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

Last update: 04-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 pentagon (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 0.809016994375 1.2360679775 0.864806265977 5 1 C3
2 0.437152698242 2.2875302017 0.505009863813 5 2 D1
3 0.389795792433 2.5654458550 0.602280894881 7 3 D1 [1]
4 0.354680462504 2.8194392015 0.664872028015 9 4 D1 [1]
5 0.340440645254 2.9373695942 0.765695954932 15 5 [1]
6 0.309016994375 3.2360679775 0.757041202686 15 5 [1]
7 0.277511095117 3.6034595286 0.712298912009 13 1 5 D1 [1]
8 0.257541824500 3.8828644704 0.701114681674 15 1 7 D1 [1]
9 0.243640224399 4.1044125717 0.705901377804 17 1 6 D1 [1]
10 0.230721507966 4.3342296469 0.703363296033 17 2 5 [2]
11 0.224333593340 4.4576471366 0.731450345446 21 1 7 [1]
12 0.217384640521 4.6001410109 0.749277161807 27 9 D1 [1]
13 0.207999994722 4.8076924297 0.743144963272 25 1 8 [2]
14 0.202386816023 4.9410333126 0.757697772154 25 2 7 [1]
15 0.197735768366 5.0572539721 0.774934987330 35 10 [1]
16 0.191785609135 5.2141555590 0.777598749710 40 10 [1]
17 0.183126736787 5.4606990631 0.753279195582 31 2 10 D1 [2]
18 0.178473321196 5.6030783385 0.757569801809 35 1 10 D1 [2]
19 0.173441904330 5.7656193517 0.755205607498 37 1 11 [1]
20 0.169279411749 5.9073929291 0.757254419213 41 11 [2]
21 0.165404018640 6.0458023222 0.759127889229 50 10 [2]
22 0.161673301786 6.1853131528 0.759806203277 45 12 [2]
23 0.158923367389 6.2923408711 0.767550381308 47 11 [1]
24 0.156027298450 6.4091348753 0.771997613946 41 4 10 [1]
25 0.153122210789 6.5307312038 0.774497337032 47 2 12 [1]
26 0.150683336700 6.6364338745 0.780022883175 45 4 12 [2]
27 0.148137584795 6.7504813271 0.782884729566 47 4 11 [1]
28 0.145444423784 6.8754784404 0.782628611099 53 2 13 [1]
29 0.143333013135 6.9767597717 0.787216154549 53 3 13 [1]
30 0.141698173815 7.0572539721 0.795890460898 70 15 [2]
31 0.138616362210 7.2141555590 0.787035327241 70 15 [1]
32 0.135793611068 7.3641167072 0.779672455391 66 1 15 [1]
33 0.134134799538 7.4551868974 0.784513474860 63 2 14 [1]
34 0.131960534658 7.5780232521 0.782295061266 66 1 17 [1]
35 0.130170592942 7.6822266642 0.783605271724 67 3 16 [2]
36 0.128215138245 7.7993910367 0.781960200280 70 2 17 [2]
37 0.126679668746 7.8939265464 0.784547243181 77 1 18 [2]
38 0.125056798044 7.9963665762 0.785238789166 75 1 16 [2]
39 0.124185741467 8.0524542366 0.794715374348 77 1 16 [2]
40 0.122413791780 8.1690141728 0.791998265961 80 1 17 [2]
41 0.120982726955 8.2656427506 0.792928693410 80 2 15 [2]
42 0.119848964213 8.3438351476 0.797115765780 81 3 15 [1]
43 0.118421345534 8.4444235580 0.796768171019 87 3 16 D1 [1]
44 0.117257562843 8.5282345612 0.799351771065 85 3 17 [1]
45 0.115792923917 8.6361063023 0.797223502406 84 4 16 [1]
46 0.114685668831 8.7194852695 0.799428585597 80 7 14 [2]
47 0.113728619203 8.7928615242 0.803231865120 93 3 18 [2]
48 0.112997340346 8.8497658169 0.809806426252 92 2 18 [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.
04-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: 10, 13, 17, 18, 20, 21, 22, 26, 30, 35, 36, 37, 38, 39, 40, 41, 46, and 47. Among them are two packings which show a reflection symmetry like N = 17 and 18.

References

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