The best known packings of unequal circles with radii of i-2/3, i=1,2,3,..., in a circle (complete up to N = 60)


Last update: 27-May-2026

Overview    Download    Results    History of updates    References

Overview

5-16   17-28   29-40   41-52   53-64  

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 0.613511790435690629043456884943 1.6299605249474365823836053036 0.716065217699803972997931399647 3 3 2 [1]
6 0.613511790435690629043456884943 1.6299605249474365823836053036 0.750588400373934213926326021056 3 4 2 [1]
7 0.613507177767355377324499622184 1.6299727798444835837572813708 0.778685888249537048237893508023 10 2 5 [1]
8 0.612938866788355682440699477063 1.6314840748144208215514662292 0.800724792414343253194566245020 16 8 [1]
9 0.610551304892410581219281177072 1.6378639960096667639954444461 0.814411144239406655479651634531 16 1 7 [1]
10 0.607180306029151381565813193090 1.6469572383528343903865000825 0.822554905022055328060214497082 18 1 7 [1]
11 0.605945357046045531870234623638 1.6503138251194000355539911439 0.834221014708460383082950810928 18 2 7 [31]
12 0.603587631260896413485275236502 1.6567602585079434647167912090 0.841002637319033494915334305995 18 3 9 [31]
13 0.601403181962008352686349070579 1.6627780330952284129987195690 0.846758714384872781115733843423 22 2 9 [31]
14 0.598736595717535641336680955992 1.6701835283704076762182522944 0.849890718995485453037633335766 14 7 5 [31]
15 0.597728254148429417644442070381 1.6730010553452563205027390370 0.856688472137618595773512018039 18 6 6 [31]
16 0.595368574979914121162752069512 1.6796318146851080964778027636 0.858729661115309475400518746377 28 2 8 [31]
17 0.593880310526356882505435811325 1.6838409731309305653657452988 0.862510452140131579864634494537 20 7 7 [31]
18 0.593110930354651323544395103821 1.6860252421955011288126256807 0.867734300796625387762836868571 26 5 10 [31]
19 0.591879670734612066648407095596 1.6895326017175906105752554161 0.871045066045228666220755956003 22 8 8 [31]
20 0.590738021043 1.6927977621 0.874116176903 30 5 10 [2]
21 0.590061955908 1.6947372899 0.878126042004 26 8 8 [2]
22 0.588796381994 1.6983800013 0.879987090434 30 7 10 [2]
23 0.587461695900 1.7022386429 0.881278304401 26 10 8 [2]
24 0.586783177310 1.7042070030 0.884217358227 32 8 9 [2]
25 0.586056604447 1.7063198203 0.886727482781 24 13 9 [2]
26 0.584900287214 1.7096931252 0.887673354084 46 3 12 [2]
27 0.584073153388 1.7121143031 0.889376154382 46 4 10 [2]
28 0.583301820074 1.7143783297 0.891030348395 40 8 11 [2]
29 0.582551501195 1.7165864270 0.892548448423 38 10 11 [2]
30 0.582016661609 1.7181638705 0.894544230103 44 8 12 [2]
31 0.581264988782 1.7203857437 0.895704613071 34 14 20 [2]
32 0.580540158226 1.7225337228 0.896789540211 36 14 9 [2]
33 0.580106222028 1.7238222278 0.898628682617 50 8 11 [2]
34 0.579644328954 1.7251958659 0.900248598652 48 10 10 [2]
35 0.578477449711 1.7286758550 0.899550584827 46 12 11 [2]
36 0.578261239672 1.7293222014 0.901691338320 44 14 11 [2]
37 0.577559895368 1.7314221573 0.902210999363 46 14 13 [2]
38 0.577235420074 1.7323954235 0.903805674623 38 19 9 [2]
39 0.576957659885 1.7332294370 0.905452992720 60 9 19 [2]
40 0.576240029019 1.7353879454 0.905629273385 58 11 22 [2]
41 0.575977107736 1.7361801130 0.907149599870 50 16 11 [2]
42 0.575414796569 1.7378767560 0.907647142409 60 12 23 [2]
43 0.575299125419 1.7382261780 0.909479261748 72 7 21 [2]
44 0.574811898007 1.7396995495 0.910066484749 0 21 24 [2]
45 0.573911734968 1.7424282151 0.909276181787 24 33 30 [2]
46 0.573756098718 1.7429008637 0.910780388619 0 21 27 [2]
47 0.573386819608 1.7440233465 0.911546725971 47 23 28 [31]
48 0.573236312368 1.7444812522 0.912951951253 42 27 14 [2]
49 0.572236312160 1.7475297858 0.911595706217 48 25 25 [31]
50 0.571924173507 1.7484835339 0.912377235458 56 22 13 [2]
51 0.570532248381 1.7527493018 0.909662704983 56 23 26 [31]
52 0.569961552373 1.7545043097 0.909517502422 85 9 27 [31]
53 0.569518486291 1.7558692546 0.909733229875 76 15 25 [31]
54 0.568802982762 1.7580779818 0.909033946110 103 2 22 [31]
55 0.566764688652 1.7644006764 0.904066364552 90 10 26 [31]
56 0.565825268211 1.7673300508 0.902566175641 24 44 32 [31]
57 0.566912728514 1.7639399324 0.907503891141 76 18 23 [31]
58 0.565718980668 1.7676620976 0.905111544019 59 28 29 [31]
59 0.565121714842 1.7695303042 0.904591810531 62 28 34 [31]
60 0.564730740442 1.7707553855 0.904698309482 46 37 23 [31]



Updates

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

14-Mar-2012: First complete presentation from N=5 to N=30. David W. Cantrell [1] was the first who sent me his results.
14-Mar-2012: Due to a mistake, all packings before have had a precision of only 16 decimal places. Now they are provided in full accuracy of 30 digits. Many thanks to David Cantrell who discovered this inaccuracy!
21-Mar-2012: First improvements for N=11 and 18 by Eckard Specht [31].
22-Mar-2012: Better packings for N=24, 25, 26, 29, 30 and extension up to N=40 by Eckard Specht [31].
27-Jul-2013: Thanks to Lin Lu who noticed that all ratio value were completely wrong before. Now they should be correct. Additionally, some improvements for N=34, 39 and 40 by Eckard Specht [31].
11-Aug-2013: Again, some improvements for N=19, 21, 22, 23, 26, 28, 33–39 and extension up to N=50 by Eckard Specht [31].
01-Jul-2014: Further improvements for N= 21, 23, 24, 27, 28, 29, 33, 34, 35, 39, 40, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60 and extension up to N=60 by Eckard Specht [31].
01-Jun-2024: Significant improvements for N= 20–50 by Jianrong Zhou, Jiyao He, and Kun He [2].
27-May-2026: Slightly improvements for N= 31, 39, 40, 42–47, 49, 51–60 by Eckard Specht [31].

References

[1]   , private communication, March 2012.
[2]   , 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 ccic, 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    27-May-2026