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

Last update: 08-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 tetradecagon (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 0.974927912182 1.0257168633 0.983158339383 14 1 D1
2 0.493652404307 2.0257168633 0.504139779747 5 2
3 0.454819312439 2.1986753259 0.641914975876 7 3 [1]
4 0.406873105275 2.4577687417 0.684946130985 10 4 [1]
5 0.362829076902 2.7561186897 0.680851863728 11 5 [1]
6 0.328822678438 3.0411527719 0.671047455716 11 1 4 [1]
7 0.325805629975 3.0693146711 0.768588121139 16 6 [1]
8 0.300256863661 3.3304817342 0.746026635144 21 1 7 [1]
9 0.271660289557 3.6810680046 0.687026179638 18 1 8 [1]
10 0.258000683320 3.8759587267 0.688525724570 21 8 [1]
11 0.250126183990 3.9979820747 0.711851588502 19 2 9 [1]
12 0.244457938076 4.0906832802 0.741767828048 19 3 6 [1]
13 0.231823619680 4.3136243036 0.722665257044 22 3 10 [1]
14 0.227332533938 4.3988424476 0.748392936298 30 10 [1]
15 0.217889144010 4.5894897818 0.736615584681 27 2 8 [1]
16 0.212612663912 4.7033886957 0.748129360601 33 11 [1]
17 0.205930729626 4.8560018304 0.745709535239 36 10 [1]
18 0.201644753853 4.9592165474 0.757050444385 33 2 10 [1]
19 0.201215442455 4.9697974857 0.795709742351 38 12 [1]
20 0.192142389595 5.2044736308 0.763756341557 39 1 12 [1]
21 0.187414903167 5.3357549645 0.762967430105 39 2 12 [1]
22 0.182427174352 5.4816394737 0.757321352757 45 13 [1]
23 0.179166353721 5.5814050977 0.763693695038 38 5 10 [1]
24 0.175601283264 5.6947192037 0.765499781189 45 2 12 [1]
25 0.171564075501 5.8287260726 0.761151619714 45 3 12 [1]
26 0.169066629971 5.9148277822 0.768718969660 49 2 12 [1]
27 0.166464003051 6.0073047726 0.773896515007 55 15 [1]
28 0.163544012412 6.1145619778 0.774650467026 56 2 12 [1]
29 0.160421750958 6.2335686653 0.771974510669 53 3 14 [1]
30 0.158737022368 6.2997275940 0.781908928146 62 16 [1]
31 0.155361076522 6.4366186331 0.773970835569 60 1 12 [1]
32 0.153063596127 6.5332321029 0.775482964753 63 1 15 [1]
33 0.151326341693 6.6082348176 0.781666436964 63 2 15 [1]
34 0.148809643700 6.7199945859 0.778788498877 65 2 15 [1]
35 0.146878354364 6.8083551476 0.781019898707 65 3 15 [1]
36 0.145528481503 6.8715071419 0.788636647921 67 3 16 [1]
37 0.144752841409 6.9083272582 0.801926153338 64 6 16 [1]
38 0.141312337436 7.0765229572 0.784914252892 73 2 18 [1]
39 0.139542815185 7.1662593210 0.785521409366 75 2 17 [1]
40 0.138029977789 7.2448030205 0.788288672750 69 6 17 [1]
41 0.135470519964 7.3816797947 0.778308745922 77 3 19 [1]
42 0.134000115596 7.4626801294 0.780078116711 79 3 16 [1]
43 0.132508412916 7.5466906440 0.780969042459 83 2 18 [1]
44 0.131173841559 7.6234711747 0.783115121410 85 2 17 [1]
45 0.129981471472 7.6934042112 0.786418770063 85 3 16 [1]
46 0.128779750534 7.7651959710 0.789098920260 85 4 17 [1]
47 0.127551084323 7.8399960714 0.790941979245 85 5 19 [1]
48 0.126176578488 7.9254011480 0.790455069229 94 2 18 [1]



Updates

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

08-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.