The best known packings of equal circles in a regular heptadecagon (in honour of Carl Friedrich Gauß, complete up to N = 48)

Last update: 10-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 heptadecagon (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 0.982973099684 1.0173218375 0.988590371264 17 1 D1
2 0.494656244315 2.0216059364 0.500691360286 5 2
3 0.457822884584 2.1842507958 0.643353043366 7 3 [1]
4 0.408795972255 2.4462080546 0.683921596576 9 4 [1]
5 0.365326873040 2.7372746814 0.682757302356 11 5 [1]
6 0.331419733741 3.0173218375 0.674281258552 15 5 [1]
7 0.328614661434 3.0430778579 0.773401521605 13 1 5 [1]
8 0.298690247275 3.3479499553 0.730239266296 15 1 7 [1]
9 0.273307969935 3.6588761032 0.687828572245 17 1 8 [1]
10 0.260282391844 3.8419809842 0.693142755067 21 8 [1]
11 0.251533840030 3.9756082119 0.712063356965 19 2 9 [1]
12 0.244967342626 4.0821767885 0.736767971950 25 9 [1]
13 0.233629941745 4.2802732926 0.725994722900 27 9 [1]
14 0.228083724171 4.3843549277 0.745160348301 29 10 [1]
15 0.218502689075 4.5766027147 0.732719823749 27 2 8 [1]
16 0.214846111817 4.6544942868 0.755628082319 33 11 [1]
17 0.206295840269 4.8474074838 0.740223678942 31 2 8 [1]
18 0.203176655691 4.9218252786 0.760244401007 35 1 10 [1]
19 0.202580564260 4.9363077038 0.797778382713 37 1 12 [1]
20 0.192996131978 5.1814509946 0.762184856733 39 1 12 [1]
21 0.188333785043 5.3097217781 0.762094582622 39 2 12 [1]
22 0.181721979977 5.5029116463 0.743311259126 45 13 [1]
23 0.178546501921 5.6007818089 0.750176815696 43 2 13 [1]
24 0.175169497155 5.7087564687 0.753461938326 45 2 14 [1]
25 0.171895472764 5.8174888723 0.755791488008 47 2 13 [1]
26 0.169873509696 5.8867330273 0.767640319585 49 2 13 [1]
27 0.167566473576 5.9677808971 0.775659533171 51 2 15 [1]
28 0.164410242053 6.0823461331 0.774370640984 53 2 12 [1]
29 0.161452859535 6.1937583693 0.773432808825 55 2 14 [1]
30 0.159600237657 6.2656548304 0.781846384833 57 2 16 [1]
31 0.156702903271 6.3815026979 0.778841150616 64 12 [1]
32 0.154778142755 6.4608605724 0.784336364270 55 5 12 [1]
33 0.153459032305 6.5163971451 0.795118687372 65 1 16 [1]
34 0.150010324801 6.6662078182 0.782806332199 61 4 13 [1]
35 0.148060256584 6.7540069366 0.785015351803 63 4 16 [1]
36 0.146631736276 6.8198060352 0.791938697974 66 4 14 [1]
37 0.146005022652 6.8490794484 0.806994224018 75 18 [1]
38 0.141992821653 7.0426095373 0.783879822890 71 3 18 [1]
39 0.140132664841 7.1360949364 0.783567616420 71 4 17 [1]
40 0.138730504857 7.2082200020 0.787656815074 73 4 18 [1]
41 0.136259931839 7.3389145767 0.778849062841 77 3 19 [1]
42 0.134681727650 7.4249121796 0.779470633984 81 2 17 [1]
43 0.133273763675 7.5033522910 0.781431458217 85 1 17 [1]
44 0.132010580697 7.5751503760 0.784518645163 79 5 14 [1]
45 0.130501163921 7.6627669053 0.784105308707 83 4 16 [1]
46 0.129279713382 7.7351656640 0.786595944719 81 6 16 [1]
47 0.128109061194 7.8058490998 0.789206521629 85 5 18 [1]
48 0.126879617374 7.8814865673 0.790602290425 89 4 21 [1]



Updates

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

10-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.