 0.846864122819519224282375549204 850 4 78 [1]
320 0.012996316600289142844450667479 15.3889756729649374501031156168 0.849005337282809272090640630845 876 2 78 C2 [2]
321 0.012967574969175000855752508303 15.4230841522348284698022289086 0.847895722525404017504586795514 847 6 79 [1]
322 0.012946031142124085031759012434 15.4487501076090831291668060906 0.847713390843541942320907807611 863 2 79 [1]
323 0.012934028411716011079636179332 15.4630864904266258880944723778 0.848769999359848180094917047467 870 3 79 [1]
324 0.012930375635174747963957347494 15.4674547470945992958651436397 0.850916941460300878662179444629 890 1 79 [2]
325 0.012893483450896425878002116362 15.5117118474367685396800710445 0.848679618229476273102787708248 846 2 80 [1]
326 0.012891477340906122388411903610 15.5141257057775120257364057094 0.851026054322031991628233974198 750 6 77 [1]
327 0.012874850401875841561183981462 15.5341610781637023938898707370 0.851436007510869212824282122138 816 6 78 [1]
328 0.012867280056315405690673400041 15.5433004585796476724729112119 0.853035740653589313029057209323 772 7 77 [1]
329 0.012848638325300313205203015637 15.5658517997334452290536335566 0.853159014209335342310570913013 777 3 80 [1]
330 0.012828078570567147057323019750 15.5907994248555435831634253608 0.853015729250980170669571258930 747 9 78 [1]
331 0.012816403613032198651055285573 15.6050016867939862683200735176 0.854043953397458491045913803978 739 13 80 [1]
332 0.012791440372612162416486721064 15.6354557558835455924374547724 0.853290405002647415125554809019 870 7 82 [1]
333 0.012787122106035006136100096453 15.6407359170839592410928129457 0.855282793869298183190746969755 695 44 77 [1]
334 0.012772220728125321547950533468 15.6589839979500228917469010664 0.855852994557591535199033500672 871 4 82 [1]
335 0.012760416145141007110748621968 15.6734700283389487589394603367 0.856829400117316315171274857521 751 5 78 [1]
336 0.012754744298248867829149188700 15.6804397895658733486799830646 0.858623295769171348786673471182 809 8 81 [1]
337 0.012747208925775783003805939635 15.6897091092298222717331256909 0.860161471662529276525189671020 648 11 81 [1]
338 0.012726757664537019866642427832 15.7149216848292755276051734807 0.859947872758118205888547166651 777 9 83 [1]
339 0.012714977011556303669697939035 15.7294818400556549405726604360 0.860896087051604373201986463312 780 6 80 [1]
340 0.012710713912952269611887098499 15.7347574156475333517571564716 0.862856712349098262740448901150 506 52 80 [1]
341 0.012709908506956195695178200968 15.7357545013435000163355500773 0.865284859377821535090110888302 508 37 82 [1]
342 0.012702672865517648921908910033 15.7447178335919275669452697607 0.866834545309022193160422923470 443 38 78 [1]
343 0.012682324965230263197358600471 15.7699791283000572587550971728 0.866586165121116731406219742661 911 4 84 [1]
344 0.012673283247371042691716542704 15.7812301750209978461836143512 0.867873849389701892543703941091 583 23 85 [1]
345 0.012665895664152951449938995963 15.7904348261797611234584845260 0.869382280896077383642529900325 574 13 79 [1]
346 0.012664325969963665018044443512 15.7923919894628088349791746543 0.871686131900221648574647764396 549 12 76 [1]
347 0.012661545917361545708999024096 15.7958594712956388505589675185 0.873821689737540379403089167453 789 84 D1 [2]





Updates

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

15-Jan-2002: First complete presentation from N=1 to N=200.
15-Sep-2009: Reorganization of the page, all numerical values are now given with 30 decimal places.
30-Jul-2010: More better packings for N=41, 63, 75, 79, 93, 95, 96, 97, 139, 141, 142, 145, 146, 147, 149, 160, 166, 198, 199 and 200 by David W. Cantrell [2].
09-Aug-2010: Some better packings for N=83, 84, 85, 87, 105, 106, 131, 172, 173, 178, 181 and 182 and new packings for N=201–347 by Eckard Specht [1].
11-Aug-2010: Some improvements for N=224, 228, 284, 288 and 347 by David W. Cantrell [2].
17-Aug-2010: Further improvements for N=75, 162, 217, 221, 272, 316, 320 and 324 by David W. Cantrell [2].
18-Aug-2010: Improvements as a direct consequence of the latter for N=226, 227, 286, 287, 309, 310, 311, 312, 313, 314, 315, 317, 318, 319, 321, 322 and 323 by Eckard Specht [1].

References

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


©  E. Specht     08-Nov-2011