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: 01-Jun-2024

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 0.581236900967 1.7204688800 0.895618050907 36 13 10 [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 0.576759932144 1.7338236314 0.904832487926 64 7 13 [2]
40 0.576218174594 1.7354537640 0.905560581067 66 7 15 [2]
41 0.575977107736 1.7361801130 0.907149599870 50 16 11 [2]
42 0.575405559531 1.7379046543 0.907618002027 56 14 16 [2]
43 0.575289932710 1.7382539536 0.909450196828 66 10 17 [2]
44 0.574742453381 1.7399097528 0.909846602700 58 15 14 [2]
45 0.573898900164 1.7424671832 0.909235512637 38 26 10 [2]
46 0.573691708076 1.7430964853 0.910575972687 54 19 18 [2]
47 0.573142908296 1.7447655472 0.910771370622 60 17 12 [2]
48 0.573236312368 1.7444812522 0.912951951253 42 27 14 [2]
49 0.572233307232 1.7475389625 0.911586132295 48 25 11 [2]
50 0.571924173507 1.7484835339 0.912377235458 56 22 13 [2]
51 0.569959464698223935089791900238 1.7545107361792287207773274282 0.907837117005812880835958183394 85 8 0 [31]
52 0.569947144443536364495206363611 1.7545486625366682509221582912 0.909471520028264059302297878937 87 8 0 [31]
53 0.569059659319721121058060989153 1.7572849939766313178549939740 0.908267985072315571453538067449 91 7 0 [31]
54 0.567816086581818226619217708172 1.7611336199012515478992090501 0.905882261311148545985616962914 94 6 0 [31]
55 0.565536695017064310086077139329 1.7682318562367139470960358107 0.900152976662166113075119132860 76 17 0 [31]
57 0.564156605017730126020316462576 1.7725574620695477505208941058 0.898701432748056638870259869512 86 14 0 [31]
58 0.564107715908685603417549657572 1.7727110830050657255318053813 0.899963060287605963288408000619 56 29 0 [31]
59 0.563690796292616692617223877772 1.7740222238450235121969542021 0.900016659990085245210858742576 84 17 0 [31]
60 0.564069382655388785852604425242 1.7728315536156970146531051189 0.902580560700191777827137412053 70 25 0 [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].

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, 2005–2024.
[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    01-Jun-2024