The best known packings of equal spheres in a sphere (complete up to N = 1000)


Last update: 13-Jul-2026


Overview    Density plots    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   517-528   529-540   541-552   553-564   565-576   577-588   589-600   601-612   613-624   625-636   637-648   649-660   661-672   673-684   685-696   697-708   709-720   721-732   733-744   745-756   757-768   769-780   781-792   793-804   805-816   817-828   829-840   841-852   853-864   865-876   877-888   889-900   901-912   913-924   925-936   937-948   949-960   961-972   973-984   985-996   997-1008  


Density plots

2-50   50-100   100-150   150-200   200-250   250-300   300-350   350-400   400-450   450-500   500-550   550-600   600-650   650-700   700-750   750-800   800-850   850-900   5-200   200-500   500-800  


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 figures 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 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, also known as packing fraction ϕ (for an infinite face-centered cubic packing you get the well-known value ρ = π/(3*sqrt(2))=0.74048048969)
contacts
number of contacts between spheres and container and mutually between the spheres
loose
number of spheres that have still degrees of freedom for a movement inside the container (so called "rattlers")
boundary
number of spheres that are near to the container boundary (including rattlers if any)
core
number of spheres that are at the container core
reference
for the best known packing so far, see at bottom of the page
records
the sequence of N 's that establish density records

N radius distance ratio density contacts loose boundary core 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 0.999999999999 3.0000000000 0.481481481480 24 12 1
14 0.323504674183 0.956413627226 3.0911454449 0.473990585425 37 1 13 1 [2]
15 0.318304823107 0.933862623344 3.1416426249 0.483749932003 42 1 14 1 [2]
16 0.310975923325 0.902656188230 3.2156830320 0.481171926089 45 1 15 1 [2]
17 0.305693961650 0.880574112178 3.2712455117 0.485634466608 48 1 16 1 [2]
18 0.301296589342 0.862444879318 3.3189887817 0.492328697969 51 1 17 1 [5]
19 0.295332333897 0.838217355793 3.3860159733 0.489425500215 52 1 18 1 [6]
20 0.287890825717 0.808558114721 3.4735389622 0.477214324831 52 2 19 1 [7]
21 0.286832818119 0.804391523913 3.4863514104 0.495570920187 57 3 20 1 [3]
22 0.279342624810 0.775243921527 3.5798331912 0.479550457554 61 1 21 1 [8]
23 0.275670701290 0.761175067144 3.6275164365 0.481836470826 64 1 22 1 [8]
24 0.271341339941 0.744769409367 3.6853949355 0.479467459311 68 1 22 2 [31]
25 0.271191827927 0.744206331155 3.6874267475 0.498620125704 84 1 24 1 [4]
26 0.266851272489 0.727959450724 3.7474057765 0.494061693011 77 24 2 [11]
27 0.262232080452 0.710879596426 3.8134159569 0.486879204757 78 1 24 3 [11]
28 0.260305475681 0.703818852575 3.8416402782 0.493864651913 82 25 3 [11]
29 0.257925462478 0.695147049080 3.8770891032 0.497600320836 76 3 26 3 [11]
30 0.255330557656 0.685755432242 3.9164916616 0.499378264835 86 1 27 3 [11]
31 0.253116209529 0.677792751050 3.9507544849 0.502714681184 85 2 27 4 [11]
32 0.250787448179 0.669469425117 3.9874403893 0.504739586490 93 1 28 4 [11]
33 0.248762350343 0.662273384345 4.0199009160 0.508004891658 94 1 29 4 [11]
34 0.247052663501 0.656228268632 4.0477199712 0.512681372527 102 30 4 [11]
35 0.244833658425 0.648423122022 4.0844057408 0.513666698068 100 1 30 5 [11]
36 0.243132164915 0.642469275729 4.1129893297 0.517403964757 98 4 32 4 [11]
37 0.240686558293 0.633958376272 4.1547812520 0.515890148167 110 32 5 [11]
38 0.240519372142 0.633378557187 4.1576692600 0.528729790846 116 32 6 [11]
39 0.236745240285 0.620357062361 4.2239497563 0.517497643020 97 6 33 6 [11]
40 0.234999237637 0.614376479603 4.2553329537 0.519109947837 98 7 34 6 [11]
41 0.232755981857 0.606732607497 4.2963450048 0.516995080282 100 7 35 6 [11]
42 0.232118623301 0.604568961677 4.3081420430 0.525265949758 108 6 36 6 [11]
43 0.229732967254 0.596502141433 4.3528798324 0.521360856683 127 36 7 [11]
44 0.228163038226 0.591220813529 4.3828308379 0.522623038233 127 1 36 8 [11]
45 0.226911569263 0.587026167360 4.4070031477 0.525753813707 119 5 38 7 [11]
46 0.225168199111 0.581205363158 4.4411244747 0.525144709692 138 38 8 [11]
47 0.223507049839 0.575683397493 4.4741318035 0.524773065591 133 2 38 9 [11]
48 0.222405942149 0.572036115512 4.4962827447 0.528056504400 139 1 39 9 [11]
49 0.221278177007 0.568311226097 4.5191984746 0.530898911129 142 1 40 9 [11]
50 0.219758275796 0.563308187652 4.5504543407 0.530647009782 142 2 41 9 [11]
51 0.218550295807 0.559345776534 4.5756057950 0.532383236403 145 2 42 9 [11]
52 0.216930407731 0.554051414773 4.6097732930 0.530841223195 142 4 42 10 [11]
53 0.216288057367 0.551958048873 4.6234637833 0.536257636148 142 5 43 10 [11]
54 0.214920670908 0.547513258709 4.6528795754 0.536078417223 160 44 10 [11]
55 0.213441688789 0.542723116002 4.6851203515 0.534811127622 151 4 43 12 [11]
56 0.213067869308 0.541515236189 4.6933402171 0.541678896170 154 4 44 12 [11]
57 0.211327241143 0.535906023810 4.7319976099 0.537949249396 166 1 45 12 [12]
58 0.210480101312 0.533185044891 4.7510429431 0.540830438196 165 3 45 13 [11]
59 0.209759450876 0.530874937027 4.7673656459 0.544523495682 163 4 46 13 [11]
60 0.209426996435 0.529810644914 4.7749335903 0.551123897404 156 12 48 12 [11]
61 0.209108659202 0.528792385035 4.7822027257 0.557758101350 204 48 13 [11]
62 0.206739154000 0.521238770431 4.8370131185 0.547847764413 184 1 48 14 [11]
63 0.205954921018 0.518748686868 4.8554314461 0.550372935465 204 1 48 15 [11]
64 0.204107219252 0.512901295725 4.8993857428 0.544195657340 172 6 48 16 [31]
65 0.203076051707 0.509649765557 4.9242635535 0.544364116868 192 1 50 15 [11]
66 0.202108764115 0.506607304417 4.9478309581 0.544878127391 187 3 50 16 [11]
67 0.201216632889 0.503807768599 4.9697680835 0.545841352947 178 7 52 15 [11]
68 0.200002230133 0.500006969185 4.9999442473 0.544018198088 193 3 52 16 [11]
69 0.199281532516 0.497756803692 5.0180264442 0.546072434137 184 7 52 17 [11]
70 0.198688729141 0.495908984103 5.0329981188 0.549057383073 199 3 53 17 [11]
71 0.197222141221 0.491349229589 5.0704246177 0.544659844168 196 5 54 17 [11]
72 0.196285915470 0.488447121304 5.0946090432 0.544502539950 211 1 54 18 [11]
73 0.195628445518 0.486413136884 5.1117310540 0.546536123532 205 4 55 18 [12]
74 0.195146793635 0.484925181615 5.1243475815 0.549940848873 212 3 56 18 [31]
75 0.193964427622 0.481280068198 5.1555845175 0.547302624777 212 5 56 19 [31]
76 0.192947674925 0.478154064935 5.1827522689 0.545924067536 217 3 57 19 [12]
77 0.192254117422 0.476026239362 5.2014490686 0.547164199576 217 4 58 19 [12]
78 0.191458926390 0.473591095466 5.2230523740 0.547421008606 173 20 59 19 [31]
79 0.190659742093 0.471148544090 5.2449457291 0.547525177852 233 1 58 21 [31]
80 0.189727325517 0.468304884249 5.2707220601 0.546360937080 226 4 58 22 [12]
81 0.188970653150 0.466002010615 5.2918269759 0.546598091937 232 3 60 21 [12]
82 0.188297320002 0.463956383642 5.3107500414 0.547452280066 238 2 61 21 [12]
83 0.187640284124 0.461963537721 5.3293460126 0.548348093717 247 61 22 [31]
84 0.186956031676 0.459891565425 5.3488512301 0.548905685956 241 3 61 23 [12]
85 0.186411242185 0.458244390411 5.3644833234 0.550598757611 240 4 60 25 [12]
86 0.185610957844 0.455828721252 5.3876129492 0.549932374075 250 2 63 23 [31]
87 0.184952925832 0.453845996615 5.4067811877 0.550430981392 250 3 63 24 [12]
88 0.184275885830 0.451809337566 5.4266460069 0.550665914247 249 4 64 24 [12]
89 0.183598057582 0.449773691224 5.4466807175 0.550800407839 256 3 62 27 [31]
90 0.182937330629 0.447792654069 5.4663528574 0.550997365700 265 1 64 26 [12]
91 0.182353276900 0.446044169806 5.4838608716 0.551800523774 265 2 64 27 [12]
92 0.181782421764 0.444337610433 5.5010819544 0.552641484079 259 5 65 27 [31]
93 0.181286816986 0.442857940356 5.5161208996 0.554091668190 270 2 64 29 [12]
94 0.180820375937 0.441466976535 5.5303501877 0.555737825088 265 5 60 34 [12]
95 0.180532959436 0.440610666444 5.5391547512 0.558975933361 318 60 35 [12]
96 0.179567271930 0.437737954098 5.5689435455 0.555843827280 274 4 65 31 [12]
97 0.179019633861 0.436111851744 5.5859794729 0.556510967846 288 1 66 31 [31]
98 0.178499675043 0.434569943846 5.6022510952 0.557363295220 289 1 67 31 [12]
99 0.177879839132 0.432734404528 5.6217725678 0.557205479589 289 2 68 31 [12]
100 0.177439223049 0.431431276621 5.6357325219 0.558661678031 289 3 69 31 [12]
101 0.176688867565 0.429215300521 5.6596661339 0.557120238800 298 1 70 31 [12]
102 0.176215110923 0.427818264840 5.6748822207 0.558122607519 298 2 68 34 [31]
103 0.175681317426 0.426246113645 5.6921248921 0.558488150802 296 3 69 34 [12]
104 0.175184121787 0.424783582408 5.7082798932 0.559136135309 295 5 68 36 [12]
105 0.174670246799 0.423273839630 5.7250734932 0.559559277461 289 8 72 33 [12]
106 0.174167434485 0.421798417156 5.7416014823 0.560024116069 289 9 72 34 [12]
107 0.173608258062 0.420159711797 5.7600946589 0.559879951388 307 4 70 37 [12]
108 0.173168221779 0.418871713304 5.7747315860 0.560826268623 304 6 71 37 [12]
109 0.172638823470 0.417323965318 5.7924398458 0.560843771142 313 4 73 36 [31]
110 0.172285622418 0.416292448420 5.8043148695 0.562522364746 325 1 73 37 [12]
111 0.171744763732 0.414714586064 5.8225938204 0.562307004483 323 2 73 38 [12]
112 0.171331227895 0.413509555717 5.8366475994 0.563284245797 334 75 37 [12]
113 0.170853998626 0.412120418703 5.8529505194 0.563577810665 331 2 76 37 [31]
114 0.170449574714 0.410944457428 5.8668377535 0.564537268024 337 1 77 37 [12]
115 0.169844873619 0.409188278724 5.8877255386 0.563449723354 319 8 77 38 [12]
116 0.169416014365 0.407944331446 5.9026297115 0.564054892449 334 4 76 40 [12]
117 0.169057962712 0.406906751917 5.9151310235 0.565317923345 337 4 78 39 [12]
118 0.168723599855 0.405938626011 5.9268531543 0.566773460390 334 6 78 40 [31]
119 0.168129445780 0.404220211731 5.9477981109 0.565559504766 355 76 43 [13]
120 0.167683162280 0.402931082686 5.9636279899 0.565782633526 346 4 77 43 [12]
121 0.167217751301 0.401588174009 5.9802263349 0.565760349002 346 5 77 44 [12]
122 0.166844562670 0.400512449885 5.9936025723 0.566625355335 358 2 80 42 [13]
123 0.166501095219 0.399523248954 6.0059664994 0.567749022503 355 4 77 46 [12]
124 0.166046011632 0.398213843804 6.0224270982 0.567684492255 346 8 79 45 [12]
125 0.165707392901 0.397240468131 6.0347337707 0.568768671188 355 6 78 47 [12]
126 0.165236192632 0.395887294522 6.0519428829 0.568441901278 364 4 80 46 [12]
127 0.164897399764 0.394915306736 6.0643770092 0.569436292598 325 18 80 47 [31]
128 0.164797177360 0.394627922447 6.0680650969 0.572874216366 370 4 80 48 [13]
129 0.164142581214 0.392752585608 6.0922643753 0.570497159137 355 10 75 54 [13]
130 0.163801770859 0.391777368453 6.1049401039 0.571345919602 347 12 76 54 [13]
131 0.163545333743 0.391044106371 6.1145125765 0.573041094762 368 7 77 54 [13]
132 0.163232271313 0.390149537839 6.1262395723 0.574105889756 388 2 81 51 [12]
133 0.162963979279 0.389383432122 6.1363253673 0.575607578599 373 8 79 54 [12]
134 0.162832431396 0.389007977620 6.1412827373 0.578532180933 373 9 80 54 [12]
135 0.162579700716 0.388286982905 6.1508293815 0.580139889296 373 10 81 54 [31]
136 0.162377505731 0.387710470628 6.1584884895 0.582259396941 378 9 82 54 [13]
137 0.161772816367 0.385987998303 6.1815082562 0.580012303665 394 5 82 55 [31]
138 0.161135317283 0.384174755721 6.2059641354 0.577366128375 403 3 84 54 [12]
139 0.160746481721 0.383070140833 6.2209759697 0.577350078204 385 10 85 54 [12]
140 0.160407536677 0.382108090971 6.2341210439 0.577833004816 407 3 84 56 [12]
141 0.160041657637 0.381070464011 6.2483731721 0.577987219679 414 2 85 56 [12]
142 0.159699981108 0.380102290890 6.2617415047 0.578366245309 406 6 86 56 [12]
143 0.159449054966 0.379391768954 6.2715956530 0.579698112359 416 3 87 56 [12]
144 0.159153649105 0.378555841827 6.2832363921 0.580513460218 413 5 88 56 [12]
145 0.158609167903 0.377016629734 6.3048057891 0.578565938862 406 9 89 56 [12]
146 0.158248840832 0.375999104031 6.3191616112 0.578594722871 415 7 90 56 [12]
147 0.157784117755 0.374688060582 6.3377734986 0.577440435470 422 6 91 56 [31]
148 0.157569359432 0.374082688459 6.3464115333 0.578997947014 424 6 93 55 [13]
149 0.157097566347 0.372753856377 6.3654709825 0.577689721246 412 11 92 57 [12]
150 0.156791541716 0.371892715676 6.3778950641 0.578174789018 412 12 94 56 [12]
151 0.156639601709 0.371465394928 6.3840816057 0.580338867384 267 61 95 56 [12]
152 0.156045993115 0.369797386687 6.4083670464 0.577565778025 433 7 92 60 [12]
153 0.155704877618 0.368839931655 6.4224063838 0.577561297462 433 8 93 60 [12]
154 0.155370357768 0.367901740596 6.4362341335 0.577597384764 445 5 97 57 [13]
155 0.154990721956 0.366837917602 6.4519991092 0.577096980376 430 11 96 59 [12]
156 0.154643961095 0.365867052409 6.4664665398 0.576930493762 451 5 96 60 [12]
157 0.154338501545 0.365012482718 6.4792646682 0.577194907418 445 8 95 62 [31]
158 0.154083535103 0.364299647771 6.4899860931 0.577997271637 457 5 96 62 [12]
159 0.153739285883 0.363337877603 6.5045183101 0.577765629467 457 6 97 62 [12]
160 0.153403357533 0.362400108477 6.5187621450 0.577596533305 444 11 96 64 [12]
161 0.153102131936 0.361559847319 6.5315876883 0.577789427542 451 10 98 63 [12]
162 0.152793586534 0.360699787219 6.5447773214 0.577870321358 457 9 98 64 [12]
163 0.152493953330 0.359865168937 6.5576370614 0.578023472518 475 4 99 64 [13]
164 0.152224382259 0.359114791871 6.5692498479 0.578490869867 472 6 98 66 [12]
165 0.151988754849 0.358459291001 6.5794341232 0.579319724570 460 11 100 65 [12]
166 0.151626257544 0.357451556917 6.5951637678 0.578670489850 478 6 100 66 [12]
167 0.151301831791 0.356550390842 6.6093053082 0.578427628982 472 9 101 66 [13]
168 0.150987186808 0.355677050951 6.6230785614 0.578268535084 478 8 102 66 [13]
169 0.150689896199 0.354852475026 6.6361449920 0.578281242895 481 8 104 65 [13]
170 0.150452861896 0.354195441660 6.6466000540 0.578962294963 484 8 102 68 [13]
171 0.150136920417 0.353320255989 6.6605868645 0.578706846949 496 5 103 68 [13]
172 0.149892326079 0.352643154926 6.6714556119 0.579250802913 493 7 104 68 [13]
173 0.149662426611 0.352007088231 6.6817037692 0.579941851553 499 6 104 69 [12]
174 0.149322781215 0.351068014795 6.6969017846 0.579331921687 505 5 106 68 [12]
175 0.149028363986 0.350254597636 6.7101320397 0.579221735085 492 10 106 69 [13]
176 0.148746153967 0.349475434761 6.7228628998 0.579228474881 490 12 106 70 [12]
177 0.148444299701 0.348642607052 6.7365335147 0.578980373455 508 7 108 69 [13]
178 0.148152689318 0.347838603140 6.7497930993 0.578826785831 511 7 105 73 [13]
179 0.147915023152 0.347183736766 6.7606384983 0.579281806036 529 2 107 72 [13]
180 0.147649322270 0.346452055774 6.7728045387 0.579384508090 527 3 109 71 [13]
181 0.147349702922 0.345627517933 6.7865762887 0.579063736851 532 3 108 73 [13]
182 0.147073489170 0.344867904333 6.7993219284 0.578994682541 523 7 110 72 [13]
183 0.146797966222 0.344110680496 6.8120834759 0.578910205996 529 6 107 76 [13]
184 0.146535497150 0.343389787532 6.8242850330 0.578957053392 538 4 109 75 [13]
185 0.146258233310 0.342628740953 6.8372219284 0.578805564812 528 8 109 76 [13]
186 0.145978949142 0.341862648457 6.8503027723 0.578606946290 540 5 111 75 [13]
187 0.145706907536 0.341116904307 6.8630926077 0.578471582701 539 6 111 76 [13]
188 0.145452947787 0.340421156238 6.8750755156 0.578529395747 536 7 111 77 [12]
189 0.145245497123 0.339853131242 6.8848950212 0.579121696964 529 12 112 77 [13]
190 0.144968950591 0.339096342036 6.8980288256 0.578866725795 553 5 113 77 [31]
191 0.144714894423 0.338401530623 6.9101387524 0.578859364608 560 3 113 78 [13]
192 0.144508021238 0.337836063518 6.9200310919 0.579398132977 553 7 113 79 [31]
193 0.144287551916 0.337233733690 6.9306048008 0.579754200336 538 13 112 81 [12]
194 0.144082324351 0.336673323732 6.9404765956 0.580274983346 554 8 115 79 [13]
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893 0.088279757335 0.193655363134 11.3276251565 0.614376861395 2542 39 314 579 [31]
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895 0.088219593254 0.193510614181 11.3353503810 0.614494766659 2530 47 315 580 [31]
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897 0.088162633052 0.193373591054 11.3426739355 0.614675778398 2534 48 316 581 [31]
898 0.088129605734 0.193294148572 11.3469246988 0.614669718577 2538 43 313 585 [31]
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900 0.088062636910 0.193133082324 11.3555536728 0.614635395041 2548 45 315 585 [31]
901 0.088021334373 0.193033757676 11.3608820762 0.614452952052 2539 37 314 587 [31]
902 0.087994964316 0.192970347472 11.3642866700 0.614582225951 2540 47 316 586 [31]
903 0.087969853130 0.192909967794 11.3675306303 0.614736996642 2554 38 315 588 [31]
904 0.087874111595 0.192679788420 11.3799159030 0.613410594350 2552 46 323 581 [31]
905 0.087847440430 0.192615674885 11.3833709338 0.613530158655 2580 37 322 583 [31]
906 0.087821787739 0.192554013149 11.3866960096 0.613670177376 2567 42 322 584 [31]
907 0.087814336288 0.192536102641 11.3876622232 0.614191153414 2578 43 325 582 [31]
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909 0.087742963144 0.192364563054 11.3969253393 0.614045811748 2586 41 326 583 [31]
910 0.087728508253 0.192329825160 11.3988031931 0.614417569742 2590 39 327 583 [31]
911 0.087696190077 0.192252162324 11.4030039289 0.614413224935 2553 47 328 583 [31]
912 0.087652682056 0.192147618198 11.4086640197 0.614172641462 2580 41 327 585 [31]
913 0.087628719449 0.192090043422 11.4117837883 0.614341951900 2605 35 329 584 [31]
914 0.087595383452 0.192009952303 11.4161267477 0.614313203920 2620 31 329 585 [31]
915 0.087554076144 0.191910717895 11.4215127843 0.614115704344 2660 19 329 586 [31]
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917 0.087485958699 0.191747095912 11.4304056888 0.614022664123 2615 37 330 587 [31]
918 0.087452167172 0.191665935803 11.4348223988 0.613980262369 2621 32 330 588 [31]
919 0.087496243125 0.191771798123 11.4290621435 0.615578905356 2591 46 329 590 [31]
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924 0.087333159263 0.191380151803 11.4504045020 0.615473677973 2583 49 329 595 [31]
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927 0.087238763494 0.191153524065 11.4627942895 0.615471912198 2645 35 331 596 [31]
928 0.087211120328 0.191087166529 11.4664276326 0.615550336196 2611 46 331 597 [31]
929 0.087185370699 0.191025358053 11.4698141670 0.615667983086 2611 41 332 597 [31]
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931 0.087131502241 0.190896065435 11.4769053015 0.615850483032 2653 38 331 600 [31]
932 0.087115088012 0.190856671784 11.4790677806 0.616163618206 2627 50 334 598 [31]
933 0.087076183677 0.190763308219 11.4841964562 0.615998712020 2639 40 334 599 [31]
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936 0.087004129144 0.190590411022 11.4937073658 0.616446571968 2627 54 337 599 [31]
937 0.086973830667 0.190517716991 11.4977113499 0.616460687724 2639 49 336 601 [31]
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942 0.086788529874 0.190073236513 11.5222599283 0.615797467219 2710 37 342 600 [31]
943 0.086746564287 0.189972598831 11.5278340787 0.615557378901 2673 48 342 601 [31]
944 0.086705758128 0.189874750442 11.5332594004 0.615340944877 2709 30 344 600 [31]
945 0.086674624964 0.189800102643 11.5374020991 0.615329479545 2732 32 343 602 [31]
946 0.086657652876 0.189759410913 11.5396617242 0.615618840177 2714 39 344 602 [31]
947 0.086614542393 0.189656057411 11.5454053369 0.615350311406 2719 37 345 602 [31]
948 0.086585683533 0.189586876343 11.5492534008 0.615384575559 2742 24 345 603 [31]
949 0.086548783028 0.189498424043 11.5541774825 0.615246439685 2719 38 346 603 [31]
950 0.086484800910 0.189345072739 11.5627253515 0.614529839032 2709 42 346 604 [31]
951 0.086438900735 0.189235073176 11.5688653083 0.614197752024 2737 34 348 603 [31]
952 0.086393774066 0.189126938091 11.5749081553 0.613881135002 2752 30 355 597 [31]
953 0.086363394585 0.189054147071 11.5789797843 0.613877920874 2735 23 355 598 [31]
954 0.086332666666 0.188980526098 11.5831010279 0.613866370257 2790 19 356 598 [31]
955 0.086316627916 0.188942100849 11.5852533184 0.614167411601 2759 26 356 599 [31]
956 0.086278255510 0.188850174641 11.5904058802 0.613990933789 2748 36 357 599 [31]
957 0.086252082479 0.188787478089 11.5939229669 0.614073995298 2769 26 357 600 [31]
958 0.086220305989 0.188711363481 11.5981959067 0.614036501295 2767 27 358 600 [31]
959 0.086114423491 0.188457780065 11.6124565370 0.612415681321 2778 30 360 599 [31]
960 0.086143888184 0.188528340666 11.6084846073 0.613683778165 2774 22 358 602 [31]
961 0.086117451607 0.188465031439 11.6120482125 0.613757619454 2682 38 316 645 [31]
962 0.086082384080 0.188381058819 11.6167786323 0.613646033402 2776 30 317 645 [31]
963 0.086039345246 0.188278006933 11.6225896088 0.613363004280 2774 29 314 649 [31]
964 0.085989262766 0.188158102007 11.6293589203 0.612928351483 2794 22 313 651 [31]
965 0.085959637651 0.188087181249 11.6333668606 0.612930230332 12 954 357 608 [31]
966 0.085934361266 0.188026674726 11.6367886520 0.613024294617 2783 29 314 652 [31]
967 0.085894494967 0.187931249717 11.6421896465 0.612805233506 2792 30 314 653 [31]
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969 0.085842538690 0.187806898315 11.6492360927 0.612959013534 2844 18 368 601 [31]
970 0.085803694870 0.187713939312 11.6545097681 0.612759006930 2823 26 369 601 [31]
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972 0.085711768687 0.187493977832 11.6670092721 0.612051033881 2851 19 371 601 [31]
973 0.085687380067 0.187435628031 11.6703299741 0.612157864157 2819 26 370 603 [31]
974 0.085656096116 0.187360785700 11.6745922981 0.612116079305 2837 22 370 604 [31]
975 0.085627301671 0.187291903679 11.6785181885 0.612126796077 2796 37 371 604 [31]
976 0.085601380426 0.187229898632 11.6820546003 0.612198304457 2825 29 371 605 [31]
977 0.085560957376 0.187133211483 11.6875737564 0.611957793409 2870 16 372 605 [31]
978 0.085530419149 0.187060173329 11.6917467487 0.611928465129 2853 24 372 606 [31]
979 0.085486672195 0.186955552414 11.6977298838 0.611614715033 2872 18 374 605 [31]
980 0.085456844251 0.186884224574 11.7018128713 0.611598806394 2867 17 367 613 [31]
981 0.085435858128 0.186834043052 11.7046872579 0.611771956401 2880 17 368 613 [31]
982 0.085408631809 0.186768943551 11.7084184446 0.611810297216 2879 17 369 613 [31]
983 0.085384408890 0.186711028589 11.7117400354 0.611912389279 2879 22 369 614 [31]
984 0.085360872491 0.186654757978 11.7149692923 0.612028483956 2873 18 370 614 [31]
985 0.085341420113 0.186608253592 11.7176395550 0.612231719827 2873 26 371 614 [31]
986 0.085306077217 0.186523765146 11.7224942539 0.612092177603 2877 23 372 614 [31]
987 0.085277722370 0.186455986599 11.7263919838 0.612102185683 2914 12 374 613 [31]
988 0.085248129597 0.186385253434 11.7304626474 0.612084697117 2853 34 374 614 [31]
989 0.085218509818 0.186314460300 11.7345398568 0.612065778859 2886 25 375 614 [31]
990 0.085187095860 0.186239383976 11.7388671360 0.612007343199 2869 29 373 617 [31]
991 0.085158445512 0.186170916907 11.7428165109 0.612007620810 2890 23 374 617 [31]
992 0.085137901776 0.186121825227 11.7456500470 0.612181921943 2904 22 376 616 [31]
993 0.085108696641 0.186052039906 11.7496805787 0.612168625696 2895 25 376 617 [31]
994 0.085076933648 0.185976147672 11.7540672556 0.612099282287 2928 13 377 617 [31]
995 0.085059124634 0.185933598387 11.7565282303 0.612330380968 2900 23 378 617 [31]
996 0.085026217584 0.185854981242 11.7610782699 0.612234667663 2907 24 378 618 [31]
997 0.084993734086 0.185777381538 11.7655732008 0.612147228737 2939 15 379 618 [31]
998 0.084958883804 0.185694133958 11.7703994594 0.612007768995 2932 10 378 620 [31]
999 0.084930865271 0.185627210115 11.7742824921 0.612015095496 2962 10 379 620 [31]
1000 0.084866089682 0.185472505663 11.7832694277 0.611227064109 2936 17 379 621 [31]
1055    0.083447520229 0.182089999363 11.9835795870 0.613045580732 3087 20 377 [31]
1061    0.083260872596 0.181645726919 12.0104434270 0.612404337722 3068 31 380 [31]
1157    0.080893782384 0.176027059405 12.3618895115 0.612461489305 3364 28 426 [31]
1205    0.079803276753 0.173448295863 12.5308137797 0.612419793834 3404 45 377 [31]
1253    0.078827079542 0.171145021290 12.6859958000 0.613730040158 3583 42 421 [31]
1289    0.078167551985 0.169591669621 12.7930320780 0.615647968543 3649 54 393 [31]
1409    0.076154441092 0.164864008616 13.1312105461 0.622295554446 4056 34 463 [31]
1433    0.075771336694 0.163966645274 13.1976027299 0.623391730215 4140 38 470 [31]
1481    0.074822911994 0.161748303031 13.3648901566 0.620381555455 4264 39 496 [31]
1505    0.074408668802 0.160780824742 13.4392943201 0.620022007168 4352 32 496 [31]
1553    0.073607102421 0.158911197643 13.5856455031 0.619342127044 4507 34 513 [31]
1601    0.072845698692 0.157138242446 13.7276464905 0.618875188136 4664 40 526 [31]
1721    0.070978933493 0.152803711459 14.0886873159 0.615416702369 4990 52 563 [31]
1745    0.070734553387 0.152237562786 14.1373621818 0.617575812766 5017 55 561 [31]
1865    0.069573892291 0.149552751615 14.3732076368 0.628084010835 5332 49 493 [31]
1961    0.068197437850 0.146377442218 14.6633074720 0.621987661744 5667 47 615 [31]
1985    0.068194753127 0.146371258062 14.6638847440 0.629525600068 5599 64 593 [31]
2000    0.067800362008 0.145463180299 14.7491837858 0.623341488610 5792 54 624 [31]
2093    0.067066559198 0.143775657008 14.9105606722 0.631374889582 5966 58 622 [31]
2123    0.066647726127 0.142813657807 15.0042628325 0.628501045803 6149 43 630 [31]
2171    0.066106943195 0.141572833663 15.1270040887 0.627192780478 6232 58 649 [31]
2243    0.065343938138 0.139824563932 15.3036383863 0.625813875897 6497 48 681 [31]
2315    0.064672251950 0.138287893382 15.4625820170 0.626188296896 6746 44 695 [31]
2347    0.064360750075 0.137575994338 15.5374199156 0.625714731748 6770 64 695 [31]
2491    0.063250369853 0.135042209396 15.8101842301 0.630322682375 7051 67 697 [31]
2587    0.062638988538 0.133649656369 15.9644978845 0.635814807150 7339 54 698 [31]
2659    0.062109131997 0.132444262154 16.1006919248 0.637066424323 7421 82 710 [31]
2731    0.061511257082 0.131085764312 16.2571868539 0.635602370147 7727 65 751 [31]
2779    0.061214170962 0.130411365550 16.3360866330 0.637447560061 7948 62 749 [31]
2899    0.060321948580 0.128388544329 16.5777138097 0.636318136328 8287 84 795 [31]
3000    0.059419702784 0.126346900865 16.8294345671 0.629379626189 8772 45 834 [31]
3043    0.059355142043 0.126200959992 16.8477399865 0.636322087330 9554 133 680 [31]
3055    0.059202488875 0.125855964062 16.8911817561 0.633915117930 8736 81 824 [31]
3103    0.058933552990 0.125248441653 16.9682625476 0.635140270350 8828 55 825 [31]
3151    0.058662130326 0.124635653607 17.0467726699 0.636094871963 8946 69 820 [31]
3319    0.057609871322 0.122263316580 17.3581363237 0.634596931727 9481 85 878 [31]
3367    0.057404156158 0.121800146973 17.4203414339 0.636902755703 9584 76 882 [31]
3463    0.056864051634 0.120585058246 17.5858028274 0.636745512566 10766 66 901 [31]
3511    0.056641776761 0.120085404178 17.6548134113 0.638030490521 9968 97 903 [31]
3559    0.056405574270 0.119554700053 17.7287442410 0.638695804753 10160 74 928 [31]
3589    0.056324917885 0.119373540644 17.7541315203 0.641320555498 10036 94 907 [31]
3781    0.055466051920 0.117446391488 18.0290459729 0.645191205422 12893 102 918 [31]
3805    0.055358103561 0.117204421633 18.0642026311 0.645503011109 10856 101 920 [31]
3925    0.054738886050 0.115817492632 18.2685485979 0.643765244814 10934 106 958 [31]
3997    0.054426950313 0.115119504159 18.3732506460 0.644430592651 10852 91 976 [31]
4000    0.054116708972 0.114425763697 18.4785811812 0.633948713105 11770 67 1022 [31]
4093    0.054033689415 0.114240198219 18.5069724245 0.645707173226 11319 130 971 [31]
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]
4321    0.053056320422 0.112058027454 18.8478958218 0.645350496353 11928 120 1002 [31]
4417    0.052684186064 0.111228347060 18.9810277943 0.645904371518 12141 122 1023 [31]
4537    0.052223435092 0.110201997023 19.1484914433 0.646197218181 12701 112 1068 [31]
4585    0.052091737452 0.109908816096 19.1969024055 0.648105741752 12954 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]
5000    0.050363856639 0.106069797345 19.8555088259 0.638744158561 14617 93 1198 [31]
5017    0.050574237403 0.106536475827 19.7729130749 0.648981220175 18155 181 1142 [31]
5089    0.050442769936 0.106244823037 19.8244466208 0.653174514388 18443 175 1135 [31]
5233    0.049858809487 0.104950316826 20.0566361349 0.648599258405 14317 127 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]
5473    0.049133604238 0.103344916714 20.3526693290 0.649174299814 15484 146 1226 [31]
5481    0.049137823467 0.103354249816 20.3509217430 0.650290709276 15562 124 1221 [31]
5721    0.048362204502 0.101639940597 20.6773039049 0.647128035614 17862 137 1288 [31]
5775    0.048230928332 0.101350064354 20.7335839177 0.647931138638 16301 127 1295 [31]
5895    0.048058151421 0.100968670498 20.8081245414 0.654312163478 16298 171 1200 [31]
5979    0.047700111454 0.100178760972 20.9643116026 0.648913388665 16845 146 1324 [31]
6000    0.047187699337 0.099049307622 21.1919634576 0.630431146176 17494 128 1355 [31]
6051    0.047556166041 0.099861355275 21.0277674432 0.650800169036 16724 137 1345 [31]
6099    0.047420954441 0.099563295376 21.0877240196 0.650383486283 17126 132 1341 [31]
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]
6459    0.046705520909 0.097987603901 21.4107450369 0.658066745676 24730 166 1276 [31]
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]
7000    0.044845910117 22.2985774488 0.631344739890 20469 110 1495 [31]
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]
10000    0.039927779623 0.083176616874 25.0452193796 0.636539677077 29162 179 1936 [31]
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    0.024017994773 0.049218109851 41.6354491477 0.694709486079 114198 1207 8469 [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

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].
26-May-2023: Many significant improvements for N= 57, 73, 76, 77, 80–82, 84, 85, 87, 88, 90, 91, 93–96, 98–101, 103–106, 108, 111, 112, 114–117, 119–134, 136, 138–141, 143–146, 148–179, 181–189, 191–268, 276, 278, 280–337, 339–343, 345–370, 372–375, 380, 384, 392, 395, 396, and 398 by Jianrong Zhou, Shuo Ren, Kun He, Yanli Liu, Chu-Min Li [12].
04-Jul-2023: Further significant improvements and expansion for N= 119, 128, 130, 131, 150, 264, 266–269, 272, 274, 277, 332, 334, 336, 359, 366–368, 370–379, 381–386, 388–422, 425–437, 439–445, 449–600 by Eckard Specht [31].
11-Jul-2023: Many more slightly improvements for N= 167, 183, 197, 200, 202–204, 206–216, 218–238, 240–248, 254, 255, 258–260, 264–267, 269, 272–277, 279, 280, 282, 285–288, 290, 292, 293, 296–305, 307–314, 319–326, 329, 330, 333, 335, 336, 338–364, 366, 367, 372, 373, 376, 377, 379–382, 386, 388, 391–396, 399–408, 410–430, 432–449, 453, 454, 456, 459, 461, 466–478, 481, 486–488, 490, 491, 494–498, 503–509, 511, 512, 519, 525, 526, 530, 532–535, 542, 558–560, 573, and 591–599 by Eckard Specht [31].
23-Jul-2023: Some last improvements for N= 95, 361, 364, 445, 453, 470, 472, 478, 482, 498, 510, 511, 518–525, 527–541, 553, 554, 559, 567, 568, 579, 587–589, and 592 before expanding this table beyond N= 600 by Eckard Specht [31].
06-Jun-2024: Jianrong Zhou et al. sent me on December 10, 2023 some improvements for N= 77, 80, 94, 98, 101, 107, 110, 112, 115–117, 121, 124, 131, 139, 142, 144, 150, 152, 153, 158, 159, 161–164, 166–168, 170, 173–176, 178–180, 184, 193, 197, 219, 230, 241, 244, 252, 254, 256, 262, 268, 283, 284, 289, 294–296, 298, 303, 310, 313, 317, 367, 372, 373 [12].
07-Jun-2024: Xiangjing Lai et al. sent me on May 24, 2024 improvements for over hundred packings, but unfortunately only 88 survived due to [12]. These are for N= 119, 122, 128–131, 136, 148, 154, 155, 163, 167–172, 175, 177–187, 189–191; 194, 199–201, 203, 206–209, 213, 214, 218–221, 224, 227, 229–231, 245–247, 249, 251, 252, 254–256, 260, 261, 274, 279–282, 287, 288, 291, 292, 294–298, 300, 303–306, 312, 366, 368–370, 392 [13].
08-Jun-2024: Jianrong Zhou et al. sent me on May 31, 2024 improvements for 109 instances, but unfortunately only 27 survived due to [13]. I regret for the delays in processing the incoming data. Thus, the following packings are new candidates: N= 155, 188, 206, 209, 210, 222, 232, 241, 245, 250, 264, 286, 300, 303, 315, 340, 349, 366, 368, 369, 371, 375, 387, 389, 390, 397, 400 [12].
08-Jun-2024: Inspired by the latest improvements, I found 307 packings on my notebook waiting for a year for publication. These new candidates range from N = 600–753, 755–887, 904–934. They won't affect the recent improvements [12] and [13].
29-Jul-2024: Many small and some significant improvements for N = 127, 190, 196, 198–202, 204, 205, 208, 209, 211–218, 221, 223, 226–228, 230, 231, 233–251, 253–256, 260–279, 282, 284–288, 290–299, 301–303, 307–311, 313–318, 320–361, 363–386, 388–396, 398–400 by Eckard Specht [31].
30-Jul-2024: An addendum for N = 76, 81, 82, 99, 100, 105, 114, 120, 123, 145, 146, 156, 157, 165, 178, 192, 195, 198, 205, 211, 228, 235, 241, 242, 244, 257, 265, 273, 289, 300, 303, 307, 308, 311, 315, 318, 319, 325, 328, 339, 360 by Eckard Specht [31].
16-Aug-2024: Large scale improvements for N = 401–498, 500 by Eckard Specht [31].
18-Aug-2024: Further improvements for N = 188, 192, 198, 211, 215, 227, 238, 248, 263, 275, 290, 307, 308, 313, 315, 321, 322, 327, 330, 334, 335, 338, 342, 343, 345, 346, 353, 357, 358, 364–369, 374, 375, 377–380, 384, 385, 388, 389, 400, 402–405, 409, 410, 412, 414, 416–420, 423, 424, 426, 430–436, 439, 440, 443, 444, 446, 448, 453, 454, 457, 458, 462–469, 472, 474, 475, 478–480, 482, 487–489, 495, 498–500 by Eckard Specht [31].
21-Aug-2024: Further improvements for N = 462, 463, 470, 471, 477, 496, 497, 501–525, 527–600 by Eckard Specht [31].
24-Aug-2024: Further improvements for N = 366–368, 370, 592, 600–700 by Eckard Specht [31].
03-Sep-2024: Further improvements for N = 220, 314, 339–343, 346–348, 350, 352, 357–361, 367, 368, 372–376, 385–388, 390, 391, 401, 402, 405, 412, 413, 421, 423, 424, 427, 434–438, 445, 452, 457, 458, 462, 465, 475, 478, 479, 481, 504, 521, 526, 537, 545, 551–555, 557, 558, 569, 572–575, 587–589, 591, 592, 604–607, 609, 614, 615, 619–621, 624, 626–633, 637, 641, 642, 644–647, 650, 651, 655–658, 660, 664, 666–672, 675, 680–683, 687, 688, 690, 692–695, 697–700 by Eckard Specht [31].
13-Sep-2024: Further improvements for N = 532, 614, 631, 639–641, 648, 649, 663 665, 669, 673, 689, 701–800 by Eckard Specht [31].
26-Sep-2024: Further improvements for N = 509, 766, 772–774, 792, 801–900 by Eckard Specht [31].
13-Jul-2026: Extension up to N = 1000 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).
[12]   , An efficient solution space exploring and descent method for packing equal spheres in a sphere, Computers & Operations Research 164 (2024) 106522.
[13]   , private communication, May 2024.
[31]   , program ssp, April 2010–July 2026.