The best known packings of equal circles in a regular nonagon (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 nonagon (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 0.469846310393 2.1283555450 0.959050541874 5 2
2 0.480860690389 2.0796043844 0.502271353194 5 2
3 0.450678162524 2.2188783109 0.661796013624 9 3 C3 [1]
4 0.395232683712 2.5301551243 0.678633909137 9 4 [1]
5 0.352206160693 2.8392461904 0.673649145168 11 5 [1]
6 0.321188475785 3.1134367370 0.672265805197 11 1 4 [1]
7 0.316435787354 3.1601988143 0.761270657441 18 6 C3 [1]
8 0.288933606801 3.4610027233 0.725363949019 15 1 7 [1]
9 0.264727783683 3.7774652365 0.685032883034 17 1 8 D1 [2]
10 0.254814656089 3.9244210492 0.705210336498 21 8 [1]
11 0.250753377701 3.9879821726 0.751200954116 32 8 [1]
12 0.243317228279 4.1098610529 0.771608191304 21 3 6 C3 [1]
13 0.227702746932 4.3916905416 0.732065186087 23 2 7 [2]
14 0.221049182018 4.5238801197 0.742977602962 25 2 8 [1]
15 0.211711448133 4.7234101359 0.730213414238 31 10 [1]
16 0.207403301040 4.8215240306 0.747517171648 33 11 [2]
17 0.199389671625 5.0153049145 0.734047391663 35 10 [2]
18 0.196449564621 5.0903650610 0.754474400370 32 3 8 [2]
19 0.194579251484 5.1392941044 0.781297653383 42 12 C3 [1]
20 0.187357378977 5.3373931972 0.762502834249 39 1 12 [1]
21 0.182666711750 5.4744512036 0.761040867532 39 2 12 [1]
22 0.176922137610 5.6522039215 0.747923017463 43 1 11 [1]
23 0.173790392372 5.7540580141 0.754482595143 43 2 13 [2]
24 0.170185826540 5.8759299780 0.754966846205 54 12 C3 [2]
25 0.167429888951 5.9726492460 0.761159810310 47 1 14 [1]
26 0.165098269544 6.0569986758 0.769711991735 51 1 13 [1]
27 0.163332030349 6.1224978216 0.782305435548 57 15 C3 [1]
28 0.159363996686 6.2749430285 0.772339629104 55 1 14 [1]
29 0.157102559089 6.3652686869 0.777381818277 53 3 14 [2]
30 0.154316890493 6.4801720460 0.775921961392 59 1 15 [1]
31 0.151664942387 6.5934815539 0.774465301815 61 1 14 [2]
32 0.149548357796 6.6868002748 0.777290101603 63 1 14 [2]
33 0.147453006374 6.7818217111 0.779275577024 64 1 13 [1]
34 0.145733308872 6.8618492762 0.784271495056 63 3 14 [2]
35 0.143989066189 6.9449717709 0.788128326370 65 3 14 [2]
36 0.142492834502 7.0178967489 0.793886495203 66 4 15 [2]
37 0.140856382056 7.0994298264 0.797305288252 90 18 C3 [2]
38 0.137559398280 7.2695869021 0.780969356854 70 4 16 [2]
39 0.136132915409 7.3457620223 0.784983919119 75 2 17 [1]
40 0.134180360633 7.4526554801 0.782181901937 73 4 19 [1]
41 0.132473464591 7.5486815649 0.781468556225 77 3 15 [2]
42 0.131089702202 7.6283642667 0.783892133613 84 1 17 [2]
43 0.129501438595 7.7219219404 0.783226735185 87 18 [2]
44 0.128415078842 7.7872474869 0.788051471195 84 2 15 [2]
45 0.127201761734 7.8615263371 0.790803583316 89 2 17 [2]
46 0.125622435834 7.9603614861 0.788428135980 88 4 18 [1]
47 0.124425326049 8.0369490019 0.790287831695 92 3 18 [1]
48 0.123505835511 8.0967834100 0.795217732043 93 3 21 C3 [2]



Updates

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

03-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: Paolo Amore (Universidad de Colima, Mexico) sent me packings from his own repository which improve my result for several values of N: 9, 13, 16–18, 23, 24, 29, 31, 32, 34–38, 41–45, and 48. He found the simple configuration for N = 9 and such exciting symmetric packings for N = 37 and 48.

References

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