The best known packings of equal circles in a regular hendecagon (complete up to N = 48)

Last update: 07-Dec-2020


Overview    Download    Results    History of updates    References

Overview

1-12   13-24   25-36   37-48  


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 regular hendecagon (supposed the radius of its circumcircle is 1 and remains fixed for all cases of N)
ratio
= 1/radius
density
ratio of total area occupied by the circles to container area (packing fraction)
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
1 0.959492973614 1.0422171162 0.972662000919 11 1 D1
2 0.487210637272 2.0525003428 0.501581917220 5 2
3 0.449007526756 2.2271341579 0.639008814987 7 3 [1]
4 0.400722084048 2.4954951070 0.678617254976 9 4 [1]
5 0.358901283716 2.7862814801 0.680453386586 11 5 [1]
6 0.328707637159 3.0422171162 0.684934735810 15 5 [1]
7 0.321977224941 3.1058097360 0.766702179897 13 1 5 [1]
8 0.293328415490 3.4091480647 0.727238080114 15 1 7 [1]
9 0.268418161914 3.7255303176 0.685085313842 17 1 8 [1]
10 0.256586353008 3.8973234090 0.695577395351 21 8 [1]
11 0.247283699843 4.0439381999 0.710660238402 23 8 [1]
12 0.241049891696 4.1485187691 0.736670840395 25 9 [1]
13 0.230119021079 4.3455773248 0.727322026602 23 2 7 [1]
14 0.225258087592 4.4393522589 0.750528473597 29 10 [1]
15 0.218430686025 4.5781113368 0.756130768396 31 10 [1]
16 0.213417631498 4.6856484770 0.769943623120 23 5 6 [1]
17 0.205123644807 4.8751083813 0.755716184322 35 10 [1]
18 0.199753916279 5.0061596720 0.758824706311 23 7 5 [1]
19 0.198862635374 5.0285967402 0.793849789648 32 3 10 [1]
20 0.189665172122 5.2724492790 0.760122392704 39 1 12 [1]
21 0.185222380887 5.3989155912 0.761175097745 39 2 12 [1]
22 0.179050572562 5.5850142543 0.745165022397 41 2 13 [1]
23 0.176060167306 5.6798764610 0.753231386799 45 1 12 [1]
24 0.172459859978 5.7984507243 0.754163747520 45 2 14 [1]
25 0.169517654009 5.8990906041 0.759011275630 45 3 12 [1]
26 0.166775028403 5.9961015122 0.764035867049 53 14 [1]
27 0.164436788130 6.0813642213 0.771329758122 47 4 15 [1]
28 0.161134239681 6.2060056384 0.768089895427 53 2 13 [1]
29 0.158798636271 6.2972832984 0.772627010958 53 3 13 [1]
30 0.156893263963 6.3737599355 0.780204054539 55 3 16 [1]
31 0.153391023890 6.5192862962 0.770619401296 59 2 15 [1]
32 0.151631494568 6.5949359851 0.777333101314 61 2 15 [1]
33 0.149947508841 6.6690004237 0.783918298392 61 3 16 [1]
34 0.147329755360 6.7874951503 0.779719156990 63 3 15 [1]
35 0.145497957877 6.8729486970 0.782816898647 65 3 16 [1]
36 0.144002526401 6.9443226101 0.788716771438 67 3 15 [1]
37 0.142934221474 6.9962251845 0.798642683217 67 4 17 [1]
38 0.139677729861 7.1593374334 0.783278642085 73 2 18 [1]
39 0.138535844662 7.2183484530 0.790801115184 77 1 18 [1]
40 0.136314304400 7.3359872568 0.785273979182 73 4 19 [1]
41 0.134374576970 7.4418838932 0.782161488941 79 2 16 [1]
42 0.132825501679 7.5286747451 0.782871660825 79 3 16 [1]
43 0.131618416086 7.5977209705 0.787009777625 81 3 17 [1]
44 0.130192777808 7.6809176119 0.787961167677 84 2 18 [1]
45 0.128819186343 7.7628187880 0.788954521212 88 1 17 [1]
46 0.127654451466 7.8336476990 0.791968870527 89 2 18 [1]
47 0.126495439168 7.9054233621 0.794558621475 86 5 15 [1]
48 0.125276215339 7.9823611951 0.795896946990 83 7 18 [1]



Updates

Please note that the results are taken from a running search. For updates look at the list below.

07-Dec-2020: First complete presentation from N=1 to N=48 by P. Amore [1].
The symmetry groups are still wrong and have to be corrected.

References

[1]   , private communication, December 2020.