The best known packings of equal circles in a regular decagon (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 decagon (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 0.475528258148 2.1029244484 0.966882799046 10 2
2 0.487457184532 2.0514622242 0.508000488876 5 2
3 0.446672272110 2.2387778746 0.639823850885 7 3 [1]
4 0.399888622321 2.5006963044 0.683753039327 10 4 [1]
5 0.363271264003 2.7527638409 0.705331496240 15 5 [1]
6 0.327711741622 3.0514622242 0.688805232459 15 5 [1]
7 0.318272363712 3.1419630292 0.757978840385 14 1 5 [1]
8 0.291381365584 3.4319284557 0.726063690358 15 1 7 [1]
9 0.267080810777 3.7441851292 0.686260598058 18 1 8 [1]
10 0.253797712807 3.9401458309 0.688551757272 21 8 [1]
11 0.245912809724 4.0664819418 0.711076247882 19 2 9 [1]
12 0.239508524406 4.1752167380 0.735841672469 25 9 [1]
13 0.233234508668 4.2875302017 0.755944916409 30 3 10 [1]
14 0.227176173360 4.4018700782 0.772351116056 22 4 6 [1]
15 0.217430954423 4.5991611574 0.758045347398 35 10 [1]
16 0.209167036152 4.7808680488 0.748286053443 33 11 [1]
17 0.202348453391 4.9419700682 0.744063298308 36 10 [1]
18 0.197916840583 5.0526271390 0.753701166192 31 3 8 [1]
19 0.196742459061 5.0827869326 0.786160057086 39 12 [1]
20 0.188578704372 5.3028256999 0.760285090570 39 1 12 [1]
21 0.184347905722 5.4245259586 0.762881167739 39 2 12 [1]
22 0.178203275655 5.6115691270 0.746818778254 43 1 11 [1]
23 0.175298553497 5.7045536318 0.755519510467 43 2 13 [1]
24 0.172069669621 5.8115994655 0.759593210955 43 3 9 [1]
25 0.168694134478 5.9278883827 0.760503393793 51 13 [1]
26 0.166892314387 5.9918876653 0.774118070815 45 4 12 [1]
27 0.163673253308 6.1097337518 0.773179582539 51 2 13 [1]
28 0.160493896012 6.2307665578 0.770967824967 50 4 14 [1]
29 0.157972155276 6.3302295158 0.773606780941 55 2 13 [1]
30 0.156362761909 6.3953846030 0.784059648850 62 16 [1]
31 0.152858996040 6.5419767623 0.774292196963 59 2 16 [1]
32 0.150803659853 6.6311387998 0.777919977294 57 4 13 [1]
33 0.148349698972 6.7408293170 0.776333740740 65 1 16 [1]
34 0.146024252954 6.8481774758 0.774979271162 65 2 15 [1]
35 0.144665247316 6.9125102162 0.782992595457 64 4 16 [1]
36 0.143055643571 6.9902869614 0.787541910350 61 6 14 [1]
37 0.141569957672 7.0636455393 0.792693161442 70 3 17 [1]
38 0.139018002886 7.1933129468 0.785031114233 71 3 15 [1]
39 0.137467724844 7.2744347892 0.787820504577 77 1 18 [1]
40 0.135576893097 7.3758881558 0.785945676145 79 1 17 [1]
41 0.134256277432 7.4484412880 0.789976627245 77 3 18 [1]
42 0.133318962162 7.5008084655 0.797984253966 79 3 16 [1]
43 0.131069130656 7.6295615527 0.789642430187 81 3 17 [1]
44 0.129604642532 7.7157729882 0.790050726298 85 2 17 [1]
45 0.128450718174 7.7850868739 0.793682437453 83 4 17 [1]
46 0.126797839627 7.8865696998 0.790574356095 89 2 20 [1]
47 0.125313743564 7.9799706845 0.788962659556 88 4 18 [1]
48 0.124153849577 8.0545227023 0.790902230882 98 18 [1]



Updates

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

05-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.