The best known packings of equal circles in a regular heptagon (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 heptagon (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 0.900968867903 1.1099162642 0.931940623499 7 1 C7
2 0.468245218314 2.1356331274 0.503437279335 5 2
3 0.428969548413 2.3311678036 0.633786312847 7 3 [1]
4 0.380581895389 2.6275553622 0.665158146701 9 4 [1]
5 0.341846932230 2.9252858684 0.670813898229 11 5 [1]
6 0.314167684366 3.1830135617 0.679896720384 11 1 4 [1]
7 0.308425907488 3.2422697825 0.764484028117 14 1 5 [1]
8 0.292853112904 3.4146811351 0.787695474600 21 1 7 C7 [1]
9 0.254822558167 3.9242993524 0.670945007893 17 1 8 [1]
10 0.243931374794 4.0995136474 0.683131011143 21 8 [1]
11 0.237517122658 4.2102227781 0.712444775415 23 8 [1]
12 0.228353523658 4.3791748162 0.718398397484 25 9 [1]
13 0.222092931626 4.5026196587 0.736175737109 27 9 [1]
14 0.214636889499 4.6590313638 0.740466545043 29 10 [1]
15 0.206627612425 4.8396242315 0.735252733681 31 10 [1]
16 0.199792679302 5.0051883958 0.733242785448 33 9 [2]
17 0.194450028480 5.1427094551 0.737961351348 35 11 [1]
18 0.191466926357 5.2228341418 0.757580365076 37 10 [2]
19 0.187920598990 5.3213964056 0.770319782358 35 2 11 [1]
20 0.182622215241 5.4757850718 0.765783285064 39 1 12 [1]
21 0.180355045441 5.5446189351 0.784232003098 37 3 11 [1]
22 0.176438418982 5.6676998455 0.786280798030 43 1 10 [2]
23 0.169990525912 5.8826807825 0.763037610251 41 3 11 [1]
24 0.166584178817 6.0029710330 0.764623110357 41 4 11 [1]
25 0.162741103626 6.1447291294 0.760156819932 47 2 13 [1]
26 0.159744194565 6.2600084011 0.761714441912 51 1 13 [1]
27 0.156454264310 6.3916442572 0.758764926361 53 1 14 [1]
28 0.154724984954 6.4630802859 0.769569068234 51 3 14 [1]
29 0.152728780845 6.5475543933 0.776619769802 51 4 14 [1]
30 0.149156265431 6.7043781038 0.766254310420 57 2 15 [2]
31 0.147209872580 6.7930226585 0.771266097139 59 2 14 [1]
32 0.145041181714 6.8945935781 0.772860855244 61 2 14 [2]
33 0.143043653236 6.9908729075 0.775210774271 62 3 14 [2]
34 0.140883472636 7.0980646721 0.774760885719 64 3 13 [1]
35 0.139632414239 7.1616608898 0.783446261126 67 2 15 [1]
36 0.137786873991 7.2575853638 0.784669675764 66 4 16 [1]
37 0.135923616742 7.3570732149 0.784802254495 70 3 17 [2]
38 0.134107351038 7.4567127921 0.784616503678 74 2 17 [2]
39 0.133041908166 7.5164285734 0.792519964229 80 1 16 [2]
40 0.131172331145 7.6235589569 0.790156539361 73 4 18 [2]
41 0.129403691461 7.7277548168 0.788217116662 79 2 16 [2]
42 0.128410781423 7.7875080964 0.795098514256 81 2 18 [1]
43 0.126792802210 7.8868830294 0.793645082748 85 1 16 [1]
44 0.125733507990 7.9533293549 0.798589168884 87 2 17 [2]
45 0.124809399321 8.0122170721 0.804777403437 79 6 18 [2]
46 0.122806228708 8.1429094478 0.796466098768 87 4 18 [2]
47 0.120996771599 8.2646833200 0.789976363809 89 4 19 [2]
48 0.119710199946 8.3535070566 0.789718339441 87 5 20 [1]



Updates

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

01-Dec-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: 16, 18, 22, 30, 32, 33, 37, 38, 39, 40, 41, 44, 45, 46, and 47. Many thanks for your contribution!

References

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