The best known packings of equal circles in a 1x0.10000 rectangle (complete up to N = 200)

Last update: 26-Jan-2011


Overview    Download    Results    History of updates    References

Overview

1-16   17-32   33-48   49-64   65-80   81-96   97-112   113-128   129-144   145-160   161-176   177-192   193-200  


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
= 0.1/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 0.050000000000000000000000000000 2.0000000000000000000000000000 0.078539816339744830961566084582 2 1 D2
2 0.050000000000000000000000000000 2.0000000000000000000000000000 0.157079632679489661923132169164 4 2 D2
3 0.050000000000000000000000000000 2.0000000000000000000000000000 0.235619449019234492884698253746 6 3 D2
4 0.050000000000000000000000000000 2.0000000000000000000000000000 0.314159265358979323846264338328 8 4 D2
5 0.050000000000000000000000000000 2.0000000000000000000000000000 0.392699081698724154807830422910 10 5 D2
6 0.050000000000000000000000000000 2.0000000000000000000000000000 0.471238898038468985769396507492 12 6 D2
7 0.050000000000000000000000000000 2.0000000000000000000000000000 0.549778714378213816730962592074 14 7 D2
8 0.050000000000000000000000000000 2.0000000000000000000000000000 0.628318530717958647692528676656 16 8 D2
9 0.050000000000000000000000000000 2.0000000000000000000000000000 0.706858347057703478654094761238 18 9 D2
10 0.050000000000000000000000000000 2.0000000000000000000000000000 0.785398163397448309615660845820 31 10 D2
11 0.045643942682142794248922756314 2.1908712114635714411502154487 0.719960815984392515367507517837 23 11 D1 [1]
12 0.042312238590422353175938879282 2.3633824002551271758957576689 0.674936825618705729264994327240 25 12 C2 [1]
13 0.039712799117777202973428794132 2.5180798689970857044305854436 0.644101370045354260424742480349 27 13 D1 [1]
14 0.037649019478642637583367116190 2.6561116699660010947831424362 0.623426484981587444597298097611 29 14 C2 [1]
15 0.035985190969373898484256267118 2.7789209201392737975021275314 0.610223256632352833136454375382 31 15 D1 [1]
16 0.034625486140018408805415896361 2.8880460940135361958393334282 0.602645078885349368685649532565 33 16 C2 [1]
17 0.033500838578400603540269053317 2.9849999057777077973550259491 0.599390907442211052484317975936 35 17 D1 [1]
18 0.032560541433392497309139877392 3.0712020008808850902509881874 0.599522675234264771491877868919 37 18 C2 [1]
19 0.031766740293079279839820752073 3.1479465339346151622792621932 0.602349811310615678399215787725 39 19 D1 [1]
20 0.031090744862968577007014920191 3.2163912585802196462466798334 0.607354316086856738203044352895 41 20 C2 [1]
21 0.030510509541279720020277464557 3.2775591592366976223983778028 0.614140995625614934825068290187 43 21 D1 [1]
22 0.030008880996422559890264688476 3.3323468479854770750543242375 0.622403686168412751584361942584 45 22 C2 [1]
23 0.029572358793615755933758166187 3.3815361398086564550730302625 0.631901872334814162504195906595 47 23 D1 [1]
24 0.029190205344862249065173969274 3.4258066642070039905204976808 0.642444203005302372526928557141 49 24 C2 [1]
25 0.028853797638876881520527924879 3.4657482959976995928094362174 0.653876673582603849110332366773 51 25 D1 [1]
26 0.028556148876199049515568134776 3.5018727641999338455439686547 0.666074021728276797166782324775 53 26 C2 [1]
27 0.028291551193424638003773960773 3.5346241468456997609276405174 0.678933372764814113852105732515 55 27 D1 [1]
28 0.028055305821173471855154755560 3.5643881637721991087162772132 0.692369484227167780782759389759 57 28 C2 [1]
29 0.027843517146688891162356398880 3.5915002933417787832053232399 0.706311143376157990464449911530 59 29 D1 [1]
30 0.027652934015977579434775963773 3.6162527977038904323300839123 0.720698407008148423106557962711 61 30 C2 [1]
31 0.027480826327644507247191016635 3.6389007669468942197101123593 0.735480464205170790200973950106 63 31 D1 [1]
32 0.027324888254563587920931388531 3.6596672992175470843385134255 0.750613965102698803606285203928 65 32 C2 [1]
33 0.027183161743543284673001225005 3.6787479301870625165596796717 0.766061702033233794980990318672 67 33 D1 [1]
34 0.027053975592314372536891208358 3.6963144162963056560963198921 0.781791559798297366419224890115 69 34 C2 [1]
35 0.026935896591124938045092682779 3.7125179650767157663042964113 0.797775673426925525890687666250 71 35 D1 [1]
36 0.026827690080607782519907373298 3.7274919942617175009050594000 0.813989747313117107516671122107 73 36 C2 [1]
37 0.026328369891951368505581575102 3.7981842556295209686135549214 0.805748560645876759742338959291 71 2 22 D1 [1]
38 0.025834884216759061025925436657 3.8707353654455362256003633136 0.796794831565577547986247819703 95 38 C2 [1]
39 0.025139633102022970412820515026 3.9777827939721627480381880457 0.774341169329474287432446148465 81 24 D1 [1]
40 0.025000000000000000000000000000 4.0000000000000000000000000000 0.785398163397448309615660845820 102 40 D2
41 0.024206181921367144980397128443 4.1311760906716379558838595150 0.754720792986998893730575820942 84 24 D1 [1]
42 0.023905223877975357228585192547 4.1831860898041276774775347001 0.754023347854767152475669021013 86 23 D1 [1]
43 0.023341132822376655414916737171 4.2842822051949463821264528928 0.735973518838247209893773894600 97 30 [1]
44 0.023045900107062934224034898706 4.3391665995008262541499293245 0.734158614956579686372116377376 90 24 C2 [1]
45 0.022652281115903905716781996299 4.4145664398359930100098575083 0.725414565842127451267461442350 99 1 31 [1]
46 0.022349337865046844349370566234 4.4744054881551811640701275754 0.721833487917812870808838634642 100 2 32 C2 [1]
47 0.022065140449831294478652996947 4.5320355076536383167722749164 0.718887815853580747338602738424 105 1 33 [1]
48 0.021802524111653096638341745858 4.5866249012219469021128607949 0.716810999502163930234363775152 105 2 33 [1]
49 0.021559612860383980147765647875 4.6383022110638671288095645815 0.715530035867038902686623388473 110 1 34 [1]
50 0.021334674870096596907045031346 4.6872052472739290669315973086 0.714976775100725462311240639149 110 2 34 C2 [1]
51 0.021121768287917630599209164907 4.7344520892790685073946037109 0.714793503551296829051175203815 112 2 35 [1]
52 0.020926705878604717441097867673 4.7785829542450411432038118808 0.715409918848456798606155924140 117 1 36 [1]
53 0.020745366554348011345542420152 4.8203534865495566937069890298 0.716585420817425246739769247601 117 2 36 D1 [1]
54 0.020568322012919738011897602716 4.8618453142257415206661701758 0.717697375879678444945987767740 119 2 37 [1]
55 0.020409946124026912175768429510 4.8995719730136090122033967789 0.719774205504370863099280494022 124 1 38 [1]
56 0.020262091292948954567326441018 4.9353247181745350645441754148 0.722281406814296066679014516352 124 2 38 C2 [1]
57 0.020113279994291099443690378754 4.9718395024771562520669919499 0.724420158851887361780249049926 126 2 39 [1]
58 0.019983322528060620669425972499 5.0041728476122928986864054551 0.727634465733667913583901332803 131 1 40 [1]
59 0.019861435123634015202527181682 5.0348828963021660547505210156 0.731178034841515497031428213961 131 2 40 D1 [1]
60 0.019735625997310823653651689333 5.0669788743273712086096360007 0.734180653027229416339697793501 133 2 41 [1]
61 0.019627834643297516607135083297 5.0948055054125824974740442231 0.738285754853791620628579749645 138 1 42 [1]
62 0.019526211384342412188488953021 5.1213211836981103331035768656 0.742638628256881549773674593369 138 2 42 C2 [1]
63 0.019419727844809211699156273774 5.1494027516317361553294119976 0.746408713456313217935110389161 145 44 D1 [2]
64 0.019329293344718676707595351906 5.1734948720886839337609810752 0.751210761433571484166465495006 145 1 44 [2]
65 0.019243525572863601058057120052 5.1965529716142937217629314665 0.756192754782252745410512183162 145 2 44 D1 [1]
66 0.019135777962237830648577554978 5.2258131442232472048456909904 0.759252192015490101851763621268 150 1 45 [1]
67 0.019064518820557297182452668467 5.2453461291753065803955911680 0.765026311577335605826702818725 152 1 46 [1]
68 0.019003524769436192583731117527 5.2621816854119772360519600705 0.771484327535963827968371092170 152 2 46 C2 [1]
69 0.018908734403404868456016418556 5.2885612472293832966207312573 0.775039588959309581233775888897 157 1 47 [1]
70 0.018840914067366362193182289851 5.3075981155928222467934482553 0.780641886448490254683197349309 159 1 48 [1]
71 0.018798561705407300270151648882 5.3195559089627353722694114888 0.788238178098726446887317686003 159 2 48 D1 [1]
72 0.018714741590524303135161036449 5.3433812866875095569529007761 0.792227728266246882184460077377 164 1 49 [1]
73 0.018651176554292729982853590093 5.3615920533969817060443422709 0.797783774333468021183681257132 166 1 50 [1]
74 0.018622620163839183743350386645 5.3698136524406401639633147638 0.806237813044634823343335131648 166 2 50 C2 [1]
75 0.018548177252985653525789734844 5.3913653420528559515291083831 0.810613087295528403461181321066 171 1 51 [1]
76 0.018487515307611263027733770529 5.4090556971076721357819056720 0.816057119757124042765574870305 153 52 D1 [2]
77 0.018472689307519436147072696419 5.4133969524022854256338887220 0.825469154741981550802767749242 156 52 D2 [1]
78 0.018404772939680528148457200140 5.4333731976883497371518521012 0.830052196610624655986216944239 157 53 D1 [2]
79 0.018350558157125400657541681785 5.4494255239408431720365678516 0.835748336581824231463629022530 159 54 D1 [1]
80 0.018340582007022506853433157438 5.4523896767131248140159283569 0.845407479834528834153574664350 162 54 D2 [1]
81 0.018191451931196074069275438963 5.4970873341073153352818599801 0.842111538348173788116886015875 189 55 D1 [1]
82 0.017920937409838304974430679122 5.5800652450859862019060897613 0.827342189173360166239497794087 170 1 37 [1]
83 0.017873831044412848954159341623 5.5947714707339607125776612405 0.833035025171651880861722723058 196 2 42 D1 [2]
84 0.017610119234363481581219074743 5.6785532607221159307771210803 0.818377634008918890578667480522 196 57 D1 [1]
85 0.017386101597307540853683259440 5.7517206741438962335385076987 0.807185263767653538291312390562 206 1 45 [1]
86 0.017338300365725516593721064059 5.7675780146063657402109086186 0.812196976762282818772519758429 214 45 D1 [1]
87 0.017106893009055824348708259907 5.8455968566041362188465080565 0.799855257342683685262533174846 212 1 46 [1]
88 0.016925817276761866746190410694 5.9081342049753786445874153916 0.792012144675916196735318132695 212 1 46 [1]
88 0.016925951389697573000232842887 5.9080873918182016676741588556 0.792024695856508294735312263061 212 1 47 [1]
89 0.016879890649629974636688351991 5.9242089937467756987658734238 0.796671235901430678564015424904 212 2 46 [1]
90 0.016690752816848198219679622654 5.9913414989320727836913389259 0.787669869885998711385271737621 219 1 48 [1]
91 0.016568554989508606121533283341 6.0355293544501083964524390558 0.784802778624656822497559310118 218 1 48 [2]
92 0.016494712088120664351829460321 6.0625489833204820990079036035 0.786370438321437804525298269518 229 48 C2 [1]
93 0.016329322717461600015977310368 6.1239527033822405491916927703 0.779056876791610494490468775048 225 1 49 [1]
94 0.016232256714478737814121228512 6.1605728494179543703546592562 0.778100207729797321498986586307 225 2 49 [1]
95 0.016150028511289405172634315923 6.1919395331157889787131769327 0.778430895880561213750734336251 227 2 49 [1]
96 0.016015760496033866473673986188 6.2438496145571071251741496400 0.773599600841444843190575037538 234 1 51 [1]
97 0.015919736977724074240241815368 6.2815108151552044218767841912 0.772313067696470623755192689615 231 2 51 [1]
98 0.015862723862605694540156343922 6.3040875492850868736083679425 0.774696290871947883727808583793 244 51 D1 [1]
99 0.015740937755705240168093954846 6.3528616624988176743642610500 0.770630636564841513493653818423 242 1 52 [1]
100 0.015663144827503088568054839358 6.3844139284474276373150470906 0.770739824730677601906472383783 240 2 52 [1]
101 0.015594967615502998828958195126 6.4123249541467295991665823510 0.771685253080844027360547641902 242 2 52 [1]
102 0.015496556090249976480348727142 6.4530466909946111840039916164 0.769520917518716788295747820355 249 1 54 [1]
103 0.015416585611667828117601680271 6.4865205901569017119306273172 0.769065793759835936691667255795 244 3 54 [1]
104 0.015374839581184214234903763068 6.5041329031088149558717284107 0.772332656017749942107057263052 259 54 C2 [1]
105 0.015283595872613060033382602075 6.5429628494163294883223735013 0.770531260834312429443527033754 257 1 55 [1]
106 0.015218520931467203180526677929 6.5709407931509868987798133096 0.771259683590994021842197901070 255 2 55 [1]
107 0.015163324506615832861467460120 6.5948598512397139215345483817 0.772898579271239733146843044471 264 56 [1]
108 0.015088939675490851115769237786 6.6273709187419723440519954458 0.772486810010903142102929150346 264 1 57 [1]
109 0.015022871389919673503524215028 6.6565170801568510729146448327 0.772826969275640753549111177502 266 1 57 [1]
110 0.014990593852533876799465995172 6.6708497997960826921859621549 0.776569328148554286396931526651 274 57 D1 [1]
111 0.014921709409507949793567743176 6.7016450498815568229343994520 0.776443766645352498018762967408 272 1 58 [1]
112 0.014866428673912000560461805318 6.7265650811941556661964365406 0.777644667798884005130193708254 270 2 58 [1]
113 0.014820598259898961742632627594 6.7473659461221872494785603729 0.779757905161574281074235724041 277 1 59 [1]
114 0.014764453337248310516219994554 6.7730242167325146296659412184 0.780709505619895316801165387176 279 1 60 [1]
115 0.014712407178893779024184903225 6.7969842585283155926812137950 0.782015182536532625403963220341 281 1 60 [1]
116 0.014683642209334296325030617839 6.8102994185210156622681210468 0.785733824655424719183643529949 289 60 C2 [1]
117 0.014631743937262938163570603971 6.8344553068160326753960759703 0.786915172062109019535549156484 287 1 61 [1]
118 0.014584274364531196943982563135 6.8567004089829080849115797967 0.788499698586769284475798665590 285 2 61 [1]
119 0.014542229707683455511455566718 6.8765245777382081465483729065 0.790603686280801197293902890092 292 1 62 [1]
120 0.014503642710985073595858162047 6.8948195975801237361161127025 0.793022124734581687537045442750 294 1 63 [1]
121 0.014461008156356117735218101973 6.9151471957400501338845413518 0.794936402637264876850314421310 297 1 63 [2]
122 0.014435877138738034259914683157 6.9271855834554346229032588536 0.798722755805907625651634588953 304 63 D1 [1]
123 0.014392960742984780826761780553 6.9478408081353675649979213424 0.800488810957859697913931686314 302 1 64 [2]
124 0.014356170989619735717589378037 6.9656456496864827495694728402 0.802876603875274876847471066943 300 2 64 [1]
125 0.014317183179559742666597018283 6.9846141343478310929620131889 0.804961387842007314360052163727 307 1 65 [1]
126 0.014291442804238161363434599840 6.9971941510583355584317908909 0.808486121003446628375333600989 310 1 65 [1]
127 0.014251652043788668516836390564 7.0167303897644089885575432502 0.810371230638240395349682300617 310 1 66 [1]
128 0.014234360386397769986695236205 7.0252541937577415452700433542 0.814771363497207610140187593147 319 66 C2 [1]
129 0.014193221776619118052870388225 7.0456166734978193385085092096 0.816397301987455520231056805418 317 1 67 [1]
130 0.014170593360145863978638682100 7.0568675184234245239139849289 0.820104691569225940459364758905 315 2 67 [1]
131 0.014134113249751625688428689289 7.0750812755626723481311393621 0.822163707432912794527151913826 322 1 68 [1]
132 0.014117997301952982969265377973 7.0831576080671664261348408781 0.826551642093151236031234170043 325 1 68 [1]
133 0.014084876149575906951082799971 7.0998139378748446957169854191 0.828910381142484105768667962120 318 2 68 [1]
134 0.014069571531572616742152669805 7.1075369833115712760141845005 0.833328844762219210652771664518 334 69 D1 [1]
135 0.014031684981614438213550142217 7.1267278399585577239490764437 0.835032335338574704660154802068 332 1 70 [1]
136 0.014018981859046358000350681938 7.1331856339817323518428554298 0.839695312074222637408979013547 330 2 70 [1]
137 0.013984567422517190001387017403 7.1507395959195304009359995639 0.841721681425729424551761521737 337 1 71 [1]
138 0.013975543239016882088764959512 7.1553569181354097794268268215 0.846771739988437936775358755190 340 1 71 [1]
139 0.013940590820770713708985018621 7.1732971210234145277211779032 0.848646907952850706981510196507 338 2 72 [1]
140 0.013934340491689008324039989029 7.1765147449672235516152971843 0.853985988850810513584198949756 349 72 C2 [1]
141 0.013900263494566328941524683296 7.1941082296094903293572949935 0.855884282527759284702718366649 347 1 73 [1]
142 0.013888900144003108871552080608 7.1999941653535165730949030929 0.860545678685477374966565562906 162 33 50 [1]
143 0.013744176153003955133290110479 7.2758089598657971453729483270 0.848639682408405185915477317377 386 1 74 [1]
144 0.013709677419354838709677419355 7.2941176470588235294117647058 0.850289541309786910847173302820 324 74 C2 [1]
145 0.013564814637607612112655411392 7.3720137481831979088872579605 0.838196034751164784522394686758 346 4 59 [1]
146 0.013537532365309930395605680409 7.3868706128637635407542132861 0.840585209962836475484678906540 365 2 61 [1]
147 0.013452371521141408916852044203 7.4336335301803487809145647545 0.835727923870831793605336212958 397 1 76 [1]
148 0.013384292765191927437868513169 7.4714444576456501927143228775 0.832918375733336318226575300176 376 2 62 C2 [1]
149 0.013269221056371605017323853338 7.5362374004600723693185387959 0.824189352191142721272932830288 373 3 63 [1]
150 0.013241342247559432765836443228 7.5521044717676879564665202363 0.826237978118117898529584939525 384 1 63 [1]
151 0.013169107498349052320282247419 7.5935290233249686147620686405 0.822696226991946750001414173384 345 62 D1 [2]
152 0.013108620191771324906002749531 7.6285679604000585186387019455 0.820554481031746431743837578412 379 3 64 [1]
153 0.013011001645917467326097465580 7.6858033471525647904240953915 0.813697098023693304159706590067 382 3 64 [1]
154 0.012987295376052527304559914571 7.6998325751789345111498467239 0.816033578280591290959562389173 394 66 D1 [2]
155 0.012900628235157665249195271046 7.7515604804015072282412207207 0.810407200443541487690749961366 389 3 65 [1]
156 0.012872736228055041277622283733 7.7683561776134623793333268117 0.812112530888177422193352833378 398 1 66 [1]
157 0.012792339706490209293110560688 7.8171782718735085602819476107 0.807141156346809872252297285759 394 3 66 [1]
158 0.012766039529864515344585894886 7.8332829665819850115636089766 0.808945622853396196326164459375 402 2 66 D1 [1]
159 0.012688842721739491639534489048 7.8809393569574619076761779548 0.804249918720029910551730309471 399 3 67 [1]
160 0.012662944319262117909426268597 7.8970575467102028078685879413 0.806007809585391487979025093620 410 1 67 [1]
161 0.012604513930292252014770281151 7.9336657131752934014814977487 0.803577843751888677868169494552 405 2 68 [1]
162 0.012566989295020834196649260051 7.9573553897766899381289990253 0.803761828990032146859191931436 405 3 68 [1]
163 0.012513585318257212263541073831 7.9913148355732124814904186665 0.801864515153097651830756816007 408 3 69 [1]
164 0.012480795168494501091952046526 8.0123100050894052100370935005 0.802561333689392329620200345445 394 8 70 D1 [2]
165 0.012432424782923741945196355711 8.0434832099167488993795285539 0.801208407095431885721008634433 412 2 69 [1]
166 0.012388944882544956271142748114 8.0717123974691414492111488205 0.800435980349760553172811415289 423 1 69 [1]
167 0.012344097421758699380358446209 8.1010378145373313017253272908 0.799438436501193644205131442321 417 3 70 [1]
168 0.012318873184638649087924902383 8.1176255734735296250116237619 0.800942102183611483275141373555 428 2 70 C2 [1]
169 0.012272116541646642210725757760 8.1485536468497591051016760543 0.799605033030327005595354473979 422 3 71 [1]
170 0.012240124265517043379387612989 8.1698516968263665147992083106 0.800148228236143312600785967259 436 1 71 [1]
171 0.012207980985989697832763611868 8.1913626925503460796717663111 0.800633340783767274742597110049 424 5 72 [1]
172 0.012167771090766040607278995700 8.2184320574446595145334811646 0.800019147249132697230371626506 431 3 72 [1]
173 0.012129382441543674295802541789 8.2444428215484040367319761664 0.799601048859428767481431069471 434 3 73 [1]
174 0.012098837299164698238747934857 8.2652570265494837054695983631 0.800177608050858206458298613402 446 74 D1 [2]
175 0.012065933672360879339372321953 8.2877962630498625170743576586 0.800404992065072703654646655450 436 3 73 [1]
176 0.012038570285517179717868162747 8.3066342288422314305650416461 0.801331778324837038594614543210 450 1 74 [1]
177 0.012002966070044154533246541186 8.3312740714622495372950407377 0.801125020917314349711589571874 446 3 74 [1]
178 0.011982076084663004188176512814 8.3457991163984917517360641728 0.802849277650290525091338744362 454 2 74 D1 [1]
179 0.011945803735386947079030059409 8.3711403782544073148949881385 0.802478959059287697235620972506 439 3 75 [1]
180 0.011920000563795392666667856272 8.3892613481688845659248628576 0.803479738753437213436114998084 462 1 75 [1]
181 0.011896666898546495241106554826 8.4057157229658794435302178453 0.804783476363619502117946741374 457 2 75 [1]
182 0.011864350936501813428444058915 8.4286111001943149000923854225 0.804839400982400145634792624172 464 1 76 [1]
183 0.011835306977328539887453658705 8.4492949943383686154723142092 0.805304296565591261578006071872 460 3 77 [1]
184 0.011813075320400887766444829524 8.4651961735402197040653545775 0.806665794660857942272599446472 460 2 76 D1 [2]
185 0.011783418200666870341738557327 8.4865018195094360337188791144 0.806982624352193574040574161617 464 2 78 [1]
186 0.011764617628684004918761962819 8.5000637637541341922312963356 0.808757739421146491705821848637 476 1 78 [1]
187 0.011740977667035736248150840373 8.5171782824153146396879742849 0.809841453217398298570205254320 472 3 78 [1]
188 0.011722577193654503671390949205 8.5305473658241770411085818879 0.811622212845334506338285402811 480 2 78 C2 [1]
189 0.011690734562102845987251073139 8.5537824393142934315043794834 0.811512617763224397076010532885 477 3 79 [1]
190 0.011672722664446277495680741400 8.5669815753087397191524812497 0.813294448582008207892615180049 488 1 79 [1]
191 0.011650039561364683323127254589 8.5836618385084711109447648878 0.814400516145837293207391751248 475 1 80 [1]
192 0.011629592422206074864981488973 8.5987536251963081056357403932 0.815793217527651965162361829519 487 1 80 [1]
193 0.011608204767814804108674519429 8.6145964858633847864432177660 0.817028681470697915390833428427 480 3 80 [1]
194 0.011585362521278158905864878296 8.6315814301309811169707086323 0.818033065625741182051276504312 498 82 D1 [2]
195 0.011567953195348693221249398036 8.6445716291632779615777050080 0.819780398062783358040454784821 385 28 77 [1]
196 0.011550535051345135885970070871 8.6576075961393962728839836448 0.821504882178953513847601853038 469 4 79 [1]
197 0.011534585223056493945459212696 8.6695791886915794706040581879 0.823417443894693910188641519854 464 6 79 [1]
198 0.011519937537794770081885667091 8.6806026223595935061023872447 0.825496643167952429043541942856 506 2 82 D1 [1]
199 0.011492336589362946074263064144 8.7014506773633657830083318003 0.825694940595824197879220420685 504 3 83 [2]
200 0.011479462700246063514159511808 8.7112091054450213314370419796 0.827985994819228409760822670022 514 1 83 [1]





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.
09-Jul-2010: Improvements for N = 46, 47, 63, 93, 100, 121, 123, 176, 185, 186 and 195 by Eckard Specht [1].
16-Jul-2010: Even better packings for N = 63, 64, 76, 83, 91 and 151 by David W. Cantrell [2].
19-Jul-2010: More better packings for N = 121, 123, 154, 164, 174, 184, 194 and 199 by David W. Cantrell [2].
04-Aug-2010: One better packing for N=78 by David W. Cantrell [2].
06-Aug-2010: Better packings for N=133, 142, 145, 165, 195, 196 and 197 by Eckard Specht [1].
26-Jan-2011: One better packing for N=88 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