The best known packings of equal spheres in a sphere (complete up to N = 519)

Last update: 25-Feb-2022


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   193-204   205-216   217-228   229-240   241-252   253-264   265-276   277-288   289-300   301-312   313-324   325-336   337-348   349-360   361-372   373-384   385-396   397-408   409-420   421-432   433-444   445-456   457-468   469-480   481-492   493-504   505-516   517-767   791-1253   1289-1986   2000-2779   2899-3589   3781-4585   4729-5481   5721-6603   7000-40143   50000-100281   110109-500000  


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.


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 spheres; colors correspond to active researchers in the past, see "References" at the bottom of the page
radius
of the spheres in the container sphere
distance
packing of spheres in a sphere is equivalent to distributing points in a sphere; the latter are then the sphere centers. "distance" is here the largest distance between these points
ratio
= 1/radius
density
ratio of total volume occupied by the spheres to container volume (for an infinite face-centered cubic packing you get the well-known value ρ = Pi/(3*sqrt(2))=0.74048048969)
contacts
number of contacts between circles and container and between the circles themselves, respectively
reference
for the best known packing so far
records
the sequence of N 's that establish density records

     
Nradiusdistance ratio density contacts loose boundary reference
1    1.000000000000    1.0000000000    1.000000000000    1 1
2    0.500000000000 2.000000000000    2.0000000000 0.250000000000 3 2
3    0.464101615138 1.732050807571 2.1547005384 0.299888971118 6 3 [1]
4    0.449489742783 1.632993161854 2.2247448714 0.363261480388 10 4 [1]
5    0.414213562373 1.414213562373 2.4142135624 0.355339059327 12 5 [1]
6    0.414213562373 1.414213562373 2.4142135624 0.426406871193 18 6 [1]
7    0.385913557400 1.256870468483 2.5912538723 0.402316780932 19 7 [2]
8    0.378024844801 1.215562524133 2.6453287760 0.432166419789 24 8 [2]
9    0.366025403785 1.154700538382 2.7320508076 0.441342951092 27 9 [2]
10    0.353049430432 1.091426291402 2.8324645611 0.440054580178 29 10 [5]
11    0.344576516755 1.051462224237 2.9021130326 0.450038547852 36 11 [5]
12    0.344576516755 1.051462224237 2.9021130326 0.490951143111 42 12 [1]
13    0.333333333333 1.000000000000 3.0000000000 0.481481481481 24 12
14    0.323504674183 0.956413627226 3.0911454449 0.473990585425 37 1 13 [2]
15    0.318304823107 0.933862623344 3.1416426249 0.483749932003 42 1 14 [2]
16    0.310975923325 0.902656188230 3.2156830320 0.481171926089 45 1 15 [2]
17    0.305693961650 0.880574112178 3.2712455117 0.485634466608 48 1 16 [2]
18    0.301296589342 0.862444879318 3.3189887817 0.492328697969 51 1 17 [5]
19    0.295332333897 0.838217355793 3.3860159733 0.489425500215 52 1 18 [6]
20    0.287890825717 0.808558114721 3.4735389622 0.477214324831 52 2 19 [7]
21    0.286832818119 0.804391523913 3.4863514104 0.495570920187 57 3 20 [3]
22    0.279342624810 0.775243921527 3.5798331912 0.479550457554 61 1 21 [8]
23    0.275670701290 0.761175067144 3.6275164365 0.481836470826 64 1 22 [8]
24    0.271341339941 0.744769409368 3.6853949355 0.479467459313 68 1 22 [31]
25    0.271191827927 0.744206331155 3.6874267475 0.498620125704 84 1 24 [4]
26    0.266851272489 0.727959450724 3.7474057765 0.494061693011 77 24 [11]
27    0.262232080452 0.710879596426 3.8134159569 0.486879204757 78 1 24 [11]
28    0.260305475681 0.703818852575 3.8416402782 0.493864651913 82 25 [11]
29    0.257925462478 0.695147049080 3.8770891032 0.497600320836 76 3 26 [11]
30    0.255330557656 0.685755432242 3.9164916616 0.499378264835 86 1 27 [11]
31    0.253116209529 0.677792751050 3.9507544849 0.502714681184 85 2 27 [11]
32    0.250787448179 0.669469425117 3.9874403893 0.504739586490 93 1 28 [11]
33    0.248762350343 0.662273384345 4.0199009160 0.508004891658 94 1 29 [11]
34    0.247052663501 0.656228268632 4.0477199712 0.512681372527 102 30 [11]
35    0.244833658425 0.648423122022 4.0844057408 0.513666698068 100 1 30 [11]
36    0.243132164915 0.642469275729 4.1129893297 0.517403964757 98 4 32 [11]
37    0.240686558293 0.633958376272 4.1547812520 0.515890148167 110 32 [11]
38    0.240519372142 0.633378557187 4.1576692600 0.528729790846 116 32 [11]
39    0.236745240285 0.620357062361 4.2239497563 0.517497643020 97 6 33 [11]
40    0.234999237637 0.614376479603 4.2553329537 0.519109947837 98 7 34 [11]
41    0.232755981857 0.606732607497 4.2963450048 0.516995080282 100 7 35 [11]
42    0.232118623301 0.604568961677 4.3081420430 0.525265949758 108 6 36 [11]
43    0.229732967254 0.596502141433 4.3528798324 0.521360856683 127 36 [11]
44    0.228163038226 0.591220813529 4.3828308379 0.522623038233 127 1 36 [11]
45    0.226911569263 0.587026167360 4.4070031477 0.525753813707 119 5 38 [11]
46    0.225168199111 0.581205363158 4.4411244747 0.525144709692 138 38 [11]
47    0.223507049839 0.575683397493 4.4741318035 0.524773065591 133 2 38 [11]
48    0.222405942149 0.572036115512 4.4962827447 0.528056504400 139 1 39 [11]
49    0.221278177007 0.568311226097 4.5191984746 0.530898911129 142 1 40 [11]
50    0.219758275796 0.563308187652 4.5504543407 0.530647009782 142 2 41 [11]
51    0.218550295807 0.559345776534 4.5756057950 0.532383236403 145 2 42 [11]
52    0.216930407731 0.554051414773 4.6097732930 0.530841223195 142 4 42 [11]
53    0.216288057367 0.551958048873 4.6234637833 0.536257636148 142 5 43 [11]
54    0.214920670908 0.547513258709 4.6528795754 0.536078417223 160 44 [11]
55    0.213441688789 0.542723116002 4.6851203515 0.534811127622 151 4 43 [11]
56    0.213067869308 0.541515236189 4.6933402171 0.541678896170 154 4 44 [11]
57    0.211316523700 0.535871563308 4.7322376049 0.537867407396 157 4 45 [31]
58    0.210480101312 0.533185044891 4.7510429431 0.540830438196 165 3 45 [11]
59    0.209759450876 0.530874937027 4.7673656459 0.544523495682 163 4 46 [11]
60    0.209426996435 0.529810644914 4.7749335903 0.551123897404 156 12 48 [11]
61    0.209108659202 0.528792385035 4.7822027257 0.557758101350 204 48 [11]
62    0.206739154000 0.521238770431 4.8370131185 0.547847764413 184 1 48 [11]
63    0.205954921018 0.518748686868 4.8554314461 0.550372935465 204 1 48 [11]
64    0.204107219252 0.512901295725 4.8993857428 0.544195657340 172 6 48 [31]
65    0.203076051707 0.509649765557 4.9242635535 0.544364116868 192 1 50 [11]
66    0.202108764115 0.506607304417 4.9478309581 0.544878127391 187 3 50 [11]
67    0.201216632889 0.503807768599 4.9697680835 0.545841352947 178 7 52 [11]
68    0.200002230133 0.500006969185 4.9999442473 0.544018198088 193 3 52 [11]
69    0.199281532516 0.497756803692 5.0180264442 0.546072434137 184 7 52 [11]
70    0.198688729141 0.495908984103 5.0329981188 0.549057383073 199 3 53 [11]
71    0.197222141221 0.491349229589 5.0704246177 0.544659844168 196 5 54 [11]
72    0.196285915470 0.488447121304 5.0946090432 0.544502539950 211 1 54 [11]
73    0.195628064638 0.486411959537 5.1117410063 0.546532931293 208 3 55 [31]
74    0.195146793635 0.484925181615 5.1243475815 0.549940848873 212 3 56 [31]
75    0.193964427622 0.481280068198 5.1555845175 0.547302624777 212 5 56 [31]
76    0.192947595405 0.478153820759 5.1827544049 0.545923392558 217 3 57 [31]
77    0.192254073574 0.476026104953 5.2014502549 0.547163825196 217 4 58 [11]
78    0.191458926390 0.473591095466 5.2230523740 0.547421008606 173 20 59 [31]
79    0.190659742093 0.471148544090 5.2449457291 0.547525177852 233 1 58 [31]
80    0.189727102250 0.468304204119 5.2707282625 0.546359008245 229 3 58 [11]
81    0.188968327254 0.465994938556 5.2918921098 0.546577909205 235 2 60 [31]
82    0.188293592202 0.463945067805 5.3108551826 0.547419766291 238 2 61 [31]
83    0.187640284124 0.461963537721 5.3293460126 0.548348093717 247 61 [31]
84    0.186955560139 0.459890138775 5.3488647209 0.548901532648 238 4 61 [11]
85    0.186411240258 0.458244384588 5.3644833789 0.550598740536 244 3 60 [31]
86    0.185610957844 0.455828721252 5.3876129492 0.549932374075 250 2 63 [31]
87    0.184934302257 0.453789928270 5.4073256708 0.550264723511 252 3 60 [11]
88    0.184275052856 0.451806833912 5.4266705368 0.550658446830 247 5 64 [11]
89    0.183598057582 0.449773691224 5.4466807175 0.550800407839 256 3 62 [31]
90    0.182935895973 0.447788356070 5.4663957267 0.550984402483 259 3 64 [11]
91    0.182348262626 0.446029169366 5.4840116687 0.551755005487 265 2 64 [11]
92    0.181782421764 0.444337610433 5.5010819544 0.552641484079 259 5 65 [31]
93    0.181267379888 0.442799945758 5.5167123871 0.553913462523 268 3 64 [31]
94    0.180806842580 0.441426642648 5.5307641333 0.555613053163 268 4 60 [31]
95    0.180125543166 0.439397865526 5.5516834671 0.555200074327 274 3 66 [31]
96    0.179567219984 0.437737799751 5.5689451565 0.555843344889 274 4 65 [31]
97    0.179019633861 0.436111851744 5.5859794729 0.556510967846 288 1 66 [31]
98    0.178498681405 0.434566999138 5.6022822809 0.557353987401 289 1 67 [31]
99    0.177878571988 0.432730654930 5.6218126153 0.557193571755 289 2 68 [11]
100    0.177431405921 0.431408169965 5.6359808164 0.558587845356 282 6 69 [11]
101    0.176679757440 0.429188420998 5.6599579629 0.557034067440 298 1 70 [31]
102    0.176215110923 0.427818264840 5.6748822207 0.558122607519 298 2 68 [31]
103    0.175673094380 0.426221910706 5.6923913336 0.558409731670 286 7 69 [31]
104    0.175163370419 0.424722579326 5.7089561454 0.558937462075 300 4 72 [31]
105    0.174632819112 0.423163952131 5.7263005035 0.559199653725 295 6 72 [31]
106    0.174152753738 0.421755365841 5.7420854884 0.559882513029 307 3 72 [31]
107    0.173608230754 0.420159631823 5.7600955649 0.559879687186 307 4 70 [31]
108    0.173168180989 0.418871593975 5.7747329463 0.560825872313 307 5 71 [31]
109    0.172638823470 0.417323965318 5.7924398458 0.560843771142 313 4 73 [31]
110    0.172285229701 0.416291301987 5.8043281002 0.562518518025 325 1 73 [31]
111    0.171743879601 0.414712008450 5.8226237949 0.562298320367 325 2 73 [31]
112    0.171326945524 0.413497083316 5.8367934883 0.563242009507 334 75 [31]
113    0.170853998626 0.412120418703 5.8529505194 0.563577810665 331 2 76 [31]
114    0.170449483384 0.410944191993 5.8668408971 0.564536360556 337 1 77 [31]
115    0.169745356914 0.408899506501 5.8911773387 0.562459882260 331 4 77 [31]
116    0.169367420514 0.407803461354 5.9043232575 0.563569665507 328 6 77 [31]
117    0.169057633289 0.406905797711 5.9151425496 0.565314618649 337 4 78 [31]
118    0.168723599855 0.405938626011 5.9268531543 0.566773460390 334 6 78 [31]
119    0.168090670453 0.404108150932 5.9491701550 0.565168293742 337 6 79 [31]
120    0.167593810095 0.402673147141 5.9668074819 0.564878661167 346 4 77 [31]
121    0.167146216576 0.401381898966 5.9827857339 0.565034573186 355 2 79 [31]
122    0.166776063926 0.400315105473 5.9960642820 0.565927750478 355 3 77 [31]
123    0.166414865560 0.399275031870 6.0090785558 0.566867380471 334 11 78 [31]
124    0.166043033766 0.398205280342 6.0225351062 0.567653950274 343 9 79 [31]
125    0.165617291650 0.396981600871 6.0380168643 0.567841393747 349 8 80 [31]
126    0.165230418840 0.395870723057 6.0521543613 0.568382314745 355 7 80 [31]
127    0.164830084333 0.394722274452 6.0668536575 0.568739200886 367 4 80 [31]
128    0.164456462770 0.393651450684 6.0806366813 0.569328342446 340 14 75 [31]
129    0.164115580588 0.392675295236 6.0932666869 0.570215673737 361 8 76 [31]
130    0.163729008890 0.391569265538 6.1076531690 0.570584869402 385 1 83 [11]
131    0.163455162313 0.390786374978 6.1178856994 0.572093771176 352 13 78 [31]
132    0.163213627869 0.390096285754 6.1269393558 0.573909199081 382 4 79 [31]
133    0.162962154416 0.389378222922 6.1363940823 0.575588241935 370 9 79 [31]
134    0.162832399256 0.389007885903 6.1412839494 0.578531838360 373 9 80 [31]
135    0.162579700716 0.388286982905 6.1508293815 0.580139889296 373 10 81 [31]
136    0.162377283465 0.387709837041 6.1584969194 0.582257005914 376 10 82 [31]
137    0.161772816367 0.385987998303 6.1815082562 0.580012303665 394 5 82 [31]
138    0.161135258979 0.384174590013 6.2059663809 0.577365501645 402 3 84 [31]
139    0.160745199639 0.383066500352 6.2210255874 0.577336263825 373 14 85 [31]
140    0.160407273660 0.382107344734 6.2341312659 0.577830162437 406 4 84 [31]
141    0.160024538913 0.381021937726 6.2490415957 0.577801767737 406 5 85 [31]
142    0.159699976468 0.380102277748 6.2617416866 0.578366194897 406 6 86 [31]
143    0.159446486235 0.379384497534 6.2716966903 0.579670095928 409 6 87 [31]
144    0.159153628674 0.378555784032 6.2832371987 0.580513236652 411 5 88 [31]
145    0.158600757402 0.376992869431 6.3051401291 0.578473905632 415 4 90 [31]
146    0.158245529341 0.375989756769 6.3192938478 0.578558400878 400 12 90 [31]
147    0.157784117755 0.374688060582 6.3377734986 0.577440435470 422 6 91 [31]
148    0.157391538060 0.373581669707 6.3535817257 0.577039912876 427 5 90 [31]
149    0.157074962827 0.372690229618 6.3663869913 0.577440399581 424 7 91 [31]
150    0.156703678076 0.371645586497 6.3814711453 0.577203332042 418 10 94 [31]
151    0.156284188488 0.370466420934 6.3985999459 0.576397479835 418 11 93 [31]
152    0.156015192670 0.369710903123 6.4096321832 0.577223844761 428 8 93 [31]
153    0.155581386131 0.368493502099 6.4275041177 0.576188173801 415 14 93 [31]
154    0.155232190987 0.367514456235 6.4419628019 0.576057828083 430 10 93 [31]
155    0.154882847307 0.366535803500 6.4564928744 0.575892828008 439 8 94 [31]
156    0.154590492826 0.365717422183 6.4687030989 0.576332278156 426 13 95 [31]
157    0.154304605357 0.364917691014 6.4806879723 0.576814696187 442 9 95 [31]
158    0.154079300217 0.364287811509 6.4901644711 0.577949615306 449 7 96 [31]
159    0.153695958755 0.363216884865 6.5063519438 0.577277285755 460 5 97 [31]
160    0.153336259774 0.362212889223 6.5216146623 0.576838952415 439 13 98 [11]
161    0.153095571235 0.361541553060 6.5318675905 0.577715152779 457 8 98 [31]
162    0.152766222693 0.360623541660 6.5459496371 0.577559904144 481 1 99 [31]
163    0.152333352287 0.359418062980 6.5645506055 0.576199135780 462 8 100 [31]
164    0.152097125131 0.358760725170 6.5747462297 0.577041255291 459 10 99 [31]
165    0.151818107200 0.357984787199 6.5868295847 0.577370594951 463 10 99 [31]
166    0.151510034723 0.357128642467 6.6002228950 0.577340841944 472 8 101 [31]
167    0.151275315465 0.356476766193 6.6104638217 0.578123566156 477 7 101 [31]
168    0.150959747967 0.355600921406 6.6242823896 0.577953326846 481 6 102 [31]
169    0.150633585245 0.354696353960 6.6386257645 0.577633195437 490 5 103 [31]
170    0.150414852078 0.354090116678 6.6482796492 0.578523605552 480 9 102 [31]
171    0.150126724785 0.353292024030 6.6610392083 0.578588956932 484 9 102 [31]
172    0.149823942661 0.352453921438 6.6745006321 0.578458372460 484 10 104 [31]
173    0.149550842703 0.351698491132 6.6866891682 0.578645655624 481 12 104 [31]
174    0.149295833059 0.350993538908 6.6981105869 0.579018323663 483 12 104 [31]
175    0.149008670736 0.350200209125 6.7110188626 0.578992142861 487 12 105 [31]
176    0.148745774624 0.349474387768 6.7228800450 0.579224043324 490 12 106 [31]
177    0.148341239250 0.348358394433 6.7412137384 0.577775304058 505 8 106 [31]
178    0.148066413782 0.347600836913 6.7537260778 0.577816149121 505 9 106 [31]
179    0.147779767882 0.346811216896 6.7668261653 0.577694152012 519 5 109 [31]
180    0.147648144480 0.346448813419 6.7728585654 0.579370643052 522 4 109 [31]
181    0.147222317786 0.345277135780 6.7924484211 0.577563217473 517 8 109 [31]
182    0.146956187105 0.344545461518 6.8047492229 0.577610412993 524 6 107 [31]
183    0.146713480572 0.343878585287 6.8160062463 0.577911252205 529 6 108 [31]
184    0.146448929587 0.343152119805 6.8283189424 0.577931582438 517 11 109 [31]
185    0.146147309943 0.342324411798 6.8424112657 0.577489651564 532 7 109 [31]
186    0.145873754857 0.341574224388 6.8552427473 0.577356992910 547 3 110 [31]
187    0.145641283299 0.340937080531 6.8661850359 0.577690329525 523 12 112 [31]
188    0.145383154153 0.340230022049 6.8783760115 0.577696996722 532 10 111 [31]
189    0.145201386787 0.339732387354 6.8869865649 0.578594228038 539 8 112 [31]
190    0.144943777745 0.339027479066 6.8992268282 0.578565229663 540 9 112 [31]
191    0.144676708866 0.338297133648 6.9119625946 0.578401258999 559 4 114 [31]
192    0.144447612062 0.337670992679 6.9229251057 0.578671813415 527 15 114 [31]
193    0.144165689697 0.336900935056 6.9364631911 0.578286496167 556 7 115 [31]
194    0.143870419235 0.336094961481 6.9506991452 0.577718476528 562 6 112 [31]
195    0.143614018530 0.335395538081 6.9631085477 0.577597247624 568 5 116 [31]
196    0.143322983399 0.334602144383 6.9772480051 0.577036906367 563 5 115 [31]
197    0.143104594387 0.334007145912 6.9878958414 0.577333764513 577 4 115 [31]
198    0.142899447497 0.333448501648 6.9979276863 0.577772462939 559 11 116 [31]
199    0.142650446091 0.332770794457 7.0101428170 0.577660243157 575 6 117 [31]
200    0.142396339034 0.332079596940 7.0226524557 0.577466064231 581 5 118 [31]
201    0.142224083682 0.331611277436 7.0311579735 0.578249799131 583 6 118 [31]
202    0.141939049418 0.330836753081 7.0452775617 0.577639717069 585 6 118 [31]
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3781    0.055466051920 0.117446391488 18.0290459729 0.645191205420 12895 103 918 [31]
3805    0.055358103561 0.117204421633 18.0642026312 0.645503011095 10821 101 920 [31]
3925    0.054738886050 0.115817492632 18.2685485979 0.643765244814 10934 106 958 [31]
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4189    0.053572966969 0.113210982145 18.6661306360 0.644091336399 11614 99 990 [31]
4213    0.053462475289 0.112964301770 18.7047082013 0.643781728683 11900 80 984 [31]
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4417    0.052684186064 0.111228347060 18.9810277943 0.645904371518 12141 122 1023 [31]
4537    0.052223435092 0.110201997022 19.1484914435 0.646197218166 12701 112 1068 [31]
4585    0.052091737452 0.109908816095 19.1969024056 0.648105741741 12945 102 1053 [31]
4729    0.051529819168 0.108658807012 19.4062392639 0.647060917536 13019 110 1111 [31]
4753    0.051456493109 0.108495799582 19.4338933647 0.647572456045 11545 185 1096 [31]
4897    0.051085445393 0.107671328562 19.5750471060 0.652862398976 12909 155 1011 [31]
4921    0.050865749062 0.107183465377 19.6595944902 0.647634085980 13590 101 1134 [31]
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5017    0.050574237403 0.106536475827 19.7729130749 0.648981220182 18163 181 1142 [31]
5089    0.050442769936 0.106244823037 19.8244466208 0.653174514388 18443 175 1135 [31]
5233    0.049858809486 0.104950316824 20.0566361353 0.648599258364 14245 128 1190 [31]
5281    0.049766460614 0.104745746285 20.0938541271 0.650918225875 15662 119 1186 [31]
5425    0.049435571481 0.104013089482 20.2283491430 0.655418083246 14933 106 1113 [31]
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5721    0.048362204502 0.101639940597 20.6773039049 0.647128035614 17862 137 1288 [31]
5775    0.048230928332 0.101350064353 20.7335839179 0.647931138621 16291 127 1295 [31]
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5979    0.047700111454 0.100178760972 20.9643116025 0.648913388677 16845 146 1324 [31]
6000    0.047187699337 0.099049307622 21.1919634575 0.630431146186 17494 128 1355 [31]
6051    0.047556166041 0.099861355275 21.0277674432 0.650800169036 16724 137 1345 [31]
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6339    0.046983761482 0.098600127853 21.2839493573 0.657452076017 16848 167 1303 [31]
6363    0.046766323549 0.098121425427 21.3829081294 0.650821082854 17994 124 1404 [31]
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6531    0.046382821560 0.097277655245 21.5597060801 0.651705184715 18827 162 1413 [31]
6603    0.046253017236 0.096992217164 21.6202111723 0.653373479790 22534 167 1414 [31]
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8000    0.043021472466 0.089911050726 23.2442067337 0.637009338072 23348 132 1647 [31]
9000    0.041348250525 0.086263339211 24.1848200904 0.636229681916 26241 181 1794 [31]
9909    0.040692765852 0.084837817132 24.5743925010 0.667700123925 37798 267 1813 [31]
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10005    0.040579698283 0.084592119242 24.6428643463 0.668564841143 28329 348 1788 [31]
20000    0.031511804440 0.065074214811 31.7341395636 0.625820540730 58668 333 3513 [31]
20113    0.032360307172 0.066885034609 30.9020552458 0.681576738151 52293 384 3275 [31]
30000    0.027569300128 0.056701830026 36.2722301748 0.628634886153 88002 416 6699 [31]
30093    0.028353902321 0.058362612455 35.2685139660 0.685968229323 71212 733 4981 [31]
40000    0.025100123336 0.051492720302 39.8404416828 0.632539364395 117150 649 9016 [31]
40143    0.025842262077 0.053055600897 38.6963028631 0.692789672540 99231 1012 6827 [31]
50000    0.023292291593 0.047695521173 42.9326584723 0.631839335086 146380 719 10666 [31]
50141    0.024014688109 0.049211167046 41.6411820741 0.694422594361 58221 14485 8515 [31]
60000    0.021962913560 0.044912230558 45.5312997189 0.635654472887 175839 986 12268 [31]
60047    0.022656242323 0.046362893598 44.1379459905 0.698320767542 143629 1088 9762 [31]
70000    0.020874072304 0.042638176998 47.9063205989 0.636677629119 204695 1135 13752 [31]
70009    0.021542499588 0.044033592831 46.4198685912 0.699909928541 133282 3628 11006 [31]
80000    0.019966505753 0.040746578296 50.0838760858 0.636789934232 233889 1322 15085 [31]
80341    0.020583242024 0.042031631287 48.5832114705 0.700613618715 190100 1080 12567 [31]
90000    0.019199467294 0.039150605355 52.0847784309 0.636956899751 263186 1448 16480 [31]
90183    0.019807115279 0.040414729770 50.4869076549 0.700790758367 167090 6098 14020 [31]
100000    0.018537044699 0.037774313537 53.9460316484 0.636973685874 292221 1566 17729 [31]
100281    0.019150716641 0.039049254490 52.2173670441 0.704326043302 241716 1440 15346 [31]
110109    0.018570913333 0.037844636124 53.8476477742 0.705216732460 259230 2168 16636 [31]
120119    0.018039268987 0.036741324612 55.4346188152 0.705128894497 160346 30268 17944 [31]
130055    0.017580785179 0.035790800737 56.8802809328 0.706711006654 225072 14314 19668 [31]
180000    0.015060083171 0.030580714445 66.4006957097 0.614829385217 155200 106084 30217 [31]
500000    0.010670618461 0.021571417286 93.7152802956 0.607489504278 1320897 1909 81156 [31]
1030719    0.008879072967 0.017917234365 112.6243701022 0.721511332383 6102348 148028 [31]
1565879    0.007732295227 0.015585099041 129.3277055056 0.723908176976 9286272 224212 [31]
2005059    0.007124757201 0.014351767259 140.3556601002 0.725165882674 11901828 286814 [31]
2519011    0.006606203613 0.013300271528 151.3728698934 0.726249638858 14964828 360170 [31]
2982989    0.006246686016 0.012571904774 160.0848829985 0.727110946851 17730708 427090 [31]





Updates

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

14-Feb-2022: First presentation for N=1–519, 531, 555, 600, 603, 627, 675, 683, 700, 767, 791, 800, 887, 900, 935, 959, 1000, 1055, 1061, 1157, 1205, 1253, 1289, 1409, 1433, 1481, 1505, 1553, 1601, 1721, 1745, 1865, 1961, 1985, 2000, 2093, 2123, 2171, 2243, 2315, 2347, 2491, 2587, 2659, 2731, 2779, 2899, 3000, 3043, 3055, 3103, 3151, 3319, 3367, 3463, 3511, 3559, 3589, 3781, 3805, 3925, 3997, 4000, 4093, 4189, 4213, 4321, 4417, 4537, 4585, 4729, 4753, 4897, 4921, 5000, 5017, 5089, 5233, 5281, 5425, 5473, 5481, 5721, 5775, 5895, 5979, 6000, 6051, 6099, 6339, 6363, 6459, 6531, 6603, 7000, 8000, 9000, 9909, 10000, 10005, 20000, 20113, 30000, 30093, 40000, 40143, 50000, 50141, 60000, 60047, 70000, 70009, 80000, 80341, 90000, 90183, 100000, 100281, 110109, 120119, 130055, 180000, and 500000.
16-Feb-2022: First improvements for N= 94, 138, 148, 149, 156, 162, 165, 167, 190, 199, 210, 217, 219, 220, 223, 231, 232, 234, 236, 238, 239, 242, 243, 245, 250, 252, 253, 256, 258, 261, 270, 271, 272, 275, 281, 287, 293, 297, 299, 307, 312, 313, 323, 332, 355, 356, 358, 362, 363, 364, 376, 401, 407, 408, 410, 413, 419, 420, 424, 430, 431, 437, 439, 446, 449, 451, 458, 462, 482, 488, 489, 490, 501, 503, 507, 509, 510, 511, 513, 515, 516, and 518 by Eckard Specht [31].
17-Feb-2022: Further small improvements for N= 120, 161, 165, 170, 182, 183, 190, 191, 206, 213, 221, 232, 233, 237, 239, 252, 266, 275, 290, 295, 299, 316, 332, 342, 351, 355, 357, 362, 369, 391, 397, 404, 412, 414, 424, 427, 428, 430, 433, 452, 461, 471, 478, 484, 490, 503, 504, 509, 510, and 514 by Eckard Specht [31].
22-Feb-2022: Contact figures added. They are displayed at the right hand side. The colors indicate the number of contacting neighbors for each sphere. Furthermore, the size of the (equal) spheres are scaled linearly such that innermost spheres are larger than those near the container. This allows a pretty look at the spheres near the center.
25-Feb-2022: Small improvements for N= 935, 1055, 9909, 100281, 110109, 120119, 130055, and 500000. New high density packings added for N= 1030719, 1565879, 2005059, 2519011, and 2982989 by Eckard Specht [31].


References

[1]   L. Fejes, Über eine Abschätzung des kürzesten Abstandes zweier Punkte eines auf einer Kugelfläche liegenden Punktsystems, Jahresbericht der Deutschen Mathematiker-Vereinigung 53 (1943), 66–68.
[2]   K. Schütte, B.L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Annalen 123 (1951), 96–124.
[3]   B.L. van der Waerden, Punkte auf der Kugel. Drei Zusätze, Math. Annalen 125 (1952), 213–222.
[4]   Raphael M. Robinson, Arrangement Of 24 Points On A Sphere, Math. Annalen 144 (1961), 17–48.
[5]   Ludwig Danzer, Endliche Punktmengen auf der 2-Sphäre mit möglichst großem Minimalabstand (Finite Point-Set on S2 with Minimum Distance as Large as Possible), Habilitationsschrift, Univ. Göttingen, Germany (1963). Engl. transl.: Discrete Math. 60 (1986), 3–66.
[6]   T. Tarnai, Ts. Gáspár, Improved packing of equal circles on a sphere and rigidity of its graph, Math. Proc. Cambridge Phil. Soc. 93 (1983), 191–218.
[7]   D. Lazić, V. Šenk, I.P. Šeškar, Arrangements of Points on a Sphere Which Maximizes the Distance, Bulletin for Applied Mathematics, Technical University Budapest XLVII-4/9/87 (1987), 7–21.
[8]   D.A. Kottwitz, The Densest Packing of Equal Circles on a Sphere, Acta Cryst. A47 (1991), 158–165.
[9]   Th. Gensane, Dense Packings of Equal Spheres in a Larger Sphere, Les Cahiers du LMPA J. Liouville 188 (2003).
[10]   H. Pfoertner, Densest Packings of n Equal Spheres in a Sphere of Radius 1, August 25, 2013.
[11]   , Serial Symmetrical Relocation Algorithm for the Equal Sphere Packing Problem, Jisuanji Yingyong Yanjiu (Application Research of Computers) 29 (2012) 5, 1695–1697 (in Chinese), arxiv:1202.4149 (2012) (in English).
[31]   , program ssp, April 2010–February 2022.