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

Last update: 05-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 octagon (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 0.923879532511 1.0823922003 0.948059448969 8 1 D8
2 0.480216935052 2.0823922003 0.512282851198 5 2 D2
3 0.435387434219 2.2968049177 0.631652057640 7 3 D1 [1]
4 0.400543816310 2.4966057627 0.712795473791 12 4 D4 [1]
5 0.348203325454 2.8718852662 0.673349764307 11 5 D1 [1]
6 0.317727658062 3.1473495449 0.672769311537 13 5 D1 [2]
7 0.310945216831 3.2160005875 0.751745228811 16 6 D2 [1]
8 0.285852433677 3.4983085053 0.726070269949 15 1 7 D1 [1]
9 0.270598050073 3.6955181300 0.731975785070 24 1 8 D8 [1]
10 0.251932403490 3.9693186988 0.704973738556 13 4 4 D2 [1]
11 0.239556874251 4.1743740526 0.701156298830 23 8 [1]
12 0.234100964947 4.2716611622 0.730453430275 23 1 8 [2]
13 0.223537235211 4.4735276387 0.721519348455 23 2 7 [1]
14 0.219348902644 4.5589469012 0.748176127867 30 10 D2 [1]
15 0.210236890635 4.7565391449 0.736400352696 31 10 [1]
16 0.204286601358 4.8950836391 0.741659609991 33 11 [2]
17 0.198194125496 5.0455582248 0.741712086622 31 2 8 [1]
18 0.193696359493 5.1627196434 0.750101960981 37 12 D1 [1]
19 0.191508243904 5.2217073251 0.773986575243 39 12 D1 [1]
20 0.185530744908 5.3899422465 0.764656969740 39 1 12 [2]
21 0.180670260259 5.5349452564 0.761373074646 39 2 12 [1]
22 0.175500940767 5.6979751540 0.752638547652 45 13 [2]
23 0.172561559662 5.7950333896 0.760712985857 46 1 14 D2 [1]
24 0.168652864098 5.9293389730 0.758234537567 45 2 11 [2]
25 0.166450891366 6.0077779806 0.769337918776 49 1 11 [2]
26 0.163606136259 6.1122401816 0.772996280163 50 2 10 [2]
27 0.161395047685 6.1959769791 0.781176288426 56 12 D1 [2]
28 0.160404294689 6.2342470440 0.800193270075 64 12 D2 [1]
29 0.154927003952 6.4546526718 0.773138182171 56 2 14 [1]
30 0.152319579409 6.5651441783 0.773103425594 58 2 14 C2 [1]
31 0.149878153406 6.6720864734 0.773469584440 61 1 15 [1]
32 0.147282442434 6.7896755613 0.771004322284 61 2 14 [1]
33 0.145306599845 6.8819998614 0.773908292152 68 14 D1 [2]
34 0.143477532364 6.9697323582 0.777412633108 63 3 15 [1]
35 0.142256369270 7.0295622272 0.786713067631 69 2 16 [1]
36 0.139934447382 7.1462032309 0.782990790742 69 2 16 [1]
37 0.138229583693 7.2343413999 0.785251190287 68 4 16 C2 [1]
38 0.136097981333 7.3476475566 0.781793117866 73 2 14 [2]
39 0.134961309874 7.4095309310 0.789020079500 78 1 18 [1]
40 0.132682454149 7.5367915556 0.782153264267 78 2 15 [2]
41 0.131274622862 7.6176185328 0.784784278851 82 17 [1]
42 0.129318110933 7.7328689136 0.780140587971 85 17 [1]
43 0.127890596942 7.8191831449 0.781179006956 83 5 14 [2]
44 0.126725223499 7.8910888644 0.784844623208 88 2 17 [2]
45 0.125455094344 7.9709796181 0.786672549018 85 3 18 [1]
46 0.124405540951 8.0382271750 0.790755386432 84 6 16 [1]
47 0.123164266115 8.1192380838 0.791903356905 87 4 17 [2]
48 0.122028320701 8.1948189917 0.793902895812 94 2 16 C2 [2]



Updates

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

30-NOV-2020: First complete presentation from N=1 to N=48 by E. Specht [1]. The symmetry groups are still wrong and have to be corrected.
05-Dec-2020: Only one day after the unexpected publication of packings in hexagonal containers, Paolo Amore (Universidad de Colima, Mexico) sent me packings from his own repository which improve my result for several values of N: 6, 12, 16, 20, 22, 24, 25, 26, 27, 33, 38, 40, 43, 44, 47, and 48. I have failed to find the simple configuration for N = 6.

References

[1]   , program coc, 2020.
[2]   , private communication, December 2020.