The best known packings of unequal circles with radii of i-1/5, i=1,2,3,..., in a circle (complete up to N = 64)


Last update: 27-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 0.445510532110326177870183871898 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 0.396224628403000922595969430717 2.5238209043959222218298085800 0.766868096625366170592551294296 14 1 6 [31]
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 0.285509511920761629619924050879 3.5025102781077682857610928268 0.833465931658215551250476763055 42 3 12 [31]
25 0.282067906970205549085670150202 3.5452455784189182690870514264 0.835448308102096287528150317799 46 2 13 [31]
26 0.278463588511064693645936551790 3.5911337828653501132735049413 0.835298003058289621133924453395 44 4 13 [31]
27 0.275278312568226913773927451343 3.6326871909030355025102381291 0.836574520840411417079017490226 50 2 13 [31]
28 0.272186808157356246098141610024 3.6739473406877288413887010428 0.837427372283272176122399586834 52 2 15 [31]
29 0.269569771902835041175738218116 3.7096147425626214637855300212 0.840297884594353664436621612603 52 3 15 [31]
30 0.266452046032066055612368639288 3.7530205336822800559489138911 0.839186591312992829625797770963 54 3 15 [31]
31 0.263529825567755447761770249642 3.7946368986719974807449018962 0.838464437702450168988962478192 58 2 17 [31]
32 0.261344939275722380090203142747 3.8263606816774313956174466594 0.841694200549941097957395723320 47 8 17 [31]
33 0.258663648810174620899366114599 3.8660244862387662508874880721 0.841034030218636417634024868753 55 5 13 [31]
34 0.256296726853183082538342344143 3.9017275494619937600187821626 0.841741118879748666559960485536 58 5 14 [31]
35 0.254020589131114872105694297164 3.9366887677118233738980149872 0.842420341271633692721727928884 62 4 13 [31]
36 0.251907064799438688931161907127 3.9697179624405208290628428019 0.843594508667380487043494274855 62 5 16 [31]
37 0.249615702646172256149096556815 4.0061582240180215181175471635 0.843015696085215588171418945909 68 3 18 [31]
38 0.247613323555749713081702598300 4.0385548953501757885799801761 0.843854691022589857773038700326 68 4 17 [31]
39 0.245421745192149956028115264254 4.0746185682000680962031907089 0.842895543390490938740908738758 76 1 18 [31]
40 0.243565440443616475250555118781 4.1056727841957213262397204753 0.843757501238062618961364979409 76 2 17 [31]
41 0.242131301059408407765738785973 4.1299906109811214937227833606 0.847124073035736394039070532648 70 6 19 [31]
42 0.239804915541527366716480224585 4.1700563049001743360762760887 0.843818875788168993163403167759 76 4 17 [31]
43 0.238291248725841449632905066133 4.1965452166080960578424011796 0.845813300268140472256546709106 76 5 18 [31]
44 0.236405812282068966952884292252 4.2300144414674720277160035525 0.844782415475940031077652380784 78 5 16 [31]
45 0.234806830674121768147593372528 4.2588198866661451363168823705 0.845419749210419849130273065468 81 4 15 [31]
46 0.233646830587184377238429509007 4.2799638988762315475686072336 0.848890903014059148485965774708 80 6 20 [31]
47 0.231516835487794302562250755498 4.3193403101465619374204124266 0.844974118166665128175924948565 84 5 19 [31]
48 0.230398912994942919981452614355 4.3402982548878137961498509801 0.848117620767833290938399742434 86 4 17 [31]
49 0.228836990017242957831964330386 4.3699228867004831864625020993 0.847697586509526343022236327702 84 7 19 [31]
50 0.227365751428029386681422643654 4.3981997891909465367553831335 0.847643498497776775382230813868 84 8 19 [31]
51 0.226192880643510832953142460204 4.4210056353455289180997292690 0.849536122016254424482486200523 88 7 17 [31]
52 0.224360980017201659857357637092 4.4571030128471110708457418436 0.846194511448489459079778372144 98 2 19 [31]
53 0.223159727243272159875924005967 4.4810952780466265452089970096 0.847332228908881620222424550399 94 6 19 [31]
54 0.221865564750487885979052693972 4.5072339239512425677774925296 0.847515025733540030414654029965 89 9 19 [31]
55 0.220586952714334435516709285911 4.5333596919262253710893865049 0.847569937773715936653476211620 98 6 19 [31]
56 0.219664248260970659696836128641 4.5524021679301977399160073017 0.850137777205819918806750870250 104 3 21 [31]
57 0.218181884536199900068024189542 4.5833319394309468521160207802 0.848149354872142186485609915777 98 8 17 [31]
58 0.217189245374810335374023143557 4.6042795455837071461066861049 0.849745638314288936899387227123 108 4 23 [31]
59 0.215799455564850505922440853003 4.6339319873746723434553369349 0.848020488178801736184554917752 106 6 21 [31]
60 0.214546032482071502417834064663 4.6610043934677039809568631047 0.847147143698002821342871310406 113 3 21 [31]
61 0.213686180138230397035065868342 4.6797598204671679843220315058 0.849189431959752049758290813051 110 6 19 [31]
62 0.212587886982871157039546994184 4.7039368714388369454312345559 0.849154647009415904816509283015 112 6 22 [31]
63 0.211206913801211992889403887752 4.7346934908636574431853015221 0.846663356564031734689057782124 120 3 24 [31]
64 0.210314293460139669092401153863 4.7547885764099407064378678448 0.847902416329505139863917451633 118 5 23 [31]





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.
22-Mar-2012: 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].


References


[31]   , program ccib, 2005–2013.
[32]   Eckard Specht, A precise algorithm to detect voids in polydisperse circle packings, Proc. R. Soc. A 471 (2015), 20150421.


©  E. Specht     14-Oct-2015