The best known packings of unequal circles with radii of i1/2, i=1,2,3,..., in a circle (complete up to N = 100)


Last update: 27-Oct-2015


Overview    Download    Results    History of updates    References

Overview

5-16   17-28   29-40   41-50   53-64   65-76   77-88   89-100  


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 0.303568546668942154854937186471 13.5821239415627905430148345968 0.829384762740243399560914666006 22 6 9 [32]
18 0.296171002063510437150010727126 14.3249698909061328241475720219 0.833313993401387084393676294364 26 5 9 [32]
19 0.289909994669866366802458496248 15.0353524324138913429158350387 0.840478050094819452097260330382 32 3 10 [31]
20 0.283003970929532350253159197516 15.8023788157839519798678294919 0.840958099399777711738884813473 24 8 8 [32]
21 0.276737263931894916287144232423 16.5593011575181743436879966718 0.842418645733623936364914658294 30 6 13 [31]
22 0.271216918130980323417425326768 17.2939645216315764208811153712 0.845924091825369157823390685197 30 7 9 [32]
23 0.265697255130495150463644942363 18.0499852019821733077302951903 0.847140376606553178660406335067 34 6 9 [32]
24 0.260601309013119992166640278074 18.7987524088749558508453949124 0.848913028241895690819556742389 40 4 12 [32]
25 0.255745637821446707711406014940 19.5506755954556593220756148056 0.850275806441081705427435946796 34 8 10 [32]
26 0.251303781330229743702489689655 20.2902617963091326404541489023 0.852573471896771021131805878294 40 6 14 [32]
27 0.246708254976836789119410715650 21.0619317265832626748454813824 0.852109483032022800584190000753 36 9 10 [32]
28 0.242761490651579104097711740422 21.7971252686191282916033574007 0.854530549478962566901194892166 44 6 13 [32]
29 0.238837957153441648839483687567 22.5473575109956258660771608829 0.855653546658438433179176240176 47 5 14 [32]
30 0.235491675882987112037923013417 23.2586801827051690478195843754 0.859573105857756734118744492861 50 5 13 [32]
31 0.231630460370033787600661263943 24.0372719284649313688818884593 0.858442722739740688190270939338 44 9 12 [32]
32 0.228264554436884147357493155902 24.7820090309138111295645108042 0.859727662402442563674856985570 47 8 10 [32]
33 0.224881347176653416065996984168 25.5448605171572624321099014729 0.859717545235770930870985951652 48 9 13 [32]
34 0.221670493172500484060441121314 26.3045920609187511815261392240 0.859911632008442715075319178091 46 11 12 [32]
35 0.218839928911689172789270423271 27.0338224496819100401313400916 0.862036460749317072498194609705 44 13 11 [32]
36 0.215983054875795427214055198562 27.7799571056645981061845016253 0.863000579879395889000101240960 56 8 14 [32]
37 0.213256125793941972222585450554 28.5232722279484196859556413928 0.864085328584190448876289255208 54 10 14 [32]
38 0.210532132642538278057238598404 29.2801575018266307576354759161 0.864313688062798965319652202023 52 12 12 [32]
39 0.207989733713647901661715670723 30.0255108119724131698444161502 0.865194586605483251091086497950 44 17 9 [32]
40 0.205621450745453726164977629595 30.7582467559095075416655681244 0.866743710636633580751647002005 56 12 15 [32]
41 0.203225774335639218915772945460 31.5074417030278628105984966078 0.867315022440723280420090092038 58 12 14 [32]
42 0.200812885179778721587670003518 32.2725341683425607663917726365 0.867005019365880339171055492699 66 9 15 [32]
43 0.198645964590617195643935709047 33.0106807747945231187168037356 0.868124823459008295755544005205 60 13 13 [32]
44 0.196475209523522788639057766194 33.7612546478370892942571820196 0.868556429039524047802188987971 54 17 9 [32]
45 0.194311536656906542078845337743 34.5229318233634328469212517881 0.868410385393271701115399967027 66 12 16 [32]
46 0.192437692271807730415081217858 35.2442907782618669763303011317 0.870258237062125764251479177548 68 12 15 [32]
47 0.190474491133079663302240836829 35.9925077611110314539909977359 0.870732762537735438662438445441 71 11 13 [32]
48 0.188470574277873397402480403557 36.7601322212816433870354030604 0.870268355531469140572737755530 66 15 15 [32]
49 0.186828386188994598893524124047 37.4675398251250607026244296215 0.872621147149602694592698857658 74 12 17 [32]
50 0.184837603102815068182878445090 38.2555697172302447308654837483 0.871205957780241658285004614228 66 17 16 [32]
51 0.183171369023922940477678779650 38.9876893239256721771888571395 0.872345511182552069420580656665 80 11 17 [32]
52 0.181443019102558470864346376678 39.7430696788174040056275733364 0.872421583297862051394212299020 74 15 16 [32]
53 0.179801594382656393662743462567 40.4896848344230001198761402749 0.872872560248722970835549462560 84 11 17 [32]
54 0.178206008184517925857220002162 41.2358107519067933086136980718 0.873327987209162924366322287445 80 14 15 [32]
55 0.176648860670428038406383380694 41.9827133837675191781861570715 0.873734959332488339590600222738 82 14 19 [32]
56 0.175133045703190099890031083106 42.7293132686698430072926556672 0.874140135372356660825507620178 90 11 18 [32]
57 0.173556194429120887846667384787 43.5008065261194018733910718827 0.873530826116845556853167274678 88 13 20 [32]
58 0.172217052158318890983072517578 44.2219455647326892271917949418 0.874932035095983316919247423534 78 19 15 [32]
59 0.170758308726895401499875078247 44.9825592976184366656724825827 0.874751999978091667037267576021 82 18 18 [32]
60 0.169473802703323335301807625638 45.7059826879245299697753405003 0.876001778983111396891250779547 84 18 14 [32]
61 0.168064732480736637837007004562 46.4716752921488457169525395078 0.875618383418468751605112961763 92 15 18 [32]
62 0.166856335797149260326438581546 47.1903439350628386827473802690 0.876992659063007561156496767147 90 17 17 [32]
63 0.165496037411048899665155700006 47.9603865890740793784110793466 0.876446028760297503505567635008 96 15 18 [32]
64 0.164253152263571931982028225060 48.7053057414852438749020932978 0.876820685926904721698333623926 86 21 15 [32]
65 0.163129455448288754598257228360 49.4224524083833910498921504129 0.878170234750222446167774057884 90 20 20 [32]
66 0.161873998425335905501686408602 50.1874203619129026270907630255 0.877806910767889315666553894132 96 18 19 [32]
67 0.160800624698357688417089114680 50.9037373904931711840766535504 0.879132590714990741523171802505 94 20 18 [32]
68 0.159659100017572435358223863691 51.6488646768503941446695759674 0.879440473535531345783705049313 94 21 21 [32]
69 0.158393427977466926594870591901 52.4429830769227144944455749697 0.878096730925855088216785297626 96 21 20 [32]
70 0.157324768747260129643035063917 53.1804389859397965149062336162 0.878663441578950155576687809584 102 19 16 [32]
71 0.156227952068706937252094539404 53.9349691370891944511856435096 0.878658228272958919016224263458 98 22 15 [32]
72 0.155246749285342109724994757145 54.6567410480376660993419172119 0.879706690473804286267124931237 98 23 19 [32]
73 0.154145462005684347979334181257 55.4281886352415490233812993933 0.879150467906997444963994839883 100 23 18 [32]
74 0.153121348771001963668726383206 56.1797903172057897417838104527 0.879230529354405921361831742259 100 24 17 [32]
75 0.152249822965297142777345420175 56.8818660617973246782555892789 0.880840326532644207945262171372 113 18 20 [32]
76 0.151211815933788861417795353606 57.6528879918921842785715868197 0.880303011202770655326682822993 108 22 19 [32]
77 0.150200530438017201636200874934 58.4216604415605580010516416407 0.879847774410607542602694299157 110 22 19 [32]
78 0.149317198387173352572415203704 59.1476464983454502222279860323 0.880677216500682135485127889675 102 27 17 [32]
79 0.148462023534300718623471272709 59.8684714496165900398145908506 0.881638897275970417594635951162 106 26 17 [32]
80 0.147513552392400361670283164128 60.6335605436884042961446263868 0.881290049646730581877628688772 120 20 19 [32]
81 0.146625973913041750342316719638 61.3806664659397612506238345601 0.881466225263867995931807064059 122 20 21 [32]
82 0.145807514583589209058610151747 62.1050647766587049487024764184 0.882282999325308565049083945853 110 27 18 [32]
83 0.144882078271786006307800989878 62.8817151701394609074336812526 0.881614297382780925778847057455 126 20 24 [32]
84 0.144060147861496874449782223823 63.6203108629549090576515441921 0.882016363579744555770991308150 126 21 22 [32]
85 0.143220767724532565613589997072 64.3729579429817111329812127788 0.882024097226993889247171771643 124 23 20 [32]
86 0.142444205096573789556243332890 65.1035153673567173902066046856 0.882630293103372858820048049663 137 17 22 [32]
87 0.141634853218153793398559568101 65.8551115149762795593626877191 0.882658992429630678520974639159 128 23 20 [32]
88 0.141018867506823229515311177403 66.5218185729151843687325745596 0.884941284184359917932888059641 124 25 21 [32]
89 0.140091849673192369740059823651 67.3413989041067471488249943668 0.883157685518534808255355206356 128 25 20 [32]
90 0.139342318838405204265428742251 68.0829274235559747359870490641 0.883440822776501711273367404563 132 24 20 [32]
91 0.138577569107488247331914621787 68.8379228731468713520450207207 0.883372162348071327573597997522 128 27 18 [32]
92 0.137837569370442721543429591354 69.5867105785038085896520960274 0.883462592142749842671087602345 116 34 18 [32]
93 0.137147904510955308969024026636 70.3156989192104264978194545137 0.884048742452067432571300745700 136 25 20 [32]
94 0.136496243930316092976901225024 71.0302308375776408152585128811 0.884983168836506760520323345261 132 28 22 [32]
95 0.135703805880770751092694067425 71.8240308851211524809153246789 0.883945100665243693811044861500 142 24 23 [32]
96 0.135059209627811166179655567131 72.5456560728690440680206396921 0.884688020106518459658530508977 148 22 25 [32]
97 0.134360497832590000256288313280 73.3017364528341388145851930219 0.884584425513249685670271782281 147 23 24 [32]
98 0.133661456959656730309112689819 74.0639460454156735048410572631 0.884336561290619345084799368103 141 27 21 [32]
99 0.133018165021634478951199593509 74.8008692605849863873497249046 0.884691611286139119105371981712 138 30 21 [32]
100 0.132367940023074167893339697853 75.5469942212352626436737899839 0.884824213070584087223694137840 140 30 21 [32]





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].
06-Aug-2014: Improvements for N=42 and some new packings for N=53–59 by Eckard Specht [31].
27-Oct-2015: What a sensation! André Müller [32] found lots of new records, namely for N=17, 18, 20, 22–100, which are not only a little bit better than the previous, but significantly better. It seems that all other people will have no chance to beat these results for years (except for small improvements by rearrangements). So, congratulation André, for these milestones!


References

[19]   , private communication, July 2013.

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


©  E. Specht     27-Oct-2015