The best known packings of unequal circles with radii of i1/2, i=1,2,3,..., in a circle (complete up to N = 100)


Last update: 05-Jul-2026

Overview    Download    Results    History of updates    References

Overview

5-16   17-28   29-40   41-50   53-64   65-76   77-88   89-100  

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 container circle, the latter has always a *radius* of 1
ratio
= 1/radius, that is the radius of the circumcircle if r1=1
density
ratio of total area occupied by the circles to container area
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
5 0.494543344330826164987430700191 4.5214802769723774026005212657 0.733719358265754275821816032965 10 9 [31]
6 0.457766152975543725896857871072 5.3509629902978928103576465111 0.733424477835101149911731099699 12 10 [31]
7 0.437359195985677791666399930707 6.0493784864906122826189130251 0.765132265252954067826997893874 10 2 9 [31]
8 0.417525212693440511396933620366 6.7742666520666042223460221198 0.784472864556162318937406976748 12 2 10 [31]
9 0.396877769095830557184664994362 7.5590023770659136880140200905 0.787559818012416982420685118199 14 2 11 [31]
10 0.380838176730934890501854004978 8.3034681221114890787043811875 0.797707442706585386146805820457 14 3 10 [31]
11 0.365584079680028318043981250480 9.0721258793824480148524528436 0.801910315892959764926966683450 16 3 12 [31]
12 0.351139294856358075190282958279 9.8653203041682594493162739373 0.801442228549432447902462851328 20 2 12 [31]
13 0.340521360740693839985942029355 10.5883262877291493139726188169 0.811683579844856236139286602192 18 4 13 [31]
14 0.329226992036486149838512132232 11.3649775907780887853061779532 0.812928092140443860424903819230 22 3 11 [31]
15 0.320958643540695773711445805821 12.0669233377924097053678542714 0.824115606907867303238986868147 24 3 8 [22]
16 0.312029237532408646549487625254 12.8193115223202231613483007121 0.827579083137978522745069428458 24 4 12 [31]
17 0.303850035311780543215753833265 13.5695413738785370522827584690 0.830923595630732547306528630353 26 4 9 [22]
18 0.296239267558937331725442215656 14.3216688391089280531406065550 0.833698184616628772004240034684 30 3 11 [22]
19 0.289909994669866366802458496248 15.0353524324138913429158350387 0.840478050094819452097260330382 32 3 10 [31]
20 0.283202939096156653144592612187 15.7912766346013912050350283677 0.842140999483364852871080684721 32 4 11 [22]
21 0.277066338318439745744292414636 16.5396335143714330088263211011 0.844423314121069238109930630304 38 2 12 [22]
22 0.271348319621250663505992517388 17.2855898513406504291917152880 0.846743971454678689998123917884 36 4 11 [22]
23 0.265697255130495150463644942363 18.0499852019821733077302951903 0.847140376606553178660406335067 34 6 9 [32]
24 0.260749430051236985140932681022 18.7880736099929804180688271362 0.849878315900561616793232014202 34 7 12 [22]
25 0.256057329455 19.5268770890 0.852349627576 40 5 12 [23]
26 0.251452883007 20.2782304685 0.853585457032 44 4 13 [23]
27 0.247086238336 21.0297119650 0.854722528449 30 12 15 [23]
28 0.243169191739 21.7605798838 0.857403209262 42 7 13 [23]
29 0.239358037197 22.4983663394 0.859384049560 42 8 12 [23]
30 0.235654723031 23.2425877343 0.860763801544 50 5 12 [23]
31 0.232015474693 23.9973836668 0.861298887952 48 7 13 [23]
32 0.228638188597 24.7415109619 0.862544451204 52 6 12 [23]
33 0.225466932338 25.4785151285 0.864200738822 52 7 13 [23]
34 0.222302570300 26.2297997139 0.864822573335 54 7 14 [23]
35 0.219295676651 26.9776398397 0.865630688360 58 6 16 [23]
36 0.216468018685 27.7177203194 0.866880457596 68 2 15 [23]
37 0.213611628920 28.4758023757 0.866968632190 64 5 16 [23]
38 0.211006154277 29.2143801402 0.868210144282 60 8 15 [23]
39 0.208452337984 29.9588772128 0.869047544223 60 9 14 [23]
40 0.205996478733 30.7022496657 0.869908259637 60 10 13 [23]
41 0.203491329952 31.4663245798 0.869583148675 68 7 16 [23]
42 0.201166881542 32.2157437085 0.870064455932 64 10 17 [23]
43 0.198939575123 32.9619610389 0.870693000105 64 11 15 [23]
44 0.196791631272 33.7069698434 0.871356288117 68 10 13 [23]
45 0.194734382458 34.4479688067 0.872194033358 66 12 14 [23]
46 0.192672434037 35.2013510237 0.872382670688 70 11 15 [23]
47 0.190746857350 35.9411142896 0.873224726137 68 13 15 [23]
48 0.188858004266 36.6847211863 0.873849971492 76 10 15 [23]
49 0.186955229932 37.4421191777 0.873806449970 68 15 13 [23]
50 0.185124507457 38.1962815674 0.873912623162 86 7 16 [23]
51 0.183185538506 38.9846736089 0.872480479464 82 10 24 [32]
52 0.181443019102558470864346376678 39.7430696788174040056275733364 0.872421583297862051394212299020 74 15 16 [32]
53 0.179801594382656393662743462567 40.4896848344230001198761402749 0.872872560248722970835549462560 84 11 17 [32]
54 0.178206008184517925857220002162 41.2358107519067933086136980718 0.873327987209162924366322287445 80 14 15 [32]
55 0.176659147213 41.9802688063 0.873836720235 90 10 25 [32]
56 0.175134640180 42.7289242486 0.874156052440 92 10 26 [32]
57 0.173581132219 43.4945569185 0.873781874409 82 16 27 [32]
58 0.172253844385 44.2125000638 0.875305913708 78 19 28 [32]
59 0.170758308726895401499875078247 44.9825592976184366656724825827 0.874751999978091667037267576021 82 18 18 [32]
60 0.169473802703323335301807625638 45.7059826879245299697753405003 0.876001778983111396891250779547 84 18 14 [32]
61 0.168064732480736637837007004562 46.4716752921488457169525395078 0.875618383418468751605112961763 92 15 18 [32]
62 0.166866669852 47.1874214365 0.877101293486 96 14 29 [32]
63 0.165504015422 47.9580746906 0.876530531865 98 14 27 [32]
64 0.164272584435 48.6995442818 0.877028164905 90 19 29 [32]
65 0.163129455448288754598257228360 49.4224524083833910498921504129 0.878170234750222446167774057884 90 20 20 [32]
66 0.161873998425335905501686408602 50.1874203619129026270907630255 0.877806910767889315666553894132 96 18 19 [32]
67 0.160803432967 50.9028484084 0.879163297834 94 20 31 [32]
68 0.159665370880 51.6468361662 0.879509557706 92 22 28 [32]
69 0.158393427977466926594870591901 52.4429830769227144944455749697 0.878096730925855088216785297626 96 21 20 [32]
70 0.157344880600 53.1736414522 0.878888106507 98 21 32 [32]
71 0.156229012944 53.9346028908 0.878670161479 96 23 31 [32]
72 0.155246749285342109724994757145 54.6567410480376660993419172119 0.879706690473804286267124931237 98 23 19 [32]
73 0.154186044049 55.4135998366 0.879613438641 113 16 31 [32]
74 0.153121348771001963668726383206 56.1797903172057897417838104527 0.879230529354405921361831742259 100 24 17 [32]
75 0.152250388117 56.8816549171 0.880846865904 102 24 33 [32]
76 0.151212487388 57.6526319861 0.880310829170 113 19 33 [32]
77 0.150219391827 58.4143250794 0.880068761550 114 20 33 [32]
78 0.149346539053 59.1360263340 0.881023354721 108 24 34 [32]
79 0.148466027364 59.8668569176 0.881686451249 112 23 36 [32]
80 0.147517784208 60.6318211597 0.881340614638 120 20 34 [32]
81 0.146656930679 61.3677100587 0.881838467961 124 19 31 [32]
82 0.145807514583589209058610151747 62.1050647766587049487024764184 0.882282999325308565049083945853 110 27 18 [32]
83 0.144920165109 62.8651890667 0.882077878717 116 25 35 [32]
84 0.144070052569 63.6159370144 0.882137652011 128 20 34 [32]
85 0.143255035667 64.3575593303 0.882446225488 110 30 37 [32]
86 0.142455296739 65.0984463741 0.882767753256 138 17 31 [32]
87 0.141668816694 65.8393235066 0.883082359429 118 28 38 [32]
88 0.141024097897 66.5193513699 0.885006930354 132 22 34 [32]
89 0.140091849673192369740059823651 67.3413989041067471488249943668 0.883157685518534808255355206356 128 25 20 [32]
90 0.139358475981 68.0750339275 0.883645709664 128 26 38 [32]
91 0.138592732090 68.8303915388 0.883565487853 124 29 38 [32]
92 0.137856574620 69.5771171821 0.883706235212 126 29 35 [32]
93 0.137151768302 70.3137180105 0.884098554768 144 21 32 [32]
94 0.136500857082 71.0278303161 0.885042989247 138 25 37 [32]
95 0.135708344311 71.8216289079 0.884004226350 144 23 35 [32]
96 0.135059209627811166179655567131 72.5456560728690440680206396921 0.884688020106518459658530508977 148 22 25 [32]
97 0.134360497832590000256288313280 73.3017364528341388145851930219 0.884584425513249685670271782281 147 23 24 [32]
98 0.133666933143 74.0609117291 0.884409026287 138 29 37 [32]
99 0.133028825649 74.7948748893 0.884833422674 148 25 37 [32]
100 0.132383192802 75.5382899320 0.885028141691 148 26 39 [32]



Updates

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

20-Mar-2012: First complete presentation from N=5 to N=50.
22-Mar-2012: First improvements for N=30, 36, 43, 44, 46, 49 and 50 by Eckard Specht [31].
26-Jul-2013: After sixteen months the first new records for N=19 and 20 by Lin Lu [19], great!
28-Jul-2013: Sorry Linda, but your recent improvement for N=19 was not strong enough to survive. However, your other record is much harder to beat. Further improvements for N=33, 41 and 48 by Eckard Specht [31].
30-Jul-2013: Further improvements for N=34, 37 and 38 by Eckard Specht [31].
23-Jun-2014: Improvements for N=22, 43, 49 and some new packings for N=51, 52, 60–64 and 83–85 by Eckard Specht [31].
06-Aug-2014: Improvements for N=42 and some new packings for N=53–59 by Eckard Specht [31].
27-Oct-2015: What a sensation! André Müller [32] found lots of new records, namely for N=17, 18, 20, 22–100, which are not only a little bit better than the previous, but significantly better. It seems that all other people will have no chance to beat these results for years (except for small improvements by rearrangements). So, congratulation André, for these milestones!
03-Feb-2022: He Dong [22] from Anhui Polytechnic University, China, found nine new records on his laptop, namely for N=15, 17, 18, 20, 21, 22, 24, 25, and 27. Congratulations!
01-Jun-2024: Significant improvements for N= 8, 24–50 by Jianrong Zhou, Jiyao He, and Kun He [23].
05-Jul-2026: André Müller has published his records on github in December 2025 [32]. I could improve most of them only by a small amount.

References

[11]   André Müller, Johannes Josef Schneider, Elmar Schömer, Packing a multidisperse system of hard discs in a circular environment, Phys. Rev. E 79, 021102 (2009).
[22]   , private communication, January 2022.
[23]   , Solution-Hashing Search Based on Layout-Graph Transformation for Unequal Circle Packing, submitted to European Journal of Operational Research, under review, May 2024.
[31]   , program ccir, 2005–2024.
[32]   , github.com/muellan/packing, December 2025.
[33]   Eckard Specht, A precise algorithm to detect voids in polydisperse circle packings, Proc. R. Soc. A 471 (2015), 20150421.

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©  E. Specht    05-Jul-2026