The best known packings of unequal circles with radii of i1/2, i=1,2,3,..., in a circle (complete up to N = 50)


Last update: 23-Jun-2014


Overview    Download    Results    History of updates    References

Overview

5-16   17-28   29-40   41-50   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 0.494543344330826164987430700191 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 0.320729975064466518393196533357 12.0755266028033386256838323924 0.822941735238826517354638884969 20 5 10 [31]
16 0.312029237532408646549487625254 12.8193115223202231613483007121 0.827579083137978522745069428458 24 4 12 [31]
17 0.303317801347891014185866595930 13.5933519473479749863843242533 0.828015197530668083478556920585 24 5 13 [31]
18 0.295537799213504898442711282419 14.3556617746019047707016795836 0.829754612257638397401122225734 30 3 14 [31]
19 0.289909994669866366802458496248 15.0353524324138913429158350387 0.840478050094819452097260330382 32 3 10 [31]
20 0.282958128807697854947737972538 15.8049389633860035097191020425 0.840685677912714172388123126734 34 3 11 [19]
21 0.276737263931894916287144232423 16.5593011575181743436879966718 0.842418645733623936364914658294 30 6 13 [31]
22 0.270471923595127699218408029415 17.3415994439576760679735113961 0.841283206712358852050908835768 23 10 0 [31]
23 0.265365487723793946937515274223 18.0725518018547601923089482517 0.845026104898644431487953715020 40 3 16 [31]
24 0.259828488672890378897027392792 18.8546664401142670372965891283 0.843885544075479049123811654246 46 1 18 [31]
25 0.255332066373965405268812031781 19.5823425980381223023196471021 0.847528033544387977853191213840 42 4 16 [31]
26 0.250475307383688788329256466512 20.3573740136463378450585879327 0.846961374720870673816600989255 42 5 18 [31]
27 0.246281143456611487331599328018 21.0984582488836081094359061757 0.849161622712144661113693175709 34 10 15 [31]
28 0.242337157616546034761091299881 21.8352920954120065942952297712 0.851545820444166799497025843292 38 9 14 [31]
29 0.238040819424110179022696255360 22.6228628357219591988833163508 0.849951475681527448522018745684 50 4 17 [31]
30 0.234331344697673771513259404102 23.3738494614034195359251945483 0.851123276171210024664385407090 38 11 16 [31]
31 0.230747948799610066298583731761 24.1292041458850476767674955852 0.851913854003639498183368975137 54 4 18 [31]
32 0.228069213811199511209507547401 24.8032347503738187112911255184 0.858256843759567524906459802237 44 10 16 [31]
33 0.224668447771135117628683840003 25.5690672345317052478414075093 0.858090494206151559472195797457 56 5 16 [31]
34 0.221311143770694648152473738366 26.3473035993473432723120825819 0.857125891249128839925717258124 56 6 17 [31]
35 0.218037782505013600826874888408 27.1332780728660209025258314142 0.855728542794665071139203266121 56 7 19 [31]
36 0.215020028969772779688610523253 27.9043772282417083951290033399 0.855321837875995620362962059831 48 12 14 [31]
37 0.212232380095751685861035456750 28.6608599854267950402666007305 0.855809080061044710987595683078 60 7 16 [31]
38 0.209746250698825261554581260979 29.3898650508917147219887990284 0.857873048803181902160317553254 56 10 14 [31]
39 0.207336349200362276397789621618 30.1201309972110109542156054915 0.859767233994691335428763708721 62 8 18 [31]
40 0.205021384117949832234332326849 30.8482714988316058582038629786 0.861692242885618600022475004569 58 11 19 [31]
41 0.202479449960878142772980953896 31.6235758180399132092140076244 0.860956480785654870391316566998 69 6 16 [31]
42 0.199954656864578719747696995943 32.4110515855452485460438171829 0.859610435941379744638653398659 61 11 20 [31]
43 0.197779970240723817222173792894 33.1552205024641747703170085823 0.860572165825275172608283744027 66 10 0 [31]
44 0.195668396440232631430178468488 33.9004647729963958232048591512 0.861438043972447842840374128350 66 11 18 [31]
45 0.193609840476494608716494312274 34.6480525782664705118185998007 0.862150017451588503617699502804 76 7 23 [31]
46 0.191379432558211966004563034447 35.4391790824350148027961565451 0.860713335985894244470321886452 72 9 19 [31]
47 0.189801596678731587314750076198 36.1201102644320459306048817574 0.864591506443101444397977725419 66 14 18 [31]
48 0.187574676108073367088681994042 36.9357067490482722728519195743 0.862014348367691426264697217180 68 14 0 [31]
49 0.185807461470542558831493347617 37.6734063562337734825358504429 0.863110318453178937981402779744 80 9 0 [31]
50 0.183723735592567106513447441899 38.4875029296516696444088850441 0.860737724042943215826237766615 82 9 22 [31]
51 0.181855272586137653167687634762 39.2698453390194539464793754865 0.859854844351839154937568874375 72 15 0 [31]
52 0.180177937591113221151396791119 40.0221172877030772030943652478 0.860298363656557905945115505915 82 11 0 [31]
60 0.168552583113471171479797192709 45.9557875016378522438369363674 0.866504184863820016879009754068 93 12 0 [31]
61 0.167370041406469822070455014119 46.6645620104670833936131546781 0.868394653572506104803824361179 95 13 0 [31]
62 0.165992713813855886486518199878 47.4358644611456738911484509670 0.867937802737592879466167442083 81 21 0 [31]
63 0.164798294905043236575743966899 48.1634469444433075518472901930 0.869071296115507186782248803936 102 12 0 [31]
64 0.163488131496982847497114776862 48.9332156820670433447908787868 0.868671997062179537490724377462 107 10 0 [31]
83 0.144038367048298095161774659956 63.2500476493838760445604192202 0.871376149641490353628233970579 129 16 0 [31]
84 0.143244761532993632675090801694 63.9824541702397013843495704244 0.872060122532379024374033696131 137 14 0 [31]
85 0.142342911755199251983644328640 64.7699582902211635830280917965 0.871244694658782979988217537401 149 9 0 [31]





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].


References

[19]   , private communication, July 2013.

[31]   , program ccir, 2005–2013.


©  E. Specht     23-Jun-2014