The best known packings of unequal circles with radii of i-1/5, i=1,2,3,..., in a circle (complete up to N = 100)


Last update: 06-Jul-2026

Overview    Download    Results    History of updates    References

Overview

5-16   17-28   29-40   41-52   53-64   65-76   67-80  

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 0.445510532110326177870183871898 2.2446158461465061681365756907 0.695057227703432417181722063386 10 5 [31]
6 0.418762856923068159265446214336 2.3879863829081483893070603544 0.699742410812028340555458125491 12 5 [31]
7 0.412775680938211778412205685930 2.4226233428458436434183291482 0.758109420731057351851704962697 12 1 6 [31]
8 0.396224843326 2.5238195354 0.766868928566 16 7 [1]
9 0.380224295376059535407571212515 2.6300265715817907674242237225 0.766215322600868718277677878324 16 1 8 [31]
10 0.368217684481656716319320819988 2.7157848255107812766391671463 0.772565778836813320126648944150 20 8 [31]
11 0.361341346032030330688841606070 2.7674663057001983356153099388 0.794015905603797412906638348064 18 2 7 [31]
12 0.353479496272733992389020193735 2.8290184028904197499412701320 0.806084390413839540166375811627 20 2 7 [31]
13 0.342006880889543238267855568678 2.9239177802477210672764305206 0.796535449627403853986318240679 18 4 9 [31]
14 0.335032622223411528392029785221 2.9847839692851308416825739213 0.803439693248824770469031828267 22 3 9 [31]
15 0.328894133918061387803680706002 3.0404920516129779897597900930 0.810884472461238920116529836488 28 1 10 [31]
16 0.322954185272443564609595356354 3.0964144315281185000463318685 0.816265195197173455167992150450 26 3 8 [31]
17 0.317245884377294160290338777441 3.1521291504312159304842404949 0.820069830754265493125926677131 28 3 9 [31]
18 0.311059040079333127847243506653 3.2148237831151217267556956941 0.818845406873179625136584232669 24 6 12 [31]
19 0.306811260790019573470353226916 3.2593327814144217074335490012 0.825623556098338510867602859667 32 3 12 [31]
20 0.301947772922070894938182032518 3.3118310174060731598852539477 0.827163434918595060730462216542 32 4 10 [31]
21 0.297477939502610777803564463151 3.3615938098536668985386539829 0.829038291966683067500043827504 36 3 11 [31]
22 0.293130379095334552998896504596 3.4114512562165070561010510254 0.829937734851978894470055541589 38 3 12 [31]
23 0.289843532600795971301975763171 3.4501373586876224461066694594 0.835398274496243113526712993435 40 3 13 [31]
24 0.285778774990 3.4992101846 0.835038750854 42 3 13 [1]
25 0.282415234076 3.5408854741 0.837507049914 44 3 11 [1]
26 0.278794617379 3.5868698234 0.837285136140 30 11 15 [1]
27 0.275927749611 3.6241371207 0.840526473015 46 4 13 [1]
28 0.272685619729 3.6672267536 0.840499536021 52 2 14 [1]
29 0.269953601659 3.7043402787 0.842692522166 32 13 16 [1]
30 0.266901145291 3.7467055411 0.842017837625 50 5 13 [1]
31 0.264340788855 3.7829954444 0.843632809859 54 4 15 [1]
32 0.261920960889 3.8179456757 0.845408589325 58 3 15 [1]
33 0.259519937562 3.8532684980 0.846611621173 60 3 15 [1]
34 0.257100935289 3.8895229956 0.847031840781 66 1 17 [1]
35 0.254836796866 3.9240800869 0.847842694381 64 3 16 [1]
36 0.252564650718 3.9593822697 0.848004547109 66 3 17 [1]
37 0.250666572685 3.9893631978 0.850128748075 66 4 18 [1]
38 0.248471752172 4.0246023593 0.849715802731 68 4 18 [1]
39 0.246628908370 4.0546747201 0.851207886666 70 4 19 [1]
40 0.244751967916 4.0857689869 0.851998243278 66 7 14 [1]
41 0.243022203381 4.1148503556 0.853369389008 68 7 17 [1]
42 0.241122741426 4.1472653889 0.853118616941 76 4 18 [1]
43 0.239393044130 4.1772308115 0.853653014666 78 4 17 [1]
44 0.237629712595 4.2082279572 0.853552130899 76 6 14 [1]
45 0.236114811212 4.2352277473 0.854864725391 86 2 18 [31]
46 0.234523681547 4.2639617177 0.855274448085 76 8 20 [31]
47 0.232953501820 4.2927021581 0.855493546745 76 9 21 [31]
48 0.231457903577 4.3204400651 0.855932003719 88 4 21 [31]
49 0.230058667952 4.3467173348 0.856772848510 88 5 20 [31]
50 0.228561086664 4.3751979595 0.856579598525 96 2 19 [31]
51 0.227208722253 4.4012394862 0.857183860624 88 7 20 [12]
52 0.225753261043 4.4296148608 0.856729285022 94 5 20 [12]
53 0.224586381913 4.4526297253 0.858200807861 92 7 21 [12]
54 0.223272702485 4.4788278588 0.858299504105 74 17 22 [12]
55 0.221936896967 4.5057852645 0.857975568701 92 9 21 [12]
56 0.220702237038 4.5309916810 0.858191143916 104 4 20 [12]
57 0.219436181484 4.5571336196 0.857929168237 108 3 21 [12]
58 0.218347789160 4.5798494404 0.858835344658 104 6 23 [12]
59 0.217173204711 4.6046196230 0.858851613103 104 7 22 [12]
60 0.216483839036 4.6192824575 0.862519330907 108 6 23 [12]
61 0.215077437895 4.6494881555 0.860283152849 114 4 22 [12]
62 0.214333163940 4.6656335474 0.863154442281 116 4 22 [12]
63 0.212972463891 4.6954426959 0.860877611377 115 5 23 [12]
64 0.211901099178 4.7191826936 0.860745405404 108 10 23 [12]
65 0.210900597353 4.7415702589 0.861011592870 112 9 25 [12]
66 0.210006415417 4.7617592920 0.861979669573 120 6 24 [12]
67 0.209137872376 4.7815347294 0.863000933939 114 10 26 [12]
68 0.208053046553 4.8064665073 0.862075845424 123 6 24 [12]
69 0.207228395154 4.8255935161 0.863150562046 130 4 23 [12]
70 0.206286235659 4.8476331773 0.863098409649 128 6 21 [12]
71 0.205259503930 4.8718815979 0.862185917413 128 7 24 [12]
72 0.204508581199 4.8897703663 0.863448451302 136 4 25 [12]
73 0.203490278637 4.9142396713 0.862314354655 136 5 24 [12]
74 0.202760122901 4.9319362491 0.863486959858 126 11 26 [12]
75 0.201901322266 4.9529145663 0.863436361123 124 13 25 [12]
76 0.201203403774 4.9700948455 0.864637875101 142 5 24 [12]
77 0.200481988574 4.9879792550 0.865520867477 136 9 26 [12]
78 0.199619546656 5.0095294612 0.865065602651 144 6 27 [12]
79 0.198804175960 5.0300754256 0.864896455929 136 11 27 [12]
80 0.198094658344 5.0480916970 0.865533977223 142 9 27 [12]
81 0.197338850469 5.0674258901 0.865656652600 140 10 28 [12]
82 0.196601021026 5.0864435738 0.865827576074 154 5 25 [12]
83 0.195932208486 5.1038060956 0.866501834574 154 6 25 [12]
84 0.195111119963 5.1252845055 0.865723820026 158 5 27 [12]
85 0.194460736718 5.1424262650 0.866357654181 148 11 29 [12]
86 0.193712263316 5.1622957828 0.866018378691 160 6 29 [12]
87 0.193105585740 5.1785141076 0.866850984436 162 6 29 [12]
88 0.192357833686 5.1986445305 0.866322675149 154 11 28 [12]
89 0.191793034894 5.2139536796 0.867350920672 152 13 28 [12]
90 0.191125918719 5.2321527436 0.867366260893 154 13 29 [12]
91 0.190415579645 5.2516711178 0.866898375321 162 10 30 [12]
92 0.189974150726 5.2638740385 0.868797576976 164 10 27 [12]
93 0.189239318175 5.2843141142 0.867932358166 164 11 28 [12]
94 0.188673535013 5.3001604063 0.868533504718 164 12 29 [12]
95 0.187988837973 5.3194647660 0.867958205112 178 6 29 [12]
96 0.187341724316 5.3378392008 0.867647009376 166 13 31 [12]
97 0.186867377894 5.3513888367 0.868861029394 166 14 31 [12]
98 0.186252013204 5.3690694817 0.868690610185 174 11 28 [12]
99 0.185629906266 5.3870630014 0.868380510812 178 10 28 [12]
100 0.185113426609 5.4020932912 0.868985978725 174 13 29 [12]



Updates

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

09-Aug-2013: First complete presentation from N=5 to N=64.
26-Jun-2014: First improvements for N= 21–26, 28, 29, 31, 32, 35, 36, 37, 39, 40, 42, 43, 44, 47, 48, 51–57, and 60 by Eckard Specht [31].
01-Jun-2024: Significant improvements for N= 8, 24–50 by Jianrong Zhou, Jiyao He, and Kun He [1].
26-May-2026: More slightly improvements for N= 45–50, 52, 53,55, 56, 58, 60, 61, 64, and new instances for N= 65–80 by Eckard Specht [31].
06-Jul-2026: André Müller has published his records on github in December 2025 [12]. I could improve most of them by further swapping operations, but only slightly.

References

[1]   , Solution-Hashing Search Based on Layout-Graph Transformation for Unequal Circle Packing, submitted to European Journal of Operational Research, under review, May 2024.
[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).
[12]   , github.com/muellan/packing, December 2025.
[31]   , program ccib, 2005–2026.
[32]   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    06-Jul-2026