The best known packings of equal circles in a regular tridecagon (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 tridecagon (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 0.970941817426 1.0299278309 0.980457110348 13 1 D1
2 0.490852350956 2.0372725078 0.501157154491 5 2
3 0.453721030612 2.2039974622 0.642304940783 7 3 [1]
4 0.405005977913 2.4690993579 0.682378046914 9 4 [1]
5 0.361832605105 2.7637089248 0.680812609101 11 5 [1]
6 0.328528936971 3.0438719013 0.673504800342 13 5 [1]
7 0.325795415063 3.0694109056 0.772734254637 14 1 5 [1]
8 0.295713378243 3.3816528895 0.727568889597 15 1 7 [1]
9 0.270713532032 3.6939416825 0.685969218499 17 1 8 [1]
10 0.257507390176 3.8833836936 0.689638633874 21 8 [1]
11 0.249350846232 4.0104135001 0.711306154273 19 2 9 [1]
12 0.242636309602 4.1213946983 0.734742257162 25 9 [1]
13 0.232022716624 4.3099228151 0.727857824970 27 9 [1]
14 0.225977211465 4.4252249752 0.743531743088 29 10 [1]
15 0.217182881887 4.6044144516 0.735842098211 31 10 [1]
16 0.212526386682 4.7052980837 0.751601932239 31 1 10 [1]
17 0.204556830817 4.8886169971 0.739808101293 35 10 [1]
18 0.201313210227 4.9673839033 0.758681059493 35 1 10 [1]
19 0.201215077473 4.9698065004 0.800049447442 35 1 12 [1]
20 0.193410671799 5.1703455176 0.778095763546 39 1 12 [1]
21 0.188968572542 5.2918852407 0.779903102247 31 6 8 [1]
22 0.180958379159 5.5261326093 0.749242258480 39 3 11 [1]
23 0.176958040459 5.6510571512 0.749049684184 45 1 12 [1]
24 0.173642698408 5.7589522000 0.752603930742 47 1 13 [1]
25 0.170506316818 5.8648853524 0.755897907516 47 2 11 [1]
26 0.168259635340 5.9431960492 0.765553282388 49 2 13 [1]
27 0.165751602816 6.0331241630 0.771474233853 51 2 15 [1]
28 0.162843659376 6.1408592992 0.772221565622 52 3 12 [1]
29 0.160247487817 6.2403474377 0.774502182358 55 2 14 [1]
30 0.158083305832 6.3257786440 0.779714232217 57 2 16 [1]
31 0.154803458236 6.4598040082 0.772618670644 63 12 [1]
32 0.152521656106 6.5564459863 0.774203608302 61 2 16 [1]
33 0.150798146734 6.6313812315 0.780455483862 63 2 15 [1]
34 0.148380724393 6.7394198545 0.778531302906 61 4 13 [1]
35 0.146487361215 6.8265275018 0.781107025855 67 2 15 [1]
36 0.145124151092 6.8906518486 0.788540626651 69 2 17 [1]
37 0.144398997547 6.9252558327 0.802365540907 71 2 18 [1]
38 0.140947674923 7.0948314723 0.785130081243 71 3 18 [1]
39 0.139180014681 7.1849396071 0.785706871227 75 2 17 [1]
40 0.137832158268 7.2552009093 0.790320583217 71 5 17 [1]
41 0.135458316082 7.3823448344 0.782415399939 79 2 17 [1]
42 0.134201917337 7.4514583684 0.786699582829 73 6 14 [1]
43 0.132116791400 7.5690605971 0.780596643412 85 1 17 [1]
44 0.130778409815 7.6465220935 0.782648882219 85 2 19 [1]
45 0.129575440603 7.7175118629 0.785778407384 85 3 18 [1]
46 0.128357192201 7.7907593868 0.788207270777 89 2 19 [1]
47 0.127001966463 7.8738938290 0.788425999187 89 3 17 [1]
48 0.125742342096 7.9527705889 0.789308021173 91 3 21 [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.