The best known packings of equal circles in a regular hexadecagon (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   121-132  


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 hexadecagon (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 0.980785280403 1.0195911582 0.987115800972 16 1 D1
2 0.495149721732 2.0195911582 0.503179917313 5 2
3 0.457221451812 2.1871239769 0.643568348049 7 3 [1]
4 0.410879308247 2.4338047206 0.692960944306 12 4 [1]
5 0.364795992040 2.7412581876 0.682795021344 11 5 [1]
6 0.331170661062 3.0195911582 0.675266469782 15 5 [1]
7 0.327408427053 3.0542891306 0.770012848460 14 1 5 [1]
8 0.298219404666 3.3532358537 0.730099430866 15 1 7 [1]
9 0.275276047043 3.6327170880 0.699841320741 24 1 8 [1]
10 0.259076416136 3.8598650349 0.688772797313 20 8 [1]
11 0.251164419238 3.9814556657 0.712080580393 19 2 9 [1]
12 0.244950583610 4.0824560826 0.738853658686 25 9 [1]
13 0.232797378030 4.2955810261 0.722969118072 22 3 10 [1]
14 0.228751748884 4.3715512772 0.751756352995 30 10 [1]
15 0.218093322591 4.5851931096 0.732143518418 27 2 8 [1]
16 0.213661979610 4.6802898757 0.749539808678 33 11 [1]
17 0.205917004872 4.8563254920 0.739696513691 33 1 10 [1]
18 0.202733212758 4.9325908981 0.759176114706 33 2 10 [1]
19 0.202115798420 4.9476587571 0.796479035795 39 12 [1]
20 0.192692071372 5.1896271231 0.762040250870 39 1 12 [1]
21 0.188035399172 5.3181475637 0.761936454367 37 3 11 [1]
22 0.181548742791 5.5081626269 0.744096732681 45 12 [1]
23 0.178389000729 5.6057267876 0.751076556753 45 1 12 [1]
24 0.175006175983 5.7140840566 0.754289769404 49 14 [1]
25 0.172287084916 5.8042655982 0.761492586211 47 2 13 [1]
26 0.169380978228 5.9038506594 0.765460608909 51 1 14 [1]
27 0.167333899680 5.9760753913 0.775803713169 55 15 [1]
28 0.165343666369 6.0480091071 0.785513012075 58 14 [1]
29 0.162161216251 6.1667026378 0.782550204106 51 4 12 [1]
30 0.160554721990 6.2284060388 0.793574393070 60 16 [1]
31 0.156347782078 6.3959973510 0.777616323408 64 1 12 [1]
32 0.153818259145 6.5011787649 0.776937327153 60 2 15 [1]
33 0.152117848281 6.5738505461 0.783600153330 67 17 [1]
34 0.149199102273 6.7024531969 0.776661115439 63 3 14 [1]
35 0.147680248230 6.7713862347 0.783308964505 67 2 17 [1]
36 0.146379242637 6.8315697088 0.791556131743 65 3 16 [1]
37 0.145538331458 6.8710420821 0.804223450026 71 2 16 [1]
38 0.141827451001 7.0508212123 0.784376216869 72 2 18 [1]
39 0.139984857939 7.1436297806 0.784236324005 71 4 17 [1]
40 0.138529233057 7.2186929642 0.787704045824 75 3 18 [1]
41 0.136082978898 7.3484575962 0.779133175374 75 4 14 [1]
42 0.134590794263 7.4299286625 0.780728849295 77 4 16 [1]
43 0.133206771660 7.5071258581 0.782963085575 83 2 17 [1]
44 0.131782751937 7.5882464534 0.784133568791 85 2 17 [1]
45 0.130457611226 7.6653250861 0.785907770723 83 4 18 [1]
46 0.129031520707 7.7500442878 0.785904342028 83 5 17 [1]
47 0.127962372565 7.8147972717 0.789737300033 89 3 18 [1]
48 0.126752473238 7.8893924075 0.791360459544 93 2 19 [1]
49 0.125183865267 7.9882499064 0.787976052846 96 2 19 [1]
50 0.124306538124 8.0446291490 0.792826527453 100 1 19 [2]
51 0.123031513986 8.1279988159 0.792178657870 97 3 20 [2]
52 0.122088244960 8.1907967497 0.795373780184 106 22 [2]
53 0.120819860639 8.2767849153 0.793912711582 97 6 12 [2]
54 0.120292677460 8.3130579609 0.801848586332 99 5 14 [2]
55 0.119615478307 8.3601220691 0.807528162139 113 18 [2]
56 0.117824987874 8.4871640392 0.797779843969 100 6 16 [2]
57 0.116957820523 8.5500909262 0.800117214325 103 6 19 [2]
58 0.115827982443 8.6334923471 0.798500517493 105 6 19 [2]
59 0.114878920879 8.7048171444 0.799011318587 107 6 20 [2]
60 0.114145934146 8.7607150222 0.802217937917 115 2 23 [2]
61 0.113537399228 8.8076704839 0.806915287640 109 7 22 [2]
62 0.111906704129 8.9360151189 0.796753765832 117 4 21 [2]
63 0.111107991882 9.0002526647 0.798089085260 119 4 22 [2]
64 0.110143534712 9.0790621766 0.796742930167 117 6 22 [2]
65 0.109377434244 9.1426536645 0.797974558023 121 5 24 [2]
66 0.108466266408 9.2194562708 0.796807738388 128 4 22 [2]
67 0.107836156868 9.2733275095 0.799509873556 124 6 20 [2]
68 0.107076852893 9.3390865811 0.800055904024 134 2 20 [2]
69 0.106390255857 9.3993570365 0.801443705828 130 7 18 [2]
70 0.105715874025 9.4593173374 0.802783938074 134 3 21 [2]
71 0.105065028291 9.5179149167 0.804257162744 134 5 21 [2]
72 0.104359653080 9.5822472621 0.804670305499 139 3 21 [2]
73 0.103410813106 9.6701686213 0.801078343553 137 5 22 [2]
74 0.103431632001 9.6682221933 0.812379020644 145 4 18 [2]
75 0.102137835875 9.7906910934 0.802887676211 135 8 22 [2]
76 0.101579651083 9.8444913852 0.804724550320 134 10 26 [2]
77 0.100841542310 9.9165480525 0.803507452053 144 6 26 [2]
78 0.100243148329 9.9757441448 0.804311399351 149 4 27 [2]
79 0.099578648796 10.0423134084 0.803858807161 150 10 18 [2]
80 0.099169800967 10.0837149036 0.807363469846 146 9 22 [2]
81 0.098792719465 10.1222033913 0.811250775226 148 8 20 [2]
82 0.098324929030 10.1703607606 0.813507124491 154 7 21 [2]
83 0.097999973825 10.2040843580 0.817994223694 170 5 19 [2]
84 0.097495311816 10.2569034487 0.819345318069 157 7 20 [2]
85 0.096798288478 10.3307611707 0.817286843468 162 6 20 [2]
86 0.095835138858 10.4345860185 0.810528386220 156 9 20 [2]
87 0.095239028968 10.4998970573 0.809784379951 163 6 24 [2]
88 0.094535164554 10.5780743570 0.807029975968 165 6 26 [2]
89 0.093977993336 10.6407890241 0.806608077532 165 7 29 [2]
90 0.093515192308 10.6934496452 0.807657215113 174 4 24 [2]
91 0.093094103442 10.7418189018 0.809293337222 169 7 28 [2]
92 0.092343179811 10.8291700810 0.805040451082 173 6 26 [2]
93 0.091918814827 10.8791655102 0.806328491466 177 5 27 [2]
94 0.091682318732 10.9072285020 0.810810297435 177 8 22 [2]
95 0.091291774390 10.9538894022 0.812469613058 177 7 25 [2]
96 0.090648811279 11.0315842634 0.809497822515 185 9 22 [2]
97 0.090341723713 11.0690825779 0.812397736304 184 5 27 [2]
98 0.089856055032 11.1289105631 0.811971893459 185 6 28 [2]
99 0.089611712206 11.1592555859 0.815802384929 184 7 26 [2]
100 0.089159275194 11.2158830118 0.815742864719 188 7 26 [2]
101 0.088684650568 11.2759084419 0.815151847596 194 5 25 [2]
102 0.088104000709 11.3502223731 0.812478086367 196 6 23 [2]
103 0.087621282589 11.4127523640 0.811477835087 191 8 27 [2]
104 0.087398933244 11.4417872494 0.815203110510 189 10 25 [2]
105 0.087072403714 11.4846950049 0.816903189360 199 6 25 [2]
106 0.086686308161 11.5358471391 0.817385831058 201 6 24 [2]
107 0.086247015270 11.5946041363 0.816755658136 203 6 27 [2]
108 0.085794413464 11.6557705756 0.815759236156 226 2 26 [2]
109 0.085186966215 11.7388850012 0.811695291706 226 8 19 [2]
110 0.084903141094 11.7781272532 0.813692711412 204 10 21 [2]
111 0.084601145863 11.8201708712 0.815259173862 224 10 22 [2]
112 0.084271240841 11.8664444717 0.816200816987 218 6 29 [2]
113 0.083924873045 11.9154186801 0.816732907751 215 10 25 [2]
114 0.083642065440 11.9557066740 0.818416871641 219 11 26 [2]
115 0.083351769406 11.9973457928 0.819875129586 229 10 25 [2]
116 0.083056921165 12.0399358172 0.821163941818 216 10 30 [2]
117 0.082761180852 12.0829595435 0.822355202246 230 8 32 [2]
118 0.082321279864 12.1475273665 0.820590453392 221 9 28 [2]
119 0.082037880226 12.1894909675 0.821856599242 229 9 27 [2]
120 0.081723754896 12.2363442707 0.822428395169 230 3 27 [2]
121 0.081127827779 12.3262267384 0.817231863026 230 10 25 [2]
122 0.080704922546 12.3908179137 0.815417646646 232 7 29 [2]



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.
14-Dec-2020: Extension from N=49 to N=122 by E. Specht [2].

References

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