The best known packings of equal circles in a regular hexagon (complete up to N = 120)

Last update: 18-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 hexagon (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 0.866025403785 1.1547005384 0.906899682117 6 1 D3
2 0.464101615138 2.1547005384 0.520899741122 5 2 D1
3 0.433012701892 2.3094010768 0.680174761588 9 3 D3 [1]
4 0.375015221341 2.6665584304 0.680229979624 10 4 D1 [1]
5 0.333404971008 2.9993553995 0.672066320823 11 5 D1 [1]
6 0.316987298108 3.1547005384 0.729009112318 18 6 D3 [1]
7 0.316987298108 3.1547005384 0.850510631038 24 6 D3 [1]
8 0.269585370786 3.7094000950 0.703040939766 15 1 7 D1 [1]
9 0.249261972966 4.0118433955 0.676164790379 21 7 [1]
10 0.242756322541 4.1193571790 0.712588953983 21 8 [1]
11 0.232050807569 4.3094010768 0.716237144042 29 7 D1 [1]
12 0.232050807569 4.3094010768 0.781349611682 33 9 D3 [1]
13 0.216506350946 4.6188021535 0.736855991720 36 6 D3 [2]
14 0.214290684433 4.6665584304 0.777378644490 36 10 D1 [1]
15 0.201559210543 4.9613212778 0.736876222020 33 9 C3 [2]
16 0.200025787346 4.9993553995 0.774087306713 39 11 D1 [1]
17 0.194813312769 5.1331194249 0.780160882191 39 11 [2]
18 0.193997690565 5.1547005384 0.819150331291 54 12 D3 [1]
19 0.193997690565 5.1547005384 0.864658683029 60 12 D3 [1]
20 0.176152871147 5.6768873166 0.750425242745 39 1 12 [1]
21 0.175149750125 5.7094000950 0.778997968601 49 1 13 D1 [1]
22 0.168017815756 5.9517497921 0.750985127277 45 2 13 [1]
23 0.166338331558 6.0118433955 0.769503331354 58 14 [1]
24 0.163462238508 6.1176208593 0.775432701310 49 14 [1]
25 0.159724477011 6.2607811822 0.771224725335 56 1 14 [1]
26 0.158493649054 6.3094010768 0.789759871678 70 15 D1 [1]
27 0.158493649054 6.3094010768 0.820135251358 75 15 D3 [1]
28 0.151660010858 6.5936959541 0.778750221348 60 2 15 [1]
29 0.150002435355 6.6665584304 0.789028343675 66 3 16 D1 [1]
30 0.150002435355 6.6665584304 0.816236217595 80 16 D1 [1]
31 0.144498618555 6.9204813859 0.782685107531 72 1 16 [1]
32 0.142897923362 6.9980023255 0.790132270668 65 15 [1]
33 0.142870299182 6.9993553995 0.814508900714 85 17 D1 [1]
34 0.140313381692 7.1269039912 0.809422153571 82 16 D1 [2]
35 0.139827762740 7.1516555826 0.827471126376 83 17 [1]
36 0.139768253701 7.1547005384 0.850388865365 108 18 D3 [1]
37 0.139768253701 7.1547005384 0.874010778291 114 18 D3 [1]
38 0.131624252390 7.5973840826 0.796073959391 84 3 18 [1]
39 0.129882647187 7.6992579198 0.795545193971 89 1 18 [1]
40 0.129711778826 7.7094000950 0.813798355495 101 1 19 D1 [1]
41 0.126161767649 7.9263315554 0.789109684720 92 2 18 [2]
42 0.125108623258 7.9930541473 0.794916966655 98 1 19 [2]
43 0.124815220497 8.0118433955 0.810030811101 115 19 [1]
44 0.123304899729 8.1099778046 0.808930725545 103 1 19 [2]
45 0.121261865975 8.2466156360 0.800127125304 101 2 20 [1]
46 0.120631837547 8.2896855452 0.809430760397 112 20 [2]
47 0.120345617062 8.3094010768 0.823107199495 130 21 D1 [1]
48 0.120345617062 8.3094010768 0.840620118633 135 21 D3 [1]
49 0.116441613111 8.5879950757 0.803360534217 105 4 21 [2]
50 0.115694712596 8.6434373496 0.809272930292 118 2 21 [1]
51 0.115386056418 8.6665584304 0.821059865157 128 3 22 D1 [1]
52 0.115386056418 8.6665584304 0.837159078199 142 22 D1 [1]
53 0.112253527371 8.9084060290 0.807558233841 121 3 21 [1]
54 0.111600904574 8.9605008474 0.813255811881 128 1 22 [1]
55 0.111119395932 8.9993289796 0.821183887407 115 20 [1]
56 0.111119069712 8.9993553995 0.836109594295 149 23 D1 [1]
57 0.109704794905 9.1153718565 0.829514652236 137 2 23 [2]
58 0.109491615458 9.1331194249 0.840790327271 149 23 [1]
59 0.109239712177 9.1541801060 0.851355786751 146 23 [2]
60 0.109233502047 9.1547005384 0.865687111165 180 24 D3 [1]
61 0.109233502047 9.1547005384 0.880115229685 186 24 D3 [1]
62 0.104279435263 9.5896184850 0.815242884313 145 3 23 [2]
63 0.103572509148 9.6550716810 0.817198443781 151 3 24 [1]
64 0.102994077927 9.7092961083 0.820923083934 121 1 23 [1]
65 0.102992974872 9.7094000950 0.833732148468 171 1 25 D1 [1]
66 0.100973952318 9.9035442017 0.813693100041 152 2 24 [2]
67 0.100511805910 9.9490800205 0.818477869442 164 1 25 [1]
68 0.099927194138 10.0072858907 0.821058846924 169 25 [1]
69 0.099881706145 10.0118433955 0.832374910950 189 25 [1]
70 0.098953578375 10.1057487402 0.828817732109 162 3 26 [2]
71 0.097726952185 10.2325917021 0.819945608574 172 2 26 [1]
72 0.097458466587 10.2607811822 0.826931680606 183 1 26 [1]
73 0.097083518356 10.3004095539 0.831978034483 183 1 26 [1]
74 0.096998845283 10.3094010768 0.841904507160 208 27 D1 [1]
75 0.096998845283 10.3094010768 0.853281595095 213 27 D3 [1]
76 0.094732105837 10.5560832957 0.824718927025 188 2 26 [1]
77 0.094395761812 10.5936959541 0.829647679761 193 2 27 [1]
78 0.093851924172 10.6550825551 0.830766452983 194 2 27 [1]
79 0.093750951305 10.6665584304 0.839607760498 208 3 28 D1 [1]
80 0.093750951305 10.6665584304 0.850235706833 222 28 D1 [1]
81 0.091952819516 10.8751423313 0.828157817780 197 3 28 [1]
82 0.091571054852 10.9204813859 0.831434943006 212 1 28 [1]
83 0.091031724997 10.9851812654 0.831690275451 162 9 26 [2]
84 0.090914418976 10.9993553417 0.839542734288 135 3 23 [1]
85 0.090914418499 10.9993553995 0.849537281723 231 29 D1 [1]
86 0.090058838635 11.1038518280 0.843430150423 213 3 29 [2]
87 0.089825721593 11.1326687086 0.848825989250 209 1 29 [2]
88 0.089674476085 11.1514451342 0.855693739323 221 1 28 [2]
89 0.089648390877 11.1546898970 0.864914126296 126 14 20 [1]
90 0.089648305354 11.1547005384 0.874630593785 270 30 D3 [1]
91 0.089648305354 11.1547005384 0.884348711494 276 30 D3 [1]
92 0.086570142383 11.5513267331 0.833723565649 228 4 29 [2]
93 0.086226341464 11.5973840826 0.836105064489 236 3 30 [1]
94 0.085555340943 11.6883410080 0.831993804927 236 3 30 [1]
95 0.085401704341 11.7093681879 0.837827608482 170 4 28 [1]
96 0.085401471629 11.7094000950 0.846642232399 259 1 31 D1 [1]
97 0.084100942843 11.8904731171 0.829605169914 234 5 30 [2]
98 0.083777268130 11.9364121357 0.831718670935 245 1 31 [1]
99 0.083455964247 11.9823670965 0.833773215467 240 5 31 [2]
100 0.083270100777 12.0091124025 0.838448065032 252 1 31 [1]
101 0.083251168624 12.0118433955 0.846447520538 280 32 [1]
102 0.082638340561 12.1009206284 0.842289396306 251 4 31 [1]
103 0.081795176612 12.2256597690 0.833279312543 258 2 32 [2]
104 0.081603679381 12.2543494066 0.837434420196 225 11 31 [2]
105 0.081386753184 12.2870118401 0.840997552314 255 2 32 [2]
106 0.081274604310 12.3039663926 0.846668844703 266 1 32 [1]
107 0.081238721020 12.3094010768 0.853901780123 304 33 D1 [1]
108 0.081238721020 12.3094010768 0.861882170591 309 33 D3 [1]
109 0.079667687518 12.5521404116 0.836544227479 274 3 32 [2]
110 0.079406286275 12.5934613859 0.838688024221 272 3 33 [1]
111 0.079118946282 12.6391976510 0.840198601494 271 2 32 [2]
112 0.078978151950 12.6617295455 0.844753390307 279 2 33 [1]
113 0.078948043030 12.6665584304 0.851646111960 306 3 34 D1 [1]
114 0.078948043029 12.6665584304 0.859182803216 320 34 D1 [1]
115 0.077735359934 12.8641586127 0.840297466642 282 5 34 [1]
116 0.077418301081 12.9168424783 0.840704261380 284 4 34 [1]
117 0.077162885142 12.9595983634 0.842365872064 276 2 34 [1]
118 0.076980480216 12.9903060776 0.845553760797 306 34 [2]
119 0.076926891317 12.9993553995 0.851532666082 105 55 23 [1]
120 0.076926891317 12.9993553995 0.858688402770 331 35 D1 [1]
126 0.076018454170 13.1547005384 0.880453855192 378 36 D3 [1]
127 0.076018454170 13.1547005384 0.887441584201 384 36 D3 [1]
154 0.068182321350 14.6665584304 0.865689746386 436 40 D1 [1]
161 0.066669531681 14.9993553995 0.865323844659 449 41 D1 [1]
168 0.065986127371 15.1547005384 0.884529965076 504 42 D3 [1]
169 0.065986127371 15.1547005384 0.889795024392 510 42 D3 [1]
200 0.060000389653 16.6665584304 0.870635002909 570 46 D1 [1]
208 0.058825759948 16.9993553995 0.870354970735 585 47 D1 [1]
216 0.058293060713 17.1547005384 0.887534931682 648 48 D3 [1]
217 0.058293060713 17.1547005384 0.891643889700 654 48 D3 [1]
252 0.053571739200 18.6665584304 0.874520549315 722 52 D1 [1]
261 0.052633364605 18.9993553994 0.874300567450 739 53 D1 [1]
270 0.052206506596 19.1547005384 0.889838141345 810 54 D3 [1]
271 0.052206506596 19.1547005384 0.893133838164 816 54 D3 [1]
310 0.048387350190 20.6665584304 0.877654046604 892 58 D1 [1]
320 0.047620509343 20.9993553995 0.877477596716 911 59 D1 [1]
330 0.047270818048 21.1547005384 0.891657695863 990 60 D3 [1]
331 0.047270818048 21.1547005384 0.894359688882 996 60 D3 [1]
374 0.044117857727 22.6665584304 0.880234568503 1080 64 D1 [1]
385 0.043479479430 22.9993553995 0.880090609776 1101 65 D1 [1]
396 0.043187775128 23.1547005384 0.893130324014 1188 66 D3 [1]
397 0.043187775128 23.1547005384 0.895385703621 1194 66 D3 [1]
444 0.040540718431 24.6665584304 0.882396623717 1286 70 D1 [1]
456 0.040001031387 24.9993553994 0.882277507503 1309 71 D1 [1]
468 0.039754001383 25.1547005384 0.894345932896 1404 72 D3 [1]
469 0.039754001383 25.1547005384 0.896256928479 1410 72 D3 [1]
520 0.037500152208 26.6665584304 0.884234368017 1510 76 D1 [1]
533 0.037037921284 26.9993553994 0.884134635203 1535 77 D1 [1]
546 0.036826036751 27.1547005384 0.895365989059 1638 78 D3 [1]
547 0.036826036751 27.1547005384 0.897005853508 1644 78 D3 [1]
602 0.034883852641 28.6665584304 0.885815680941 1752 82 D1 [1]
616 0.034483525107 28.9993553994 0.885731335592 1779 83 D1 [1]
630 0.034299786365 29.1547005384 0.896233897533 1890 84 D3 [1]
631 0.034299786365 29.1547005384 0.897656491021 1896 84 D3 [1]
690 0.032608810743 30.6665584304 0.887190734205 2012 88 D1 [1]
705 0.032258735290 30.9993553994 0.887118787442 2041 89 D1 [1]
720 0.032097885158 31.1547005384 0.896981158198 2160 90 D3 [1]
721 0.032097885158 31.1547005384 0.898226965362 2166 90 D3 [1]
784 0.030612346328 32.6665584304 0.888397412482 2290 94 D1 [1]
800 0.030303622234 32.9993553995 0.888335586392 2321 95 D1 [1]
816 0.030161635719 33.1547005384 0.897631171826 2448 96 D3 [1]
817 0.030161635719 33.1547005384 0.898731210027 2454 96 D3 [1]
884 0.028846243910 34.6665584304 0.889464857780 2586 100 D1 [1]
901 0.028571954786 34.9993553994 0.889411387595 2619 101 D1 [1]
918 0.028445698148 35.1547005383 0.898201677635 2754 102 D3 [1]
919 0.028445698148 35.1547005383 0.899180110835 2760 102 D3 [1]
990 0.027272807779 36.6665584304 0.890415853815 2900 106 D1 [1]
1008 0.027027497890 36.9993553994 0.890369352380 2935 107 D1 [1]
1026 0.026914494950 37.1547005384 0.898706362718 3078 108 D3 [1]
1027 0.026914494950 37.1547005384 0.899582294846 3084 108 D3 [1]
1102 0.025862141359 38.6665584304 0.891268470398 3232 112 D1 [1]
1121 0.025641449449 38.9993553994 0.891227833273 3269 113 D1 [1]
1140 0.025539717741 39.1547005383 0.899155953324 3420 114 D3 [1]
1141 0.025539717741 39.1547005383 0.899944686616 3426 114 D3 [1]
1220 0.024590229382 40.6665584304 0.892037222605 3582 118 D1 [1]
1240 0.024390627371 40.9993553994 0.892001560268 3621 119 D1 [1]
1260 0.024298560964 41.1547005384 0.899558972268 3780 120 D3 [1]
1261 0.024298560964 41.1547005384 0.900272907959 3786 120 D3 [1]



Updates

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

02-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.
04-Dec-2020: Only one day after the unexpected publication of packings in hexagonal containers, Paolo Amore (Universidad de Colima, Mexico) sent me packings from his own repository which improve my result for several values of N: 13, 15, 17, 34, 41, 42, 44, 46, 49, 57, 59, 62, 66, 70, 83, 86, 87, 88, 92, 97, 99, 103, 104, 105, 109, 111, and 118. Surprisingly I have missed the prominent case N = 13. Great work, Paolo!
18-Dec-2020: Extension by the first higher hex numbers: N=127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, and 1261 by E. Specht [1]. The hex number Hk is given by 3k(k-1)+1, where k is the number of circles along each side of the hexagon.
It is believed that these group of packings are the densest for all containers (besides equilateral triangles), exceeding a packing fraction of 0.9 for N=1261. So if you want a extremely dense packing fraction take these.
18-Dec-2020: Extension for N=126, 168, 216, 270, 330, 396, 468, 546, 630, 720, 816, 918, 1026, 1140, and 1260 by E. Specht [1], which are hex numbers (see above) minus one, Hk-1, i.e. by omitting the central circle. Conjecture: One cannot increase the radius of the circles of these packings compared to the associated packings with hex numbers.
18-Dec-2020: Extension for N=161, 208, 261, 320, 385, 456, 533, 616, 705, 800, 901, 1008, 1121, and 1240 by E. Specht [1]. These packings have been generated by taking the hex number packings (see above) as templates and scratching a straight row of circles from the center to one of the vertices of the hexagon, inclusive. So we have N= Hk-k. Finally, the whole packing is slightly expanded in their radii until all circles jam.
18-Dec-2020: Extension for N=154, 200, 252, 310, 374, 444, 520, 602, 690, 784, 884, 990, 1102, and 1220 by E. Specht [1]. These packings have been generated by taking the hex number packings (see above) as templates and scratching a straight row of circles from one of the vertices of the hexagon to its opposite vertex, inclusive. So we have N= Hk-2k+1. Finally, the whole packing is slightly expanded in their radii until all circles jam.

References

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