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

Last update: 07-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 dodecagon (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 0.965925826289 1.0352761804 0.977048616657 12 1 D1
2 0.491333809940 2.0352761804 0.505605644621 5 2
3 0.456625858814 2.1899767188 0.655044609005 9 3 C3 [1]
4 0.408248290464 2.4494897428 0.698131700798 12 4 [1]
5 0.360457215913 2.7742543521 0.680308771123 11 5 [1]
6 0.329459311299 3.0352761804 0.681998533586 18 6 C3 [1]
7 0.329459311299 3.0352761804 0.795664955850 24 6 C3 [1]
8 0.294687461094 3.3934256866 0.727514944778 15 1 7 [1]
9 0.270902167318 3.6913695077 0.691665456614 20 1 8 [1]
10 0.256656322814 3.8962609182 0.689814872232 21 8 [1]
11 0.248926770058 4.0172457135 0.713780223768 27 8 [1]
12 0.243095766159 4.1136051680 0.742616602119 27 9 C3 [1]
13 0.230908167326 4.3307259833 0.725856146978 27 9 [1]
14 0.225999527680 4.4247880085 0.748810139846 30 10 [1]
15 0.216453515072 4.6199295940 0.735951445752 31 10 [1]
16 0.211918467266 4.7187959261 0.752464835355 33 11 [1]
17 0.205484903721 4.8665375504 0.751687592636 35 10 [1]
18 0.204124145232 4.8989794856 0.785398163398 54 11 [1]
19 0.204124145232 4.8989794856 0.829031394697 60 12 C3 [1]
20 0.190550724215 5.2479464674 0.760466033783 39 1 12 [1]
21 0.186129703363 5.3725976130 0.761867211308 39 2 12 [1]
22 0.179638465950 5.5667364710 0.743446932991 43 1 11 [1]
23 0.176340572223 5.6708447035 0.748964003860 43 2 13 [1]
24 0.173878839335 5.7511310970 0.759859547709 51 1 11 [1]
25 0.170362625375 5.8698320585 0.759831466770 48 3 12 [1]
26 0.167677684105 5.9638228267 0.765512876126 49 2 13 [1]
27 0.166203679230 6.0167139779 0.781040670420 57 15 C3 [1]
28 0.162997380513 6.1350679186 0.779018729659 58 14 [1]
29 0.159263342623 6.2789087779 0.770297110594 55 2 15 [1]
30 0.158499527940 6.3091670555 0.789234059251 62 16 [1]
31 0.155891230621 6.4147290134 0.788921301764 84 12 C3 [1]
32 0.151891844500 6.5836319474 0.773121068137 59 3 16 [1]
33 0.150922619190 6.6259120427 0.787138627137 66 2 16 [1]
34 0.148120361074 6.7512662861 0.781154716184 67 2 16 [1]
35 0.146402072200 6.8305044114 0.785581251284 73 1 16 [1]
36 0.145105724217 6.8915268877 0.793780103525 71 2 16 [1]
37 0.144948974278 6.8989794856 0.814067908960 96 18 C3 [1]
38 0.140383736593 7.1233322625 0.784234258817 73 2 18 [1]
39 0.138883550489 7.2002767533 0.787761670741 74 3 16 [1]
40 0.137850626782 7.2542288950 0.795987228335 80 3 18 [1]
41 0.135307358781 7.3905810372 0.786059295811 81 1 17 [1]
42 0.133407914534 7.4958071528 0.782782483546 80 3 18 [1]
43 0.132361398066 7.5550728128 0.788896020007 100 1 20 [1]
44 0.131244741439 7.6193528901 0.793679417173 90 2 18 [1]
45 0.129552456718 7.7188810258 0.790919781479 85 4 15 [1]
46 0.128119564827 7.8052091525 0.790710233205 88 4 20 [1]
47 0.126888271657 7.8809490187 0.792445565992 110 20 [1]
48 0.125894723452 7.9431446575 0.796681828928 94 6 14 [1]



Updates

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

07-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.