The best known packings of unequal circles with radii of i-2/3, i=1,2,3,..., in a circle (complete up to N = 40)


Last update: 22-Mar-2012

Overview    Download    Results    History of updates    References

Overview

5-16   17-28   29-40  

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 5.4501014927643883946924077927 0.716065217699803972997931399649 3 3 6 D1 [1]
6 0.613511790435690629043456884943 6.2487122256527938848997352755 0.750588400373934213926326021057 3 4 6 D1 [1]
7 0.613507177767355377324499622184 7.0146156827699955211380561934 0.778685888249537048237893508024 10 2 9 [1]
8 0.612938866788355682440699477063 7.7606898791317636904370309238 0.800724792414343253194566245020 16 12 [1]
9 0.610551304892410581219281177072 8.5105909709295952067862465007 0.814411144239406655479651634532 16 1 11 [1]
10 0.607180306029151381565813193090 9.2615211594717050436578193384 0.822554905022055328060214497081 18 1 11 [1]
11 0.605945357046045531870234623638 9.9680693717705170640431488633 0.834221014708460383082950810928 18 2 11 [31]
12 0.603587631260896413485275236502 10.6818285481970069721836386155 0.841002637319033494915334305994 18 3 13 [31]
13 0.601403181962008352686349070579 11.3839188873056872225618975388 0.846758714384872781115733843423 22 2 13 [31]
14 0.598736595717535641336680955992 12.0881606488855117870759290625 0.849890718995485453037633335766 14 7 9 [31]
15 0.597728254148429417644442070381 12.7515993587723369980520608666 0.856688472137618595773512018039 18 6 10 [31]
16 0.595368574979914121162752069512 13.4370545174808647718224221084 0.858729661115309475400518746377 28 2 12 [31]
17 0.593880310526356882505435811325 14.0973591482978237445146027318 0.862510452140131579864634494536 20 7 11 [31]
18 0.593110930354651323544395103821 14.7339248755817324205747193614 0.867734300796625387762836868570 26 5 14 [31]
19 0.591817584814619928050295226494 15.3772025653454446699762088744 0.870862337019110140053358582966 36 1 14 [31]
20 0.589467028756857307078430309218 16.0440120119640184639943687912 0.870358843847203442361393750913 30 5 12 [31]
21 0.587819532576617578088707546182 16.6886212325949896075383299082 0.871464407091012520895807238210 28 7 14 [31]
22 0.586575646462604663544984327813 17.3178104853656981947944197161 0.873361597000185084993477050810 28 8 15 [31]
23 0.585721509156687414681507349032 17.9310079999388503899101273798 0.876064969048636033397858537179 26 10 12 [31]
24 0.585628166251287346739927738119 18.5155439375936984458874916548 0.880739836261994492429342326817 30 9 13 [31]
25 0.583999434874983642556683027494 19.1444361412649588500595113441 0.880513246036122999184064576298 34 8 16 [31]
26 0.582433329725615932311662488641 19.7689585800697892213942696780 0.880201193918056849363702162469 14 19 9 [31]
27 0.581006953301845839590470594883 20.3864446875541412572277794553 0.880062775513784388157334197695 36 9 14 [31]
28 0.579696449422450033964101675785 20.9975141758878683607995429357 0.880049527172803180559447353058 20 18 12 [31]
29 0.579696449422450033964101675785 21.5574733582701665339630926191 0.883821228404057024859469461742 20 19 12 [31]
30 0.579696449422450033964101675785 22.1126249861598562022640878477 0.887426236749683814864218717013 20 20 12 [31]
31 0.579583815854970758601834988234 22.6675728694831820901264624019 0.890530873937452077349475863194 22 20 13 [31]
32 0.579010407161471504263676334508 23.2367889724430348949842362314 0.892069600447035820922379935843 30 17 15 [31]
33 0.575539057399854251991335533801 23.9227253188003309987585211069 0.884534611930522335328648239886 24 21 12 [31]
34 0.574308633006290611373775850618 24.5168033249964117943858278611 0.883751086714648119459558932848 24 22 10 [31]
35 0.573785176027795895039283845707 25.0785088090548903462741503460 0.885016505273708990859945975266 22 24 11 [31]
36 0.571856417351331953737338580994 25.7003996296323749663800472582 0.881827686721508863127849515200 58 7 22 [31]
37 0.571925861103822538530692042076 26.2308013403701227429235040402 0.884694910593234958295993148009 38 18 14 [31]
38 0.573159974682735325384083692436 26.7031098685177572236040710349 0.891088479581864412196925133991 36 20 12 [31]
39 0.571032275378012506938638046397 27.3298847025182760932417453543 0.886950397858528180142204419089 42 17 15 [31]
40 0.573363925112205332049813661236 27.7405219943483169469832646355 0.896611560042512092737566692241 30 25 12 [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].

References

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

©  E. Specht     22-Mar-2012