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

Last update: 06-Mar-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 pentagon (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 0.809016994375 1.2360679775 0.864806265977 5 1 C3
2 0.437152698242 2.2875302017 0.505009863813 5 2 D1
3 0.389795792433 2.5654458550 0.602280894881 7 3 D1 [1]
4 0.354680462504 2.8194392015 0.664872028015 9 4 D1 [1]
5 0.340440645254 2.9373695942 0.765695954932 15 5 [1]
6 0.309016994375 3.2360679775 0.757041202686 15 5 [1]
7 0.277511095117 3.6034595286 0.712298912009 13 1 5 D1 [1]
8 0.257541824500 3.8828644704 0.701114681674 15 1 7 D1 [1]
9 0.243640224399 4.1044125717 0.705901377804 17 1 6 D1 [1]
10 0.230721507966 4.3342296469 0.703363296033 17 2 5 [2]
11 0.224333593340 4.4576471366 0.731450345446 21 1 7 [1]
12 0.217384640521 4.6001410109 0.749277161807 27 9 D1 [1]
13 0.207999994722 4.8076924297 0.743144963272 25 1 8 [2]
14 0.202386816023 4.9410333126 0.757697772154 25 2 7 [1]
15 0.197735768366 5.0572539721 0.774934987330 35 10 [1]
16 0.191785609135 5.2141555590 0.777598749710 40 10 [1]
17 0.183126736787 5.4606990631 0.753279195582 31 2 10 D1 [2]
18 0.178473321196 5.6030783385 0.757569801809 35 1 10 D1 [2]
19 0.173441904330 5.7656193517 0.755205607498 37 1 11 [1]
20 0.169279411749 5.9073929291 0.757254419213 41 11 [2]
21 0.165404018640 6.0458023222 0.759127889229 50 10 [2]
22 0.161673301786 6.1853131528 0.759806203277 45 12 [2]
23 0.158923367389 6.2923408711 0.767550381308 47 11 [1]
24 0.156027298450 6.4091348753 0.771997613946 41 4 10 [1]
25 0.153122210789 6.5307312038 0.774497337032 47 2 12 [1]
26 0.150683336700 6.6364338745 0.780022883175 45 4 12 [2]
27 0.148137584795 6.7504813271 0.782884729566 47 4 11 [1]
28 0.145444423784 6.8754784404 0.782628611099 53 2 13 [1]
29 0.143333013135 6.9767597717 0.787216154549 53 3 13 [1]
30 0.141698173815 7.0572539721 0.795890460898 70 15 [2]
31 0.138616362210 7.2141555590 0.787035327241 70 15 [1]
32 0.135793611068 7.3641167072 0.779672455391 66 1 15 [1]
33 0.134134799538 7.4551868974 0.784513474860 63 2 14 [1]
34 0.131960534658 7.5780232521 0.782295061266 66 1 17 [1]
35 0.130170592942 7.6822266642 0.783605271724 67 3 16 [2]
36 0.128215138245 7.7993910367 0.781960200280 70 2 17 [2]
37 0.126679668746 7.8939265464 0.784547243181 77 1 18 [2]
38 0.125056798044 7.9963665762 0.785238789166 75 1 16 [2]
39 0.124185741467 8.0524542366 0.794715374348 77 1 16 [2]
40 0.122413791780 8.1690141728 0.791998265961 80 1 17 [2]
41 0.120982726955 8.2656427506 0.792928693410 80 2 15 [2]
42 0.119848964213 8.3438351476 0.797115765780 81 3 15 [1]
43 0.118421345534 8.4444235580 0.796768171019 87 3 16 D1 [1]
44 0.117257562843 8.5282345612 0.799351771065 85 3 17 [1]
45 0.115792923917 8.6361063023 0.797223502406 84 4 16 [1]
46 0.114685668831 8.7194852695 0.799428585597 80 7 14 [2]
47 0.113728619203 8.7928615242 0.803231865120 93 3 18 [2]
48 0.112997340346 8.8497658169 0.809806426252 92 2 18 [1]
49 0.111299134462 8.9847958372 0.802016295537 100 3 18 [1]
50 0.110419276505 9.0563897143 0.805495905634 105 5 15 [1]
51 0.108971323158 9.1767262342 0.800199292510 96 4 18 [1]
52 0.107852412844 9.2719297940 0.799220496365 101 2 20 [1]
53 0.106722676369 9.3700798558 0.797614101984 99 4 20 [1]
54 0.105895776586 9.4432472403 0.800118988953 108 1 21 [1]
55 0.104863207018 9.5362332360 0.799120918102 108 4 20 [2]
56 0.104252745744 9.5920735024 0.804204630593 114 1 21 [2]
57 0.103187535998 9.6910929244 0.801923385362 109 3 21 [1]
58 0.102435670827 9.7622243495 0.804144255376 114 5 21 [1]
59 0.101468332217 9.8552915787 0.802632228934 118 4 20 [1]
60 0.100739761056 9.9265671222 0.804556636832 125 2 21 [1]
61 0.099859697138 10.0140499988 0.803736816952 126 4 21 [2]
62 0.099194901497 10.0811632947 0.806072173278 117 7 19 [1]
63 0.098454910555 10.1569337107 0.806898395459 116 7 18 [1]
64 0.097834421788 10.2213513580 0.809406854119 123 5 20 [1]
65 0.097013439104 10.3078502240 0.808315107924 127 5 17 [1]
66 0.096256109964 10.3889508975 0.807986466935 141 2 23 [2]
67 0.095785092097 10.4400379862 0.812220942913 135 1 24 [2]
68 0.095077744308 10.5177085056 0.812213475288 135 3 23 [1]
69 0.094438906523 10.5888561910 0.813119785823 141 2 22 [1]
70 0.093736337552 10.6682213762 0.812676198079 133 5 22 [1]
71 0.093001827819 10.7524768432 0.811418406850 138 4 21 [1]
72 0.092405877836 10.8218224145 0.812335118933 140 3 22 [2]
73 0.091856585595 10.8865357179 0.813854923253 144 1 23 [1]
74 0.091081045227 10.9792328086 0.811131504666 140 5 22 [2]
75 0.090526425533 11.0464982364 0.812111287464 153 2 24 [2]
76 0.089885244874 11.1252964978 0.811323289022 147 3 22 [1]
77 0.089275764804 11.2012482021 0.810889024844 161 4 23 [2]
78 0.088753888691 11.2671119514 0.811844629497 168 1 24 [1]
79 0.088155608032 11.3435778202 0.811204815763 156 4 27 [1]
80 0.087759391988 11.3947917977 0.814105590159 153 7 24 [2]
81 0.087256528187 11.4604605613 0.814862656501 162 5 25 [1]
82 0.086531018480 11.5565495191 0.811261790603 171 4 24 [1]
83 0.086062502187 11.6194623046 0.812287122268 165 5 25 [1]
84 0.085582245062 11.6846665950 0.812924429047 171 5 24 [1]
85 0.085105640203 11.7501025504 0.813465521529 166 5 24 [2]
86 0.084693665494 11.8072584787 0.815086778108 162 7 25 [1]
87 0.084353558917 11.8548643690 0.817955377425 173 4 25 [2]
88 0.083957834537 11.9107407369 0.819612678541 173 5 24 [1]
89 0.083373984546 11.9941490795 0.817437693736 177 1 27 [1]
90 0.082916317197 12.0603523384 0.817572088512 195 3 26 [2]
91 0.082486675079 12.1231701853 0.818111555839 185 3 28 [2]
92 0.082097116118 12.1806958306 0.819307950188 188 4 26 [2]
93 0.081604858730 12.2541723074 0.818311249615 186 5 28 [1]
94 0.081226027424 12.3113247282 0.819448791186 188 6 26 [2]
95 0.080767780940 12.3811746263 0.818848288723 184 6 27 [1]
96 0.080301040266 12.4531387973 0.817931839723 187 6 28 [2]
97 0.079943232591 12.5088762061 0.819103315117 191 4 26 [2]
98 0.079605103982 12.5620085896 0.820562077603 184 6 26 [1]
99 0.079303089608 12.6098491867 0.822657285419 189 7 27 [1]
100 0.078846720977 12.6828356032 0.821430477223 198 3 29 [2]
101 0.078397075020 12.7555779313 0.820209190624 214 4 28 [1]
102 0.078049093952 12.8124485419 0.820992976718 209 4 28 [1]
103 0.077631543326 12.8813618428 0.820195162227 198 7 27 [2]
104 0.077168830932 12.9585998378 0.818315391322 201 4 28 [2]
105 0.076787199063 13.0230039929 0.818032371565 208 2 31 [1]
106 0.076515935706 13.0691729869 0.819998750986 211 4 29 [2]
107 0.076149393401 13.1320809705 0.819823215620 221 5 27 [1]
108 0.075754999505 13.2004489015 0.818935868866 207 5 30 [1]
109 0.075483429846 13.2479406678 0.820603353794 223 5 30 [1]
110 0.075131105447 13.3100663707 0.820419135752 217 4 29 [2]
111 0.074877167713 13.3552060066 0.822290616350 232 4 30 [1]
112 0.074526781472 13.4179952528 0.821951691180 222 5 30 [1]
113 0.074290113999 13.4607412234 0.824031913371 247 5 31 [1]
114 0.073964717970 13.5199596165 0.824057660622 250 3 30 [1]
115 0.073627370472 13.5819056635 0.823720661427 242 4 33 [1]
116 0.073329299010 13.6371138617 0.824169606091 253 6 28 [1]
117 0.072993462481 13.6998570284 0.823677742223 249 8 29 [1]
118 0.072709457058 13.7533691002 0.824265929821 245 5 29 [1]
119 0.072392698136 13.8135478543 0.824024314767 260 7 29 [2]
120 0.072129502864 13.8639524784 0.824917776266 240 8 28 [1]
121 0.071828527609 13.9220450883 0.824864925050 230 10 28 [1]
122 0.071564375957 13.9734328236 0.825576165456 236 9 31 [1]
123 0.071264314128 14.0322686359 0.825377962935 252 5 30 [1]
124 0.070978271129 14.0888187905 0.825422032474 253 6 29 [1]
125 0.070710600803 14.1421510869 0.825814693890 255 5 31 [1]
126 0.070440622682 14.1963537789 0.826076858999 245 4 32 [1]
127 0.070211417946 14.2426976873 0.827223278429 269 5 33 [1]
128 0.069935399593 14.2989102203 0.827194483475 254 4 33 [1]
129 0.069729464282 14.3411398653 0.828754512297 272 3 34 [2]
130 0.069387326253 14.4118537779 0.827003212366 249 8 31 [2]
131 0.069105802355 14.4705649297 0.826616103437 279 1 34 [1]
132 0.068829497993 14.5286545618 0.826278921104 266 3 35 [1]
133 0.068582227196 14.5810371125 0.826567544881 282 2 34 [1]
134 0.068261352367 14.6495779138 0.825007911039 267 4 34 [1]
135 0.068040648959 14.6970967400 0.825798712309 272 5 34 [1]
136 0.067765039748 14.7568717397 0.825189782026 266 9 30 [1]
137 0.067545701581 14.8047910762 0.825884926139 264 5 33 [2]
138 0.067338459371 14.8503546018 0.826816199221 267 6 32 [1]
139 0.067090550524 14.9052287124 0.826686888953 293 2 34 [1]
140 0.066864948010 14.9555189941 0.827043964551 307 8 34 [1]
141 0.066699596393 14.9925944695 0.828836872007 308 8 31 [1]
142 0.066470805976 15.0441985067 0.828998552700 300 6 32 [1]
143 0.066176694705 15.1110599352 0.827465164955 299 4 34 [1]
144 0.065970110319 15.1583799870 0.828057416415 303 7 33 [1]
145 0.065781819983 15.2017685169 0.829054938330 310 10 32 [1]
146 0.065569457672 15.2510030661 0.829391494827 314 10 31 [1]
147 0.065412400035 15.2876212991 0.831076575011 281 12 33 [1]
148 0.065149259364 15.3493686615 0.830011710195 280 12 31 [1]
149 0.064960070341 15.3940720007 0.830773777238 312 10 34 [1]
150 0.064730870089 15.4485796132 0.830458026371 306 6 33 [1]
151 0.064486188449 15.5071965649 0.829686267464 317 7 35 [1]
152 0.064303447765 15.5512656748 0.830454121365 302 7 35 [1]
153 0.064104138705 15.5996168142 0.830743798905 333 5 34 [1]
154 0.063934247315 15.6410694111 0.831747248158 332 9 33 [1]
155 0.063792464324 15.6758327272 0.833439338665 298 6 34 [1]
156 0.063543880700 15.7371565756 0.832291779573 325 6 35 [1]
157 0.063378537105 15.7782120837 0.833273579974 331 7 36 [1]
158 0.063128021010 15.8408260547 0.831964844740 341 6 35 [1]
159 0.062943785355 15.8871919501 0.832350754032 348 5 38 [1]
160 0.062737965620 15.9393118683 0.832116981324 322 5 37 [1]
161 0.062534800211 15.9910961037 0.831903493098 308 10 34 [1]
162 0.062329247848 16.0438323023 0.831576724797 343 7 36 [1]
163 0.062130345361 16.0951946137 0.831378287765 352 6 36 [1]
164 0.061963172335 16.1386185103 0.831983426596 331 8 39 [1]
165 0.061769355491 16.1892574732 0.831828167543 322 12 36 [1]
166 0.061559614133 16.2444163123 0.831195923056 338 14 38 [1]
167 0.061377202938 16.2926942274 0.831254856567 351 7 36 [1]
168 0.061197868747 16.3404383269 0.831352899641 358 8 38 [2]
169 0.061039784632 16.3827576724 0.831986402184 372 6 35 [1]
170 0.060870040510 16.4284431491 0.832261186809 344 8 39 [1]
171 0.060703446559 16.4735292094 0.832580717201 345 6 37 [1]
172 0.060508367506 16.5266398883 0.832075735101 335 6 37 [1]
173 0.060356303309 16.5682777966 0.832712159269 381 3 39 [1]
174 0.060201479446 16.6108874600 0.833234253388 375 6 39 [1]
175 0.060015243635 16.6624333991 0.832846057289 348 8 39 [1]
176 0.059846499713 16.7094150000 0.832901636603 376 11 37 [1]
177 0.059661881434 16.7611207686 0.832474030307 393 6 37 [1]
178 0.059505580185 16.8051466248 0.832796571751 383 12 37 [1]
179 0.059361397555 16.8459645694 0.833421698988 377 8 41 [1]
180 0.059217781153 16.8868198120 0.834027373712 375 7 40 [1]
181 0.059099157200 16.9207150724 0.835304244913 384 9 38 [1]
182 0.058908887350 16.9753673000 0.834519648558 395 13 39 [1]
183 0.058783353045 17.0116189057 0.835532481866 394 13 38 [1]
184 0.058597882648 17.0654630306 0.834805321539 374 11 38 [1]
185 0.058444798229 17.1101625859 0.834962544915 394 11 39 [1]
186 0.058291372929 17.1551972402 0.835074171486 375 6 40 [1]
187 0.058148151436 17.1974512569 0.835443281080 376 12 35 [1]
188 0.058011965909 17.2378229962 0.835981282794 371 14 37 [1]
189 0.057887757934 17.2748096608 0.836833005565 405 10 38 [1]
190 0.057732101738 17.3213856744 0.836742591853 378 12 38 [1]
191 0.057580382406 17.3670260289 0.836731261856 387 7 41 [1]
192 0.057450429294 17.4063103146 0.837319728047 415 8 38 [1]
193 0.057312414847 17.4482265783 0.837641648820 425 12 39 [1]
194 0.057135549328 17.5022383045 0.836793085463 422 12 39 [1]
195 0.056983864053 17.5488274904 0.836646387750 419 14 43 [1]
196 0.056820122561 17.5993988560 0.836111009708 426 9 42 [1]
197 0.056676656007 17.6439485046 0.836138462440 436 13 41 [1]
198 0.056537340893 17.6874254113 0.836256460388 387 9 42 [1]
199 0.056408547824 17.7278096774 0.836655082719 400 7 43 [1]
200 0.056271364514 17.7710280999 0.836774480720 427 15 39 [1]


Updates

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

01-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: 10, 13, 17, 18, 20, 21, 22, 26, 30, 35, 36, 37, 38, 39, 40, 41, 46, and 47. Among them are two packings which show a reflection symmetry like N = 17 and 18.
26-Feb-2023: Extension to N = 200, inspired by the article [2] by Paolo Amore.
06-Mar-2023: Update after publication of [2] with correct authorship.

References

[1]   , program cpt, 2023.
[2]   , Circle packing in regular polygons , Phys. Fluids 35, 027130 (2023).

Datenschutzerklärung der Otto-von-Guericke-Universität Magdeburg nach DSGVO

©  E. Specht    06-Mar-2023