The best known packings of equal circles in a regular pentadecagon (complete up to N = 122)

Last update: 14-Dec-2020


Overview    Download    Results    History of updates    References

Overview

1-12   13-24   25-36   37-48   49-60   61-72   73-84   85-96   97-108   109-120  


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 pentadecagon (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 0.978147600734 1.0223405949 0.985335425987 15 1 D1
2 0.493133395003 2.0278488744 0.500880509957 5 2
3 0.459339047264 2.1770411332 0.651873452267 9 3 C3 [1]
4 0.406997390340 2.4570182113 0.682367839122 9 4 [1]
5 0.367155419957 2.7236422116 0.694137010441 15 5 [1]
6 0.330869393641 3.0223405949 0.676456307805 15 5 [1]
7 0.327244319326 3.0558208071 0.772000500703 18 6 C3 [1]
8 0.297701136798 3.3590735015 0.730173861943 15 1 7 [1]
9 0.272092054641 3.6752267585 0.686198144013 17 1 8 [1]
10 0.258769830772 3.8644381264 0.689608479493 21 8 [1]
11 0.250987901042 3.9842557982 0.713630770631 27 8 [1]
12 0.245253312580 4.0774168939 0.743337973628 27 9 C3 [1]
13 0.233258591424 4.2870875362 0.728440290891 27 9 [1]
14 0.227740446201 4.3909635582 0.747796932723 29 10 [1]
15 0.218419147385 4.5783531891 0.736966901512 35 10 [1]
16 0.213488734913 4.6840879000 0.751009134615 33 11 [1]
17 0.205729279778 4.8607568212 0.740996987108 36 10 [1]
18 0.202255567522 4.9442396679 0.758313495634 42 12 C3 [1]
19 0.201730511417 4.9571083371 0.796291517389 42 12 C3 [1]
20 0.192381611410 5.1980019955 0.762311411885 38 1 12 [1]
21 0.187632715972 5.3295609714 0.761398004197 39 2 12 [1]
22 0.181362657703 5.5138142144 0.745235840000 43 1 11 [1]
23 0.178174367236 5.6124795924 0.751958019965 43 2 13 [1]
24 0.175473627788 5.6988620604 0.761044866536 48 3 9 C3 [1]
25 0.172500077830 5.7970988337 0.766114899703 46 2 14 [1]
26 0.170298665878 5.8720366061 0.776553083913 43 5 10 [1]
27 0.168362014671 5.9395820486 0.788183431979 57 15 C3 [1]
28 0.163697719154 6.1088206065 0.772713673462 53 2 12 [1]
29 0.160868818668 6.2162450640 0.772888868575 55 2 14 [1]
30 0.158917906374 6.2925570996 0.780265193149 57 2 16 [1]
31 0.155977032566 6.4112003130 0.776708951916 66 12 C3 [1]
32 0.153127899757 6.5304885758 0.772740971486 63 1 15 [1]
33 0.151794657144 6.5878471536 0.783072946675 63 2 17 [1]
34 0.149404813236 6.6932247920 0.781597936872 65 2 15 [1]
35 0.147126416010 6.7968759596 0.780233629269 65 3 15 [1]
36 0.145963846257 6.8510115734 0.789893259438 70 2 16 [1]
37 0.145167710374 6.8885842273 0.803002851659 78 18 C3 [1]
38 0.141559726544 7.0641560591 0.784220762185 73 2 18 [1]
39 0.139698271087 7.1582847248 0.783830175979 73 3 16 [1]
40 0.138277813782 7.2318181250 0.787662754501 74 4 17 [1]
41 0.135827391017 7.3622852689 0.778993589345 77 3 17 [1]
42 0.134285460356 7.4468225923 0.779978415360 79 3 19 [1]
43 0.132829039385 7.5284742299 0.781321596229 87 17 [1]
44 0.131715241216 7.5921358133 0.786140279763 85 2 18 [1]
45 0.130375566772 7.6701488228 0.787735181109 96 18 C3 [1]
46 0.128952850181 7.7547723730 0.787762008492 91 1 19 [1]
47 0.127640470924 7.8345057234 0.788587634798 92 2 20 [1]
48 0.126536909746 7.9028324780 0.791500137729 99 21 C3 [2]
49 0.125056825729 7.9963648059 0.789198391250 95 2 20 [1]
50 0.123991417245 8.0650743593 0.791641516155 95 3 20 [2]
51 0.122907586240 8.1361942789 0.793419490706 95 4 20 [1]
52 0.121794550867 8.2105479505 0.794391099258 99 3 21 [2]
53 0.120505834060 8.2983534183 0.792624194560 103 2 16 [2]
54 0.120070139599 8.3284653732 0.801750243153 97 6 12 [1]
55 0.119472121264 8.3701535506 0.808483476623 108 3 15 C3 [2]
56 0.117745060420 8.4929252780 0.799555711862 105 4 18 [1]
57 0.116766332686 8.5641124201 0.800360129560 114 3 17 [2]
58 0.115783042154 8.6368433701 0.800743119323 109 4 19 [2]
59 0.114583936005 8.7272268249 0.797764660076 111 4 22 [1]
60 0.113824066333 8.7854882734 0.800561596653 112 5 21 [2]
61 0.113292327933 8.8267230292 0.806317611899 114 6 24 C3 [2]
62 0.111713253450 8.9514893633 0.796849673387 113 6 20 [1]
63 0.110972872243 9.0112112968 0.799005028137 119 4 22 [2]
64 0.110074911334 9.0847222849 0.798604902549 119 5 20 [2]
65 0.109402424857 9.1405652234 0.801202991374 123 4 20 [1]
66 0.108586596204 9.2092397677 0.801441236641 127 3 21 [2]
67 0.107721271816 9.2832175404 0.800669078642 130 2 19 [1]
68 0.107221100639 9.3265224293 0.805090577937 131 3 18 [1]
69 0.106745261257 9.3680973584 0.809695282698 129 5 19 [2]
70 0.105712319242 9.4596354254 0.805609448915 129 6 22 [1]
71 0.104994525812 9.5243060747 0.806059259140 139 2 23 [2]
72 0.104148216880 9.6017006335 0.804287804303 141 2 21 [1]
73 0.103385200284 9.6725643250 0.803553720675 143 2 21 [2]
74 0.102662875600 9.7406194221 0.803218825199 147 2 20 [2]
75 0.102036522432 9.8004124030 0.804170005100 139 6 23 [2]
76 0.101437223330 9.8583140111 0.805348041563 141 6 25 [2]
77 0.100835741189 9.9171185555 0.806296963668 144 6 24 [2]
78 0.100102185075 9.9897919236 0.804927985659 151 3 20 [2]
79 0.099584993010 10.0416736475 0.806845154307 162 6 24 C3 [2]
80 0.099131009720 10.0876607917 0.809625831253 151 5 23 [2]
81 0.098732481024 10.1283791274 0.813168279487 149 7 20 [2]
82 0.098348067716 10.1679679451 0.816809584047 153 6 21 C3 [1]
83 0.097705052487 10.2348852444 0.815994903898 159 5 22 [2]
84 0.097172464440 10.2909811515 0.816847585918 161 7 19 [2]
85 0.096647289563 10.3469016516 0.817661582146 177 3 21 C3 [2]
86 0.095747077056 10.4441830576 0.811941631915 164 6 21 [2]
87 0.095133182870 10.5115793441 0.810883785534 163 6 26 [1]
88 0.094539460931 10.5775936329 0.809998518813 165 6 25 [1]
89 0.093997132006 10.6386224628 0.809831231760 164 8 28 [2]
90 0.093556348244 10.6887455396 0.811267993570 166 7 21 [2]
91 0.093029645776 10.7492616108 0.811072051300 210 30 C3 [2]
92 0.092371356890 10.8258667369 0.808421366195 173 8 25 [2]
93 0.091892439966 10.8822880356 0.808756575009 174 8 21 [2]
94 0.091458586013 10.9339105664 0.809752184724 180 6 26 [2]
95 0.091081701493 10.9791537006 0.811635781150 179 6 27 [2]
96 0.090614570229 11.0357528317 0.811787972316 171 11 25 [2]
97 0.090210453617 11.0851897968 0.812944274901 183 7 24 [1]
98 0.089903567382 11.1230291424 0.815746529670 181 8 25 [2]
99 0.089644850728 11.1551304050 0.819334421723 183 9 24 C3 [2]
100 0.088945046531 11.2428970359 0.814739632232 192 4 23 [2]
101 0.088443034716 11.3067128826 0.813624382745 192 6 24 [2]
102 0.088223897337 11.3347973756 0.817613322166 201 6 21 C3 [2]
103 0.087471283686 11.4323233621 0.811602795400 197 7 25 [2]
104 0.087156185006 11.4736550244 0.813589008781 194 8 25 [2]
105 0.086843708149 11.5149389785 0.815532601136 200 7 25 [2]
106 0.086352689668 11.5804152001 0.814015947331 202 8 25 [2]
107 0.085917814998 11.6390296940 0.813440019012 209 3 27 [2]
108 0.085430088155 11.7054777959 0.811747152529 214 6 25 [2]
109 0.085128375468 11.7469644464 0.813486778290 234 6 21 C3 [2]
110 0.084769612774 11.7966800516 0.814044962539 219 9 20 [2]
111 0.084448948050 11.8414737317 0.815242432836 223 9 22 [2]
112 0.084131857401 11.8861039195 0.816421223735 218 4 28 [2]
113 0.083797734968 11.9334967750 0.817181098294 216 11 26 [2]
114 0.083592178637 11.9628416953 0.820373171378 222 11 30 [2]
115 0.083363173625 11.9957045362 0.823041301470 225 9 33 C3 [2]
116 0.082990713774 12.0495409007 0.822796239678 226 11 29 [2]
117 0.082599267928 12.1066448298 0.822079025049 229 11 26 [2]
118 0.082278809159 12.1537976816 0.822684493491 236 6 29 [2]
119 0.081872296026 12.2141438379 0.821478517307 234 6 27 [2]
120 0.081446768941 12.2779578982 0.819793132303 229 7 27 [2]
121 0.081053913101 12.3374672701 0.818669578766 246 6 24 C3 [2]
122 0.080660596770 12.3976270948 0.817443983453 242 8 25 [2]



Updates

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

08-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.
09-Dec-2020: Extension from N=49 to N=120 by E. Specht [2].
10-Dec-2020: Improvements for N=49, 51–54, 56, 57, 59, 60, 62, 64–68, 70, 72, 73, 78, 82, 83, 87, 88, 90, 97, 98, 114, 115, and extension of 121 and 122 by P. Amore [1].
13-Dec-2020: Improvements for N=52, 66, 90, 98, 102, and 113 by E. Specht [2].
14-Dec-2020: Improvements for N=48, 53, 57, 58, 60, 64, 73, 74, 76–78, 80, 81, 83, 84, 86, 92–96, 100, 114–117, 121, and 122 by E. Specht [2].

References

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