The best known packings of equal circles in a regular hexadecagon (complete up to N = 200)

Last update: 14-Jan-2023


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   133-144   145-156   157-168   169-180   181-192   193-200  


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 0.146379246751 6.8315695168 0.791556176236 67 3 16 [2]
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 0.127965573608 7.8146017855 0.789776811882 89 3 18 [2]
48 0.126752861782 7.8893682237 0.791365311190 93 2 19 [2]
49 0.125224583493 7.9856524343 0.788488742006 93 3 20 [2]
50 0.124309646840 8.0444279702 0.792866182697 96 3 18 [2]
51 0.123031513986 8.1279988159 0.792178657870 97 3 20 [2]
52 0.122088244960 8.1907967497 0.795373780184 106 22 [2]
53 0.120820011984 8.2767745474 0.793914700574 99 6 12 [2]
54 0.120293981022 8.3129678767 0.801865965032 103 4 14 [2]
55 0.119615478307 8.3601220691 0.807528162139 113 18 [2]
56 0.117825103758 8.4871556918 0.797781413249 101 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 0.111131002358 8.9983890974 0.798419688150 121 3 22 [2]
64 0.110143534712 9.0790621766 0.796742930167 117 6 22 [2]
65 0.109509650513 9.1316152989 0.799904919386 125 3 20 [2]
66 0.108787930470 9.1921961901 0.801540720657 125 4 20 [2]
67 0.107858486148 9.2714077094 0.799841011622 128 4 22 [2]
68 0.107112259286 9.3359995080 0.800585089860 135 1 20 [2]
69 0.106390898460 9.3993002642 0.801453387384 129 7 17 [2]
70 0.105736301152 9.4574898980 0.803094206598 134 3 22 [2]
71 0.105065028291 9.5179149167 0.804257162744 134 5 21 [2]
72 0.104379962641 9.5803828120 0.804983531735 133 6 22 [2]
73 0.103882362596 9.6262731710 0.808400775928 137 5 18 [2]
74 0.103431779611 9.6682083955 0.812381339379 145 4 18 [2]
75 0.102347738100 9.7706116282 0.806191076528 142 4 23 [2]
76 0.101651605712 9.8375229097 0.805865018194 138 7 23 [2]
77 0.100969657337 9.9039654721 0.805550395229 144 6 24 [2]
78 0.100434225410 9.9567651955 0.807380575642 143 7 27 [2]
79 0.099735335856 10.0265366474 0.806390542006 143 7 22 [2]
80 0.099261435768 10.0744059590 0.808856197887 154 8 20 [2]
81 0.098809547404 10.1204795111 0.811527168901 144 8 18 [2]
82 0.098387825085 10.1638591882 0.814548218625 157 7 24 [2]
83 0.098019112369 10.2020919781 0.818313749224 160 6 22 [2]
84 0.097495551199 10.2568782647 0.819349341601 157 7 19 [2]
85 0.096800013134 10.3305771107 0.817315966940 162 5 20 [2]
86 0.095884090091 10.4292589005 0.811356610560 165 5 24 [2]
87 0.095255981179 10.4980284453 0.810072683132 160 8 24 [2]
88 0.094621460822 10.5684269859 0.808504040411 166 6 23 [2]
89 0.094136751728 10.6228436997 0.809335608942 174 2 26 [2]
90 0.093630994308 10.6802240795 0.809658734395 173 4 21 [2]
91 0.093121125226 10.7387018528 0.809763221405 174 6 22 [2]
92 0.092636810667 10.7948448657 0.810168291436 183 2 24 [2]
93 0.092274065221 10.8372812838 0.812573177106 181 3 24 [2]
94 0.091745732828 10.8996894916 0.811932314955 181 6 24 [2]
95 0.091292035235 10.9538581041 0.812474255952 175 7 26 [2]
96 0.090805583335 11.0125386928 0.812300206250 185 6 25 [2]
97 0.090384196570 11.0638810539 0.813161790005 183 6 25 [2]
98 0.089982294680 11.1132973832 0.814254990578 178 11 25 [2]
99 0.089677133347 11.1511147009 0.816993974801 186 7 28 [2]
100 0.089221530647 11.2080569875 0.816882447139 198 1 25 [2]
101 0.088715610784 11.2719733445 0.815721093404 189 7 25 [2]
102 0.088230728413 11.3339198031 0.814817084200 190 8 26 [2]
103 0.087830356054 11.3855851772 0.815354995730 195 6 23 [2]
104 0.087426745495 11.4381473809 0.815722023959 195 7 24 [2]
105 0.087098806284 11.4812136086 0.817398676167 195 6 24 [2]
106 0.086743164698 11.5282858711 0.818458410330 197 8 24 [2]
107 0.086264110555 11.5923063898 0.817079473500 204 5 27 [2]
108 0.085814904455 11.6529873960 0.816148951779 200 6 25 [2]
109 0.085328022973 11.7194793124 0.814385607016 204 8 22 [2]
110 0.085050962448 11.7576564829 0.816528551526 204 10 26 [2]
111 0.084657408389 11.8123152956 0.816343882400 217 11 24 [2]
112 0.084435175294 11.8434052694 0.819379447762 229 7 25 [2]
113 0.084015455037 11.9025719680 0.818496895377 225 9 23 [2]
114 0.083719221135 11.9446882859 0.819927466743 218 10 25 [2]
115 0.083388430929 11.9920711885 0.820596517547 229 5 33 [2]
116 0.083069337897 12.0381361561 0.821409482702 230 11 24 [2]
117 0.082764487855 12.0824767472 0.822420923534 230 8 31 [2]
118 0.082450799752 12.1284451213 0.823174630737 227 8 29 [2]
119 0.082130939448 12.1756795517 0.823722193868 230 12 25 [2]
120 0.081724078256 12.2362958548 0.822434903464 227 7 26 [2]
121 0.081251525979 12.3074611578 0.819725882175 233 5 28 [2]
122 0.080858850073 12.3672300447 0.818531085455 240 5 28 [2]
123 0.080425361133 12.4338888370 0.816415752916 231 8 26 [2]
124 0.080154293197 12.4759380953 0.817514544477 232 8 28 [2]
125 0.079846836187 12.5239777523 0.817797282728 237 6 28 [2]
126 0.079563197380 12.5686251047 0.818493482462 242 6 27 [2]
127 0.079316216763 12.6077622057 0.819875536391 245 9 26 [2]
128 0.079095241745 12.6429855695 0.821733344578 252 9 24 [2]
129 0.078777049136 12.6940525314 0.821503376902 245 9 26 [2]
130 0.078475072721 12.7428999468 0.821536817091 247 7 25 [2]
131 0.078131215128 12.7989817944 0.820617318430 259 9 26 [2]
132 0.077844501567 12.8461224604 0.820823989666 246 11 26 [2]
133 0.077593823549 12.8876237085 0.821724379020 247 12 29 [2]
134 0.077365638225 12.9256349840 0.823040580060 270 10 25 [2]
135 0.077124729924 12.9660097479 0.824026742112 286 12 24 [2]
136 0.076827882410 13.0161078066 0.823752715905 278 10 25 [2]
137 0.076522500411 13.0680518100 0.823226035460 261 13 27 [2]
138 0.076244458501 13.1157072876 0.823219937861 262 6 30 [2]
139 0.075957199993 13.1653088857 0.822948995253 285 5 28 [2]
140 0.075699064836 13.2102028231 0.823245356140 279 6 32 [2]
141 0.075352995070 13.2708726318 0.821562061024 263 7 30 [2]
142 0.075127057404 13.3107835520 0.822434512573 272 12 32 [2]
143 0.074886459294 13.3535489517 0.822929928217 267 11 31 [2]
144 0.074620428867 13.4011558923 0.822807418530 274 6 33 [2]
145 0.074365483672 13.4470987160 0.822869643802 283 14 28 [2]
146 0.074093279496 13.4965007191 0.822490170725 285 7 34 [2]
147 0.073822439925 13.5460166450 0.822080506237 314 8 27 [2]
148 0.073678029750 13.5725670650 0.824437899177 311 9 24 [2]
149 0.073508311497 13.6039038258 0.826188960335 293 8 29 [2]
150 0.073262079353 13.6496262300 0.826171036861 321 10 27 [2]
151 0.073119798077 13.6761865637 0.828451596543 312 7 24 [2]
152 0.072819765466 13.7325353028 0.827108272925 301 6 30 [2]
153 0.072600658662 13.7739797190 0.827547207570 284 13 23 [2]
154 0.072377972560 13.8163582735 0.827854041042 299 13 27 [2]
155 0.072169287700 13.8563096834 0.828431799641 307 9 32 [2]
156 0.071962743253 13.8960794822 0.829010907036 301 15 28 [2]
157 0.071715869764 13.9439151096 0.828610471327 319 16 26 [2]
158 0.071510406284 13.9839787237 0.829116967795 327 11 31 [2]
159 0.071195045875 14.0459211411 0.827021686286 320 11 32 [2]
160 0.070965455754 14.0913630354 0.826864221271 332 9 34 [2]
161 0.070714453092 14.1413806694 0.826156786801 322 5 31 [2]
162 0.070423466029 14.1998123124 0.824460838127 326 14 29 [2]
163 0.070220540427 14.2408473919 0.824776284073 314 7 33 [2]
164 0.070116122236 14.2620551181 0.827370157305 315 8 28 [2]
165 0.069856303698 14.3151003855 0.826257421857 343 12 29 [2]
166 0.069684721845 14.3503478743 0.827186532166 350 13 30 [2]
167 0.069504844014 14.3874864290 0.827878954022 342 15 28 [2]
168 0.069316361498 14.4266083561 0.828325483770 353 12 32 [2]
169 0.069108833251 14.4699302963 0.828274043451 352 13 32 [2]
170 0.068917196448 14.5101665700 0.828560738950 359 14 28 [2]
171 0.068770090929 14.5412051444 0.829880447882 359 12 28 [2]
172 0.068556705334 14.5864652499 0.829561424365 361 13 29 [2]
173 0.068333866674 14.6340321231 0.828969057597 361 15 27 [2]
174 0.068146835518 14.6741956893 0.829202992385 338 8 32 [2]
175 0.067951369208 14.7164069195 0.829191226052 373 10 28 [2]
176 0.067812444407 14.7465558681 0.830523053307 378 14 29 [2]
177 0.067628984365 14.7865594817 0.830728714265 370 15 29 [2]
178 0.067484198183 14.8182837898 0.831848833134 386 12 33 [2]
179 0.067278582384 14.8635712074 0.831432353097 369 14 29 [2]
180 0.067096821787 14.9038355822 0.831565817031 343 14 27 [2]
181 0.066956583946 14.9350510595 0.832693887359 346 9 29 [2]
182 0.066809709984 14.9678841629 0.833625106940 360 15 26 [2]
183 0.066599490025 15.0151299901 0.832938856097 359 11 30 [2]
184 0.066311677918 15.0803000527 0.830267584786 350 10 31 [2]
185 0.066114478754 15.1252799515 0.829822316438 393 14 30 [2]
186 0.065970327268 15.1583301374 0.830673673169 377 17 28 [2]
187 0.065796834531 15.1982995402 0.830752835539 373 9 30 [2]
188 0.065627506828 15.2375131760 0.830902156361 416 9 33 [2]
189 0.065542183360 15.2573495226 0.833151227578 361 8 31 [2]
190 0.065405175062 15.2893100439 0.834061454187 410 9 31 [2]
191 0.065274502055 15.3199177093 0.835104314372 388 12 30 [2]
192 0.065079422726 15.3658400476 0.834466368011 365 10 31 [2]
193 0.064933261714 15.4004276638 0.835049022048 423 13 30 [2]
194 0.064754679203 15.4428994523 0.834765068760 435 1 34 [2]
195 0.064583911559 15.4837323392 0.834648324670 405 13 29 [2]
196 0.064400403935 15.5278529155 0.834167910888 387 12 33 [2]
197 0.064246835168 15.5649690352 0.834430037373 399 10 32 [2]
198 0.064085187429 15.6042299339 0.834450795536 409 12 29 [2]
199 0.063985436301 15.6285563999 0.836056394359 410 12 28 [2]
200 0.063775074264 15.6801071820 0.834741810169 416 13 34 [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].
14-Jan-2023: Extension from N=123 to N=200 by E. Specht [2]. Furthermore, many improvements were found for N=36, 47–50, 53, 54, 56, 63, 65–70, 72–122.

References

[1]   , private communication, December 2020.
[2]   , program cxd, 2023.