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-Jul-2014


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 0.589467028756857307078430309218 1.6964477251745981339689531063 0.870358843847203442361393750911 30 5 8 [31]
21 0.588601256045743007734661390529 1.6989430276076156316380605851 0.873783816790064974924903695811 18 12 0 [31]
22 0.588004231333870291404105846926 1.7006680338533099227916060158 0.877620861776414557200433458124 24 10 8 [31]
23 0.586877266296595940207993283390 1.7039337821182873317169259639 0.879525717160692670360871643790 32 7 0 [31]
24 0.586376926077468302292881861861 1.7053877046108163365429651327 0.882993430694346593083711706559 34 7 0 [31]
25 0.583999434874983642556683027494 1.7123304241108885873523377549 0.880513246036122999184064576297 34 8 12 [31]
26 0.583228688186140530496237777338 1.7145932980595163966178352309 0.882606803057547696960484372206 34 9 10 [31]
27 0.581738426396738435085741227589 1.7189856379162622619641727266 0.882280124087891732051773051488 52 1 0 [31]
28 0.580750766779114821604194930970 1.7219090480862751702020357038 0.883253601486688726744026973370 48 4 0 [31]
29 0.580575868842688474311432371901 1.7224277715733957599856265154 0.886504836731080760942914620367 34 12 0 [31]
30 0.579696449422450033964101675785 1.7250407536501167614208529028 0.887426236749683814864218717013 20 20 8 [31]
31 0.579583815854970758601834988234 1.7253759898813840773965040530 0.890530873937452077349475863195 22 20 9 [31]
32 0.579010407161471504263676334508 1.7270846734903765551501885389 0.892069600447035820922379935842 30 17 11 [31]
33 0.577403607373332892926856342005 1.7318908077992456920880950982 0.890275075532037798944727704986 20 23 0 [31]
34 0.577039171745231795181421215963 1.7329846030652306847223296403 0.892174617184978107149907278065 22 23 0 [31]
35 0.576771758303825049055724159924 1.7337880809920511940883904947 0.894253598357947835056943137089 22 24 0 [31]
36 0.576720384514224524973334579342 1.7339425254446429077876401084 0.896892383921704377600943916053 50 11 12 [31]
37 0.576588508160859794902304633261 1.7343391098613685281716621996 0.899178727792796337744142056981 52 11 12 [31]
38 0.575822018671129680315638899053 1.7366477272053257366407424653 0.899385030242909756785096124579 46 15 13 [31]
39 0.574658698871400098769653505556 1.7401633386285601733704945427 0.898251584116236036061453978371 63 7 0 [31]
40 0.574567244197631877691963829996 1.7404403228667757310944640966 0.900378947243951587697239115283 62 9 0 [31]
41 0.573707210431683976261834902590 1.7430493844544040345910575359 0.900013625450214593270143397620 26 28 7 [31]
42 0.573184606056661124751835587472 1.7446386197977319954694848362 0.900625066113980700678791215119 54 13 13 [31]
43 0.572720998504235952229096808628 1.7460508740061567899856516073 0.901346104544430442893078627318 55 15 12 [31]
44 0.572308077671500114359503945495 1.7473106514040700805286513677 0.902155443887254268915394449339 52 18 13 [31]
45 0.572402522386389535425688772571 1.7470223503399742981458115222 0.904500232395268291554338981786 50 20 12 [31]
46 0.572137850215207871993997165698 1.7478305265485461445617342334 0.905650018695338332191041752597 54 19 11 [31]
47 0.571142662109048640253732853370 1.7508760356078414416608902261 0.904425351200126014874338599140 66 14 16 [31]
48 0.570533546538931339103418855023 1.7527453136916696235507710937 0.904363247930779947683949915334 72 12 0 [31]
49 0.570178863503833393131426508041 1.7538356189755126876683462178 0.905052291983256585929876287889 66 16 0 [31]
50 0.569964924737669036650615804788 1.7544939286575539407014946959 0.906136855352102716009685184241 64 18 0 [31]
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].


References

[1]   , private communication, March 2012.
[31]   , program ccic, 2005–2013.


©  E. Specht     01-Jul-2014