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

Last update: 17-Oct-2026

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   193-204   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   997-1000  

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.

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 cube
distance
packing of spheres in a cube is equivalent to distributing points in a cube; the latter are then the sphere centers. "distance" is here the greatest distance of 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
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, 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 0.500000000000 2.0000000000 0.523598775598 6 1
2 0.316987298108 1.732050807569 3.1547005384 0.266835854699 7 2
3 0.292893218813 1.414213562373 3.4142135624 0.315745968789 12 3 [1]
4 0.292893218813 1.414213562373 3.4142135624 0.420994625052 18 4 [1]
5 0.263932022500 1.118033988750 3.7888543820 0.385065731603 18 5 [1]
6 0.257359312881 1.060660171780 3.8856180832 0.428409926205 24 6 [1]
7 0.250136153877 1.001089824549 3.9978227238 0.458897881425 24 7 [1]
8 0.250000000000 1.000000000000 4.0000000000 0.523598775598 36 8 [1]
9 0.232050807569 0.866025403784 4.3094010768 0.471064494277 32 8 1 [1]
10 0.214285714286 0.750000000000 4.6666666667 0.412162301491 34 9 1 [1]
11 0.207622238388 0.710116382462 4.8164397406 0.412385039078 36 10 1 [5]
12 0.207106785524 0.707106806467 4.8284270236 0.446532259382 39 10 2 [5]
13 0.207106781187 0.707106781187 4.8284271247 0.483743250604 57 13 [4]
14 0.207106781187 0.707106781187 4.8284271247 0.520954269881 66 14 [4]
15 0.192307692308 0.625000000000 5.2000000000 0.446858308715 48 1 14 1 [5]
16 0.188796768447 0.606667120726 5.2967008293 0.451016324213 51 13 3 [6]
17 0.188685238868 0.606091526731 5.2998316455 0.478356091364 75 16 1 [5]
18 0.187680601147 0.600925212577 5.3282011774 0.498447327617 69 17 1 [4]
19 0.183185303162 0.578209612716 5.4589532170 0.489231140531 40 6 18 1 [6]
20 0.178407199723 0.554761174904 5.6051549576 0.475724737051 46 6 20 [6]
21 0.177219044407 0.549038105677 5.6427344101 0.489597369041 66 8 13 [5]
22 0.173273103228 0.530330085890 5.7712361663 0.479407418832 72 4 16 6 [5]
23 0.171816783368 0.523539214257 5.8201531911 0.488667209462 68 1 20 3 [5]
24 0.170540688701 0.517638090205 5.8637033052 0.498636296697 57 8 23 1 [6]
25 0.167804075635 0.505135865094 5.9593308221 0.494807338849 68 3 22 3 [6]
26 0.166905167286 0.501074021252 5.9914262468 0.506373890289 90 20 6 [5]
27 0.166666666667 0.500000000000 6.0000000000 0.523598775598 108 26 1 [4]
28 0.160191005919 0.471414849839 6.2425477277 0.482126525769 80 2 24 4 [7]
29 0.160188620509 0.471404520791 6.2426406871 0.499323023283 138 25 4 [6]
30 0.160188620509 0.471404520791 6.2426406871 0.516541058569 144 26 4 [6]
31 0.160188620509 0.471404520791 6.2426406871 0.533759093855 150 27 4 [6]
32 0.160188620509 0.471404520791 6.2426406871 0.550977129140 156 28 4 [4]
33 0.154605431908 0.447619754886 6.4680780465 0.510830470115 78 8 29 4 [31]
34 0.152117093470 0.437265213712 6.5738831658 0.501304476054 89 6 28 6 [31]
35 0.151668522645 0.435414346693 6.5933259094 0.511496920126 130 26 9 [7]
36 0.149378180763 0.426037892018 6.6944181198 0.502634862800 118 32 4 [31]
37 0.149061631334 0.424751593566 6.7086344826 0.513319713166 114 31 6 [7]
38 0.149038769068 0.424658782601 6.7096635745 0.526950682001 120 32 6 [7]
39 0.147617379098 0.418912200383 6.7742701172 0.525491489893 114 7 34 5 [6]
40 0.147058823529 0.416666666667 6.8000000000 0.532870726235 152 34 6 [7]
41 0.144837878409 0.407807785810 6.9042712513 0.521817848600 100 8 36 5 [27]
42 0.143045425139 0.400738455850 6.9907863116 0.514943781726 125 34 8 [24]
43 0.141652316451 0.395292959753 7.0595386299 0.511950660652 116 4 34 9 [24]
44 0.140860610938 0.392217103519 7.0992166891 0.515121870168 127 2 36 8 [7]
45 0.140631937346 0.391331205970 7.1107603214 0.524267579580 163 31 14 [7]
46 0.140246869719 0.389841971935 7.1302839201 0.531527784154 133 5 36 10 [24]
47 0.139958859838 0.388730187263 7.1449567477 0.539743790761 145 37 10 [24]
48 0.139958844038 0.388730126323 7.1449575543 0.551227514528 184 40 8 [6]
49 0.136426759341 0.375238725197 7.3299402906 0.521174764236 142 2 40 9 [28]
50 0.135954512642 0.373454739484 7.3554013072 0.526307423600 139 6 42 8 [7]
51 0.135023077846 0.369949631470 7.4061413497 0.525875327965 131 7 42 9 [24]
52 0.133820037890 0.365448827728 7.4727224395 0.521981845434 139 6 42 10 [7]
53 0.133230838600 0.363255291396 7.5057697640 0.525023524875 142 6 43 10 [7]
54 0.132203229522 0.359446412077 7.5641117363 0.522647120970 159 45 9 [24]
55 0.131220272864 0.355822902423 7.6207736668 0.520540000702 157 1 35 20 [28]
56 0.130661065058 0.353770081343 7.6533893211 0.523257206240 161 3 45 11 [24]
57 0.130622145186 0.353627439986 7.6556697073 0.532125290725 213 3 35 22 [24]
58 0.130602161881 0.353554212840 7.6568410936 0.541212353064 202 5 38 20 [24]
59 0.130601937482 0.353553390593 7.6568542495 0.550540762723 79 28 37 22 [27]
60 0.130601937482 0.353553390593 7.6568542495 0.559871962092 288 48 12 [7]
61 0.130601937482 0.353553390593 7.6568542495 0.569203161460 297 49 12 [7]
62 0.130601937482 0.353553390593 7.6568542495 0.578534360828 306 50 12 [7]
63 0.130601937482 0.353553390593 7.6568542495 0.587865560196 318 50 13 [4]
64 0.126743552194 0.339561588122 7.8899477148 0.545815931130 172 7 48 16 [28]
65 0.126182631398 0.337551547885 7.9250209710 0.547016851657 175 7 44 21 [24]
66 0.125739648703 0.335968392771 7.9529409404 0.549603225501 199 4 47 19 [24]
67 0.124481087806 0.331490861749 8.0333488213 0.541344260396 178 8 48 19 [7]
68 0.123435656367 0.327794329054 8.1013868232 0.535697246665 196 3 47 21 [28]
69 0.123140054765 0.326752833040 8.1208344589 0.539679258630 199 3 52 17 [24]
70 0.123037729239 0.326392689088 8.1275882300 0.546136963249 196 4 51 19 [28]
71 0.123029495260 0.326363717355 8.1281321840 0.553827714565 207 2 51 20 [24]
72 0.123027579247 0.326356975932 8.1282587703 0.561601865590 321 2 51 21 [24]
73 0.121787558899 0.322008336225 8.2110193278 0.552357488411 202 14 54 19 [24]
74 0.121338760478 0.320441460104 8.2413896109 0.553756696936 201 6 55 19 [24]
75 0.121321447244 0.320381089346 8.2425657022 0.560999688104 212 2 51 24 [24]
76 0.120183555428 0.316425360582 8.3206058968 0.552633674326 184 15 57 19 [24]
77 0.119055120033 0.312525843749 8.3994707638 0.544281491412 211 7 58 19 [24]
78 0.118546035457 0.310774160126 8.4355414851 0.544307495936 212 8 57 21 [28]
79 0.118014005111 0.308948513009 8.4735705653 0.543896606506 232 2 56 23 [28]
80 0.117788131720 0.308174971777 8.4898196906 0.547624911880 234 18 62 18 [31]
81 0.117097826866 0.305816563817 8.5398681322 0.544778719257 244 5 56 25 [28]
82 0.116833115824 0.304914439764 8.5592170759 0.547772643047 240 2 59 23 [24]
83 0.116285160419 0.303050985847 8.5995495590 0.546688062441 211 6 53 30 [25]
84 0.116283622527 0.303045763366 8.5996632911 0.553252714427 255 27 58 26 [27]
85 0.116283622527 0.303045763366 8.5996632911 0.559839056266 333 12 52 33 [27]
86 0.116283622527 0.303045763366 8.5996632911 0.566425398104 387 6 53 33 [27]
87 0.116283622527 0.303045763366 8.5996632911 0.573011739943 408 4 52 35 [27]
88 0.115521432464 0.300462606289 8.6564023547 0.568275559153 338 66 22 [31]
89 0.113769756020 0.294564596620 8.7896822054 0.548983252852 240 8 59 30 [28]
90 0.113282348330 0.292932965022 8.8275006189 0.548047055414 251 6 61 29 [24]
91 0.112825882631 0.291408639085 8.8632145096 0.547464824936 256 11 63 28 [31]
92 0.112501402437 0.290327250587 8.8887780804 0.548719312572 254 6 66 26 [28]
93 0.112377874546 0.289916047527 8.8985487939 0.552858510481 243 8 64 29 [24]
94 0.112059611009 0.288857809573 8.9238218034 0.554068926558 207 14 65 29 [30]
95 0.111760878734 0.287866092344 8.9476748154 0.555496897916 209 15 56 39 [33]
96 0.111656828004 0.287521027935 8.9560129719 0.559777838107 232 3 66 30 [33]
97 0.111549039238 0.287163761983 8.9646670812 0.563972393641 241 4 62 35 [28]
98 0.111518995642 0.287064217790 8.9670821930 0.569326282759 280 6 65 33 [24]
99 0.111387457102 0.286628569091 8.9776715083 0.573102987205 289 3 67 32 [24]
100 0.111382347512 0.286611652352 8.9780833528 0.578812244763 340 73 27 [31]
101 0.110240609817 0.282842729626 9.0710673831 0.566806476447 77 57 61 40 [34]
102 0.110240607782 0.282842722929 9.0710675505 0.572418390062 106 48 54 48 [33]
103 0.110240604606 0.282842712475 9.0710678119 0.578030285101 516 72 31 [4]
104 0.110240604606 0.282842712475 9.0710678119 0.583642229616 522 73 31 [4]
105 0.110240604606 0.282842712475 9.0710678119 0.589254174132 531 74 31 [4]
106 0.110240604606 0.282842712475 9.0710678119 0.594866118647 542 74 32 [4]
107 0.110240604606 0.282842712475 9.0710678119 0.600478063163 549 75 32 [4]
108 0.110240604606 0.282842712475 9.0710678119 0.606090007678 558 76 32 [4]
109 0.108037386304 0.275631865206 9.2560550955 0.575754464368 312 10 71 38 [30]
110 0.107352134546 0.273405623693 9.3151384854 0.570050521135 298 9 72 38 [33]
111 0.106942983304 0.272080076837 9.3507771067 0.568680679143 306 10 71 40 [33]
112 0.106596216530 0.270958798591 9.3811959988 0.568240263440 311 6 64 48 [30]
113 0.106015030674 0.269083947175 9.4326247292 0.563987375658 308 9 71 42 [30]
114 0.105640379878 0.267878287958 9.4660772818 0.562967499377 333 3 75 39 [30]
115 0.105195455078 0.266449453105 9.5061141117 0.560760455488 318 8 77 38 [30]
116 0.105020830230 0.265889541191 9.5219205353 0.562824430034 344 6 79 37 [30]
117 0.104870806953 0.265408906247 9.5355421500 0.565247044772 331 3 68 49 [33]
118 0.104828888162 0.265274674746 9.5393552057 0.569394877056 358 10 76 42 [33]
119 0.104807243211 0.265205374872 9.5413252879 0.573864637748 325 10 78 41 [30]
120 0.104799467118 0.265180480284 9.5420332517 0.578558233498 334 10 75 45 [30]
121 0.104794644945 0.265165042945 9.5424723327 0.583299026102 612 77 44 [24]
122 0.104794644945 0.265165042945 9.5424723327 0.588119679211 555 3 79 43 [24]
123 0.103656141844 0.261530839222 9.6472816971 0.573824174916 304 19 77 46 [33]
124 0.103565446266 0.261242228485 9.6557301306 0.576972265002 302 19 82 42 [33]
125 0.103553928444 0.261205586015 9.6568040926 0.581431235880 303 16 75 50 [33]
126 0.103553390594 0.261203874965 9.6568542494 0.586073553624 285 35 83 43 [27]
127 0.102725314597 0.258575032267 9.7346988318 0.576666562760 317 20 83 44 [30]
128 0.102138048911 0.256717307677 9.7906706723 0.571296089644 310 26 86 42 [27]
129 0.101709786165 0.255366018627 9.8318956091 0.568547223969 349 14 83 46 [30]
130 0.101487077953 0.254664459641 9.8534712021 0.569199102534 357 13 82 48 [33]
131 0.101237776101 0.253880056920 9.8777357476 0.569360971922 373 7 83 48 [30]
132 0.101008386444 0.253159171803 9.9001680475 0.569816264227 370 11 84 48 [30]
133 0.100848081101 0.252655884454 9.9159050830 0.571403856761 378 11 86 47 [31]
134 0.100317013156 0.250991451871 9.9683988642 0.566652992450 355 16 86 48 [30]
135 0.100072456661 0.250226468088 9.9927595801 0.566716766792 414 13 83 52 [33]
136 0.099974992408 0.249921856160 10.0025013847 0.569248188371 369 9 81 55 [30]
137 0.099905277604 0.249704062592 10.0094812204 0.572235068313 399 8 80 57 [33]
138 0.099869136468 0.249591185210 10.0131035009 0.575786634851 360 16 86 52 [33]
139 0.099861492019 0.249567312386 10.0138700092 0.579825833672 334 14 76 63 [33]
140 0.099861423766 0.249567099242 10.0138768534 0.583996045098 420 44 90 50 [31]
141 0.099861423766 0.249567099242 10.0138768534 0.588167445420 575 17 81 60 [31]
142 0.099861423766 0.249567099242 10.0138768534 0.592338845742 649 11 80 62 [31]
143 0.099861423766 0.249567099242 10.0138768534 0.596510246064 705 4 79 64 [31]
144 0.099861423766 0.249567099242 10.0138768534 0.600681646386 714 4 80 64 [4]
145 0.098210363392 0.244432296017 10.1822248229 0.575345294482 369 18 88 57 [30]
146 0.097707699220 0.242877378042 10.2346079990 0.570463451917 380 19 91 55 [30]
147 0.097451054406 0.242084982392 10.2615616229 0.569856585456 383 23 90 57 [30]
148 0.097115466024 0.241050370104 10.2970210713 0.567826321505 441 13 86 62 [30]
149 0.096865267053 0.240280132513 10.3236178501 0.567256025991 410 12 95 54 [33]
150 0.096773128685 0.239996725341 10.3334470383 0.569435075060 385 25 95 55 [33]
151 0.096537083462 0.239271267581 10.3587136066 0.569046921871 392 17 95 56 [30]
152 0.096404526333 0.238864240615 10.3729569351 0.570459047790 431 9 94 58 [30]
153 0.096348281670 0.238691617786 10.3790122944 0.573207627701 447 9 102 51 [27]
154 0.096219754365 0.238297329810 10.3928762508 0.574648214678 404 15 91 63 [30]
155 0.095966893965 0.237522352826 10.4202601406 0.573831815849 572 5 99 56 [30]
156 0.095769693874 0.236918638761 10.4417165760 0.573980985087 459 9 95 61 [24]
157 0.095690305492 0.236675763139 10.4503794283 0.576224984542 388 11 80 77 [33]
158 0.095690301365 0.236675750515 10.4503798790 0.579895132343 594 92 66 [30]
159 0.095526510549 0.236174960882 10.4682982164 0.580573856155 391 14 84 75 [30]
160 0.095463734169 0.235983129900 10.4751821066 0.583074229108 419 18 94 66 [30]
161 0.095420001748 0.235849528301 10.4799830401 0.585912476281 502 15 92 69 [31]
162 0.095420001748 0.235849528301 10.4799830401 0.589551684209 624 106 56 [31]
163 0.095371784915 0.235702260396 10.4852813742 0.592292107759 293 64 93 70 [31]
164 0.095371784915 0.235702260396 10.4852813742 0.595925801671 389 59 97 67 [27]
165 0.095371784915 0.235702260396 10.4852813742 0.599559495584 329 63 96 69 [31]
166 0.095371784915 0.235702260396 10.4852813742 0.603193189497 859 103 63 [4]
167 0.095371784915 0.235702260396 10.4852813742 0.606826883409 868 104 63 [4]
168 0.095371784915 0.235702260396 10.4852813742 0.610460577322 877 105 63 [4]
169 0.095371784915 0.235702260396 10.4852813742 0.614094271235 884 106 63 [4]
170 0.095371784915 0.235702260396 10.4852813742 0.617727965147 891 107 63 [4]
171 0.095371784915 0.235702260396 10.4852813742 0.621361659060 900 108 63 [4]
172 0.095371784915 0.235702260396 10.4852813742 0.624995352973 906 110 62 [4]
173 0.093528698827 0.230099144903 10.6919053995 0.592883535222 491 9 101 72 [27]
174 0.093249714094 0.229255435891 10.7238934694 0.590990342935 440 19 103 71 [30]
175 0.093049225314 0.228649829665 10.7469997372 0.590561244723 473 11 104 71 [30]
176 0.092858397270 0.228073959152 10.7690852890 0.590289186714 476 12 96 80 [30]
177 0.092627323509 0.227377359490 10.7959505049 0.589222362673 472 20 107 70 [30]
178 0.092221922406 0.226157136622 10.8434087461 0.584805064088 525 3 106 72 [30]
179 0.091925731949 0.225267161265 10.8783468872 0.582442329286 501 4 96 83 [30]
180 0.091676214094 0.224518426940 10.9079548047 0.580939791974 518 4 111 69 [30]
181 0.091532577755 0.224087828723 10.9250719747 0.581425752715 368 49 86 95 [33]
182 0.091532577510 0.224087827989 10.9250720039 0.584638044997 408 56 96 86 [30]
183 0.091381691307 0.223635821898 10.9431110948 0.584948018827 481 11 90 93 [30]
184 0.091329890499 0.223480720453 10.9493178469 0.587144832729 500 6 99 85 [33]
185 0.091292909602 0.223370016686 10.9537531924 0.589619019023 526 17 109 76 [33]
186 0.091282593792 0.223339139478 10.9549910718 0.592605215903 524 13 109 77 [33]
187 0.091276235359 0.223320108238 10.9557542121 0.595666771783 505 10 102 85 [33]
188 0.091268473841 0.223296878269 10.9566858950 0.598699401494 971 107 81 [24]
189 0.091268473841 0.223296878269 10.9566858950 0.601883972778 977 108 81 [24]
190 0.091268473841 0.223296878269 10.9566858950 0.605068544063 984 109 81 [24]
191 0.090490250008 0.220972150260 11.0509143240 0.592826134136 541 28 111 80 [30]
192 0.090345838270 0.220541731807 11.0685784664 0.593081385559 493 18 113 79 [33]
193 0.090334034669 0.220506564650 11.0700247549 0.595936714306 510 27 111 82 [33]
194 0.090326418131 0.220483873329 11.0709582057 0.598872961358 456 30 105 89 [33]
195 0.090325523934 0.220481209377 11.0710678051 0.601942058043 385 46 107 88 [33]
196 0.090325523878 0.220481209212 11.0710678119 0.605028939275 843 17 121 75 [31]
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1183 0.051134288434 0.113918900723 19.5563491861 0.662536798008 6055 35 457 726 [31]
1372 0.049056216500 0.108785658645 20.3847763107 0.678460972010 7686 508 864 [4]
1470 0.047685892458 0.105426498230 20.9705627483 0.667691256461 7777 522 948 [24]
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1688 0.045873685240 0.101015254455 21.7989898733 0.682577459594 9498 590 1098 [4]
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2048 0.043078931470 0.094280904158 23.2132034356 0.685825026836 11568 676 1372 [4]
2304 0.041397560998 0.090268950789 24.1560124774 0.684692149301 12984 4 680 1624 [4]
2457 0.040605151571 0.088388347649 24.6274169979 0.689028480731 13926 770 1687 [4]
2916 0.038400053241 0.083189033081 26.0416305604 0.691626473494 16578 868 2048 [4]
3240 0.037058394199 0.080049824286 26.9844396017 0.690706287813 18390 4 872 2368 [4]
3430 0.036422118203 0.078567420132 27.4558441227 0.694189959299 19554 974 2456 [4]
4000 0.034637963350 0.074432292757 28.8700576848 0.696314831452 22860 1084 2916 [4]
4400 0.033542564329 0.071909164189 29.8128667263 0.695552746514 25116 4 1088 3312 [4]
4631 0.033020441266 0.070710678119 30.2842712471 0.698412507264 26526 1202 3429 [4]
5324 0.031547249214 0.067343502970 31.6984848098 0.700182328892 30558 1324 4000 [4]
5808 0.030636040488 0.065271395187 32.6412938510 0.699540916087 33306 4 1328 4480 [4]
6084 0.030199894577 0.064282434653 33.1126983722 0.701930470576 34986 1454 4630 [4]
6912 0.028962914549 0.061487546190 34.5269119345 0.703427151521 39816 1588 5324 [4]
7488 0.028193060799 0.059755502636 35.4697209760 0.702879907491 43104 4 1592 5896 [4]
7813 0.027823280050 0.058925565100 35.9411254965 0.704906337047 45078 1730 6083 [4]
8788 0.026769935040 0.056568542496 37.3553390588 0.706188448883 50778 1876 6912 [4]
9842 0.025793436663 0.054392829322 38.7695526217 0.707456272345 56946 2030 7812 [4]
10976 0.024885671378 0.052378280088 40.1837661841 0.708566789704 63588 2188 8788 [4]
12195 0.024039628994 0.050507627228 41.5979797464 0.709665506225 70734 2354 9841 [4]
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14896 0.022509135229 0.047140452079 44.4264068712 0.711597975515 86586 2702 12194 [4]
16384 0.021814713469 0.045619792335 45.8406204336 0.712454424075 95328 2884 13500 [4]
17969 0.021161856162 0.044194173824 47.2548339959 0.713302576036 104646 3074 14895 [4]
19652 0.020546939999 0.042854956436 48.6690475583 0.714063481283 114546 3268 16384 [4]
21438 0.019966750919 0.041594516540 50.0832611207 0.714817335940 125058 3470 17968 [4]
23328 0.019418427916 0.040406101782 51.4974746831 0.715497824550 136188 3676 19652 [4]
25327 0.018899415860 0.039283710066 52.9116882454 0.716172273735 147966 3890 21437 [4]
27436 0.018407425680 0.038221988173 54.3259018064 0.716784442454 160398 4108 23328 [4]
29660 0.017940400614 0.037216146378 55.7401153702 0.717391400110 173514 4334 25326 [4]
32000 0.017496487470 0.036261886215 57.1543289326 0.717945037298 187320 4564 27436 [4]
34461 0.017074012045 0.035355339059 58.5685424949 0.718494148360 201846 4802 29659 [4]
37044 0.016671458028 0.034493013716 59.9827560573 0.718997259351 217098 5044 32000 [4]
39754 0.016287448814 0.033671751485 61.3969696197 0.719496414947 233106 5294 34460 [4]
42592 0.015920731776 0.032888687497 62.8111831820 0.719955609888 249876 5548 37044 [4]
45563 0.015570164619 0.032141217327 64.2253967444 0.720411328694 267438 5810 39753 [4]
48668 0.015234703487 0.031426968053 65.6396103068 0.720832113623 285798 6076 42592 [4]
51912 0.014913392590 0.030743773095 67.0538238692 0.721249827704 304986 6350 45562 [4]
55296 0.014605355105 0.030089650263 68.4680374315 0.721636825700 325008 6628 48668 [4]
58825 0.014309785185 0.029462782549 69.8822509939 0.722021097808 345894 6914 51911 [4]
62500 0.014025940925 0.028861501273 71.2964645563 0.722378219051 367650 7204 55296 [4]
66326 0.013753138134 0.028284271247 72.7106781187 0.722732909806 390306 7502 58824 [4]
70304 0.013490744840 0.027729677693 74.1248916810 0.723063483823 413868 7812 62492 [4]
74439 0.013238176395 0.027196414661 75.5391052434 0.723391881737 438366 8122 66317 [4]
78732 0.012994891130 0.026683274761 76.9533188057 0.723698761272 463806 8436 70296 [4]
83188 0.012760386473 0.026189140043 78.3675323681 0.724003684924 490218 8758 74430 [4]
87808 0.012534195489 0.025712973861 79.7817459305 0.724289328296 517608 9084 78724 [4]
92597 0.012315883773 0.025253813613 81.1959594928 0.724573207362 546006 9418 83179 [4]
97556 0.012105046666 0.024810764252 82.6101730552 0.724839744321 575418 9756 87800 [4]
102690 0.011901306754 0.024382992454 84.0243866176 0.725104684407 605874 10102 92588 [4]
108000 0.011704311609 0.023969721396 85.4386001800 0.725353969238 637380 10452 97548 [4]
113491 0.011513731759 0.023570226039 86.8528137423 0.725601804153 669966 10810 102681 [4]
119164 0.011329258847 0.023183828891 88.2670273047 0.725835458797 703638 11172 107992 [4]
125024 0.011150603965 0.022809896167 89.6812408671 0.726067793037 738426 11542 113482 [4]
131072 0.010977496146 0.022447834323 91.0954544295 0.726287242366 774336 11916 119156 [4]
137313 0.010809680995 0.022097086912 92.5096679918 0.726505485865 811398 12298 125015 [4]
143748 0.010646919435 0.021757131728 93.9238815542 0.726711986719 849618 12684 131064 [4]
150382 0.010488986577 0.021427478217 95.3380951166 0.726917383466 889026 13078 137304 [4]
157216 0.010335670679 0.021107665110 96.7523086789 0.727112048709 929628 13476 143740 [4]
164255 0.010186772202 0.020797258270 98.1665222413 0.727305700465 971454 13882 150373 [4]
171500 0.010042102942 0.020495848730 99.5807358037 0.727489519020 1014510 14292 157208 [4]
178956 0.009901485235 0.020203050891 100.9949493667 0.727672405074 1058826 14710 164246 [4]
186624 0.009764751233 0.019918500878 102.4091629289 0.727846258703 1104408 15132 171492 [4]
194509 0.009631742231 0.019641855032 103.8233764901 0.728019252431 1151286 15562 178947 [4]
202612 0.009502308058 0.019372788525 105.2375900533 0.728183929867 1199466 15996 186616 [4]
210938 0.009376306504 0.019110994086 106.6518036155 0.728347812670 1248978 16438 194500 [4]
219488 0.009253602807 0.018856180831 108.0660171777 0.728504021566 1299828 16884 202604 [4]
228267 0.009134069166 0.018608073189 109.4802307409 0.728659494666 1352046 17338 210929 [4]
237276 0.009017584300 0.018366409900 110.8944443021 0.728807871751 1405638 17796 219480 [4]
246520 0.008904033037 0.018130943107 112.3086578654 0.728955566217 1460634 18262 228258 [4]
256000 0.008793305932 0.017901437498 113.7228714276 0.729096685961 1517040 18732 237268 [4]
265721 0.008685298920 0.017677669529 115.1370849898 0.729237171268 1574886 19210 246511 [4]
275684 0.008579912986 0.017459426695 116.5512985520 0.729371553309 1634178 19692 255992 [4]
285894 0.008477053861 0.017246506858 117.9655121142 0.729505344671 1694946 20182 265712 [4]
296352 0.008376631746 0.017038717618 119.3797256764 0.729633460203 1757196 20676 275676 [4]
307063 0.008278561046 0.016835875742 120.7939392396 0.729761024887 1820958 21178 285885 [4]
318028 0.008182760127 0.016637806616 122.2081528018 0.729883302177 1886238 21684 296344 [4]
329252 0.008089151093 0.016444343748 123.6223663640 0.730005064952 1953066 22198 307054 [4]
340736 0.007997659569 0.016255328303 125.0365799262 0.730121894118 2021448 22716 318020 [4]
352485 0.007908214511 0.016070608663 126.4507934884 0.730238241978 2091414 23242 329243 [4]
364500 0.007820748014 0.015890040026 127.8650070517 0.730349979144 2162970 23772 340728 [4]
376786 0.007735195147 0.015713484026 129.2792206139 0.730461265418 2236146 24310 352476 [4]
389344 0.007651493790 0.015540808377 130.6934341751 0.730568236335 2310948 24852 364492 [4]
402179 0.007569584479 0.015371886547 132.1076477383 0.730674784269 2387406 25402 376777 [4]
415292 0.007489410275 0.015206597444 133.5218613005 0.730777287486 2465526 25956 389336 [4]
428688 0.007410916621 0.015044825131 134.9360748637 0.730879393375 2545338 26518 402170 [4]
442368 0.007334051226 0.014886458551 136.3502884259 0.730977703023 2626848 27084 415284 [4]
456337 0.007258763945 0.014731391274 137.7645019871 0.731075638966 2710086 27658 428679 [4]
470596 0.007185006673 0.014579521261 139.1787155503 0.731170007199 2795058 28236 442360 [4]
485150 0.007112733238 0.014430750636 140.5929291125 0.731264023517 2881794 28822 456328 [4]
500000 0.007041899309 0.014284985478 142.0071426748 0.731354682675 2970300 29412 470588 [4]
515151 0.006972462304 0.014142135623 143.4213562370 0.731445010046 3060606 30010 485141 [4]
530604 0.006904381302 0.014002114478 144.8355697992 0.731532174562 3152718 30612 499992 [4]
546364 0.006837616966 0.013864838846 146.2497833624 0.731619025914 3246666 31222 515142 [4]
562432 0.006772131466 0.013730228760 147.6639969246 0.731702894005 3342456 31836 530596 [4]
578813 0.006707888407 0.013598207330 149.0782104868 0.731786466189 3440118 32458 546355 [4]
595508 0.006644852764 0.013468700594 150.4924240490 0.731867221361 3539658 33084 562424 [4]
612522 0.006582990814 0.013341637380 151.9066376112 0.731947696639 3641106 33718 578804 [4]
629856 0.006522270078 0.013216949181 153.3208511734 0.732025509028 3744468 34356 595500 [4]
647515 0.006462659266 0.013094570021 154.7350647366 0.732103056402 3849774 35002 612513 [4]
665500 0.006404128222 0.012974436352 156.1492782989 0.732178083970 3957030 35652 629848 [4]
683816 0.006346647869 0.012856486930 157.5634918611 0.732252860365 4066266 36310 647506 [4]
702464 0.006290190170 0.012740662724 158.9777054233 0.732325249966 4177488 36972 665492 [4]
721449 0.006234728073 0.012626906806 160.3919189855 0.732397401291 4290726 37642 683807 [4]
740772 0.006180235472 0.012515164268 161.8061325487 0.732467289633 4405986 38316 702456 [4]
760438 0.006126687167 0.012405382126 163.2203461109 0.732536951729 4523298 38998 721440 [4]
780448 0.006074058824 0.012297509238 164.6345596721 0.732604466238 4642668 0 780448 [4]
800807 0.006022326937 0.012191496227 166.0487732353 0.732671765733 4764126 0 800807 [4]
821516 0.005971468795 0.012087295404 167.4629867975 0.732737025322 4887678 0 821516 [4]
842580 0.005921462446 0.011984860698 168.8772003607 0.732802080400 5013354 41782 800798 [4]
864000 0.005872286670 0.011884147582 170.2914139220 0.732865196171 5141160 42492 821508 [4]
885781 0.005823920943 0.011785113019 171.7056274842 0.732928117261 5271126 43210 842571 [4]
907924 0.005776345414 0.011687715391 173.1198410474 0.732989193138 5403258 0 907924 [4]
930434 0.005729540875 0.011591914445 174.5340546096 0.733050083543 5537586 0 930434 [4]
953312 0.005683488734 0.011497671238 175.9482681718 0.733109216838 5674116 0 953312 [4]
976563 0.005638170994 0.011404948083 177.3624817340 0.733168173300 5812878 0 976563 [4]
1000188 0.005593570226 0.011313708498 178.7766952966 0.733225455233 5953878 0 1000188 [4]
1024192 0.005549669549 0.011223917161 180.1909088590 0.733282568444 6097146 0 1024192 [4]

Updates

08-Feb-2012: First raw presentation from N=1 to N=600.
13-Feb-2012: First improvements for N=47, 55, 56 and 57 by David W. Cantrell [24].
13-Feb-2012: All parameters are displayed at 30 decimal places.
13-Feb-2012: The packings for N=14, 63, 172, 365, 666, 1099, 1688, 2457, 3430, 4631, 6084, 7813, 9842, 12195, 14896, 17969 ...
13-Feb-2012: Extension up to N=1000 and improvements for N=43, 47, 51, 68, 73, 74, 77, 79, 81, 82, 84, 85, 86, 87, 92, 93, 94, 95, 97, 99, 109–122, 124, 126–139 any many others by E. Specht [31].
13-Feb-2012: Further improvements for N=65, 68, 180, 416 and 567 by David W. Cantrell [24].
14-Feb-2012: More improvements for N=43 and 51 by David W. Cantrell [24].
14-Feb-2012: One improvement for N=57 by Eckard Specht [31].
15-Feb-2012: Now all spheres are colored in the usual manner. All of these images were made by POV-Ray.
20-Feb-2012: Again, one improvement for N=57 by Eckard Specht [31].
20-Feb-2012: The case N=12 was a hard nut to crack, it was done by David W. Cantrell [24].
20-Feb-2012: Two improvements for N=47 and 71 by David W. Cantrell [24].
24-Feb-2012: Six improvements for N=56, 57, 72, 76, 77 and 79 by program Packntile created by Tomaz and Nevenka Kristan [26].
25-Feb-2012: David Cantrell has found new regular lattice packings for N=126, 221, 405, 550, 726, 750, 787, 800, 936, 968 and 1008. Furthermore he improved the cases N=46, 71 and 72 [24].
29-Feb-2012: Many, many new record packings, namely for N=41, 42, 55, 58, 59, 64, 65, 66, 68, 69, 71, 73, 74, 75, 76, 77, 78, 79, 83, 84, 85, 86, 87, 89, 90, 92, 93, 94, 95, 96, 97, 99, 101, 102, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 134, 135, 136, 137, 138, 139, 140, 145, 150, 155, 160, 165, 175 and 195 are found by Liang Yu from WenQi Huang's group [27]. That's overwhelming!
01-Mar-2012: A symmetric better packing for N=98 by David W. Cantrell [24].
01-Mar-2012: Three improvements for N=58, 71 and 82 by program Packntile [26].
01-Mar-2012: Some better packings for N=42, 55, 64 and 65 by David W. Cantrell [24].
01-Mar-2012: A remarkable improvement for N=54 by WenQi Huang and Liang Yu [27].
08-Mar-2012: One better packing for N=55 by Eckard Specht [31].
08-Mar-2012: A couple of improvements for N=73, 77, 81, 90, 99, 112 and 176 by program Packntile [26].
08-Mar-2012: More better packings for N=54, 55, 56, 57, 58, 66, 68, 69, 75 and 99 by David W. Cantrell [24].
22-Mar-2012: A couple of new records for N=42, 54, 55, 71, 73, 81 and 82 by WenQi Huang and Liang Yu [27].
29-Mar-2012: More improvements for N=74, 93, 96, 97, 122, 178, 258 and 437 by program Packntile [26].
30-Mar-2012: In [4], Michael Goldberg mentioned more packings, i.e. for N=256, 500 and 864. Many thanks to David W. Cantrell who has kindly calculated the coordinates.
31-Mar-2012: To continue this work, I have (re)calculated all these cubic close-packed Goldberg packings up to 39754, namely for N=32, 63, 108, 172, 256, 365, 500, 666, 864, 1099, 1372, 1688, 2048, 2457, 2916, 3430, 4000, 4631, 5324, 6084, 6912, 7813, 8788, 9842, 10976, 12195, 13500, 14896, 16384, 17969, 19652, 21438, 23328, 25327, 27436, 29660, 32000, 34461, 37044 and 39754. More will follow ...
01-Apr-2012: ... more packings for N=42592, 45563, 48668, 51912, 55296, 58825, 62500, 66326, 70304, 74439, 78732, 83188, 87808, 92597, 97556, 102690, 108000, 113491, 119164 and 125024 by Eckard Specht [31].
02-Apr-2012: ... even more packings for N=131072, 137313, 143748, 150382, 157216, 164255, 171500, 186624, 194509, 202612, 210938, 219488, 275684, 285894 and 389344 by Eckard Specht [31].
04-Apr-2012: ... even more packings for N=178956, 228267, 237276, 246520, 256000, 265721, 296352, 307063, 318028, 329252, 340736, 352485, 364500, 376786, 402179, 415292, 428688, 442368, 456337, 530604, 546364 and 562432 by Eckard Specht [31].
04-Apr-2012: Here comes a bundle of new records for N=74, 90, 93, 96, 97 and 112 by WenQi Huang and Liang Yu [27].
06-Apr-2012: More better packings for N=42, 54, 55, 73, 74, 76, 77, 78, 79, 81, 82 and 89 by David W. Cantrell [24].
06-Apr-2012: More packings for N=470596, 485150, 500000, 515151, 578813, 595508, 612522, 629856, 647515, 665500, 683816, 702464, 721449, 740772, 760438, 780448, 800807, 821516, 842580, 864000, 885781, 907924, 930434, 953312 and 976563 by Eckard Specht [31].
06-Apr-2012: David W. Cantrell sent me better packings for N=121, 190 and 280 [24].
06-Apr-2012: Seven improvements for N=550, 600, 726, 936, 1183, 1575 and 1920 by Eckard Specht [31].
09-Apr-2012: The last of Goldberg's packings (up to one million of spheres) for N=1000188 and 1024192 are now shown here [31].
09-Apr-2012: As it is easy to see there are many candidate packings which are derived from higher packings (i.e. from a "root" packing with a higher number of spheres) only by omitting one or more spheres. If they have exactly the same radius as the root packing and were not discovered before I have decided to give the credits who owns them for the associated root packing. According to this rule David W. Cantrell and I have calculated some improvements compared to the old packings shown here for N=146, 174, 179, 188, 189, 201, 235, 236, 237, 238, 239, 240, 241, 255, 257, 361–364, 408–415, 489–499, 564–566, 653–665, 721–725 and 857–863.
09-Apr-2012: They same once more for N=619–628, 630, 636, 637, 641–652 and 707–720.
09-Apr-2012: Here are derivatives from two more roots: N=793–799 by David W. Cantrell and N=895–935 by Eckard Specht.
09-Apr-2012: David W. Cantrell sent me four packings, namely for N=220, 445, 786 and 1008, which are members of a particularly nice family in which spheres are grouped in pyramids in four corners of the cube and in a tetrahedron in the middle of the cube. I have calculated all derivatives from these roots in the ranges N=211–219, 428–444, 733–785, 937–1007 which are better than the previous ones.
11-Apr-2012: Three improvements for N=69, 125 and 199 by program Packntile [26].
12-Apr-2012: The range between N=501 and 600 was reconsidered, improvements for N=501–538, 551–563, 568–582 and 593–599 were found by Eckard Specht [31].
13-Apr-2012: Six improvements for N=64, 69, 122, 125, 132 and 146 by WenQi Huang and Liang Yu [27].
15-Apr-2012: The range between N=200 and 299 was reconsidered, nearly for all of the non-regular packings and its derivatives improvements were found by Eckard Specht [31].
15-Apr-2012: A new family of packings was discovered by David W. Cantrell [24] which typical roots are N=122, 395, 538, 711 and 918. Their derivatives are N= 394–385, 537–530, 710–685 and 917–886. David sent me two (may be the final) improvements for N=64 and 69.
16-Apr-2012: Several better packings were found for N=93, 94, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 163, 164, 179, 181, 182, 183, 188, 189, 193, 194 by Eckard Specht [31].
21-Apr-2012: A third new family of packings was discovered by David W. Cantrell [24] which roots are N=n^2 (n+1) / 2. Improvements were found for N=405 (and 404 as its derivative) and the new packing for N=1470. David sent me two improvements for N=92 and 93.
21-Apr-2012: A couple of new records for N=92, 338, 339, 340, 341, 342, 343, 344, 346, 347, 397, 398 and 399 by Eckard Specht [31].
22-Apr-2012: Two new records for N=68 and 75 by WenQi Huang and Liang Yu [27].
22-Apr-2012: Improvements for N=163, 306, 311, 338, 339, 340, 341, 342, 343, 344, 346, 459, 462, 463, 466, 467, 468, 470, 471, 472, 474, 477, 479, 584, 586, 602, 605, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 630, 632, 634, 635, 636, 637, 638, 639, 673, 788, 812 and 879 by Eckard Specht [31]. Now all packings shown here fulfill the unproven conjecture that the radius r(N) as a function of N is a monoton decreasing function (but not a strictly monoton decreasing function!).
24-Apr-2012: Three improvements for N=236, 330 and 331 by program Packntile [26].
21-May-2012: Two improvements for N=75 and 92 by David W. Cantrell [24].
21-May-2012: Three improvements for N=78, 79 and 89 by WenQi Huang and Liang Yu [27].
21-May-2012: Further improvements for N=147, 148, 151, 152 and 153 by WenQi Huang and Liang Yu [27].
21-May-2012: Eight improvements for N=149, 154, 179, 181, 235, 237, 238 and 326 by program Packntile [26].
21-May-2012: The last improvement for today for N=179 by WenQi Huang and Liang Yu [27].
01-Jun-2012: Welcome to the list of players in this 3D game! 19 nice improvements arrived for N=42, 49, 51, 55, 64, 68, 70, 71, 75, 78, 79, 81, 89, 90, 95, 96, 97, 98 and 99 by Marco Locatelli, Mirko Maischberger and Fabio Schoen [28].
03-Jun-2012: Six improvements for N=149, 156, 157, 158, 159 abd 173 by WenQi Huang and Liang Yu [27].
03-Jun-2012: Many new records for N=101, 102, 109, 112, 115, 116, 119, 123, 124, 131, 132, 133, 134, 135, 137, 139, 140, 161, 165, 174, 175, 176, 177, 178, 182, 183, 184, 185, 186, 187, 191, 192, 197, 198, 201, 202, 203, 204, 205, 206, 207, 208, 213, 222, 223, 224, 225, 226, 227, 228, 229, 230, 232, 233, 234, 239, 240, 241, 257, 258, 259, 261, 262, 263, 264, 265, 266, 267, 268, 271, 272, 273, 275, 281, 282, 283, 289, 290, 291, 292, 293, 294, 295, 297 and 298 by Eckard Specht [31].
16-Jun-2012: Twenty-three improvements for N=42, 51, 71, 75, 90, 95, 96, 97, 98, 99, 109, 111, 112, 113, 114, 115, 116, 117, 118, 119 and 120 by David W. Cantrell [24]. Davids packings are well fine-tuned, many of them without rattlers. So it is very hard to beat these records.
14-Aug-2012: Twenty-two improvements for N=127, 129, 130, 131, 132, 134, 135, 136, 137, 138, 145, 146, 147, 148, 149, 150, 151, 154, 155, 156, 409 and 413 by David W. Cantrell [24].
14-Aug-2012: One improvement for N=340 by Eckard Specht [31].
15-Oct-2012: Three tiny improvements for N=92, 94 and 97 by Marco Locatelli, Mirko Maischberger and Fabio Schoen [28].
02-Nov-2012: Lots of slightly improvements for N=300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 321, 322, 323, 324, 325, 327, 328, 329, 332, 333, 334, 335, 336, 337, 338, 339, 341, 342, 343, 344, 345, 346, 349, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 396, 397, 406, 407, 408, 410, 411, 412, 414, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 460, 461, 462, 463, 464, 465, 466, 467, 468, 470, 471, 472, 474, 477, 478, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 631, 632, 639, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 788, 789, 790, 791, 792, 793, 794, 795, 796, 79,7 798 and 799 by Eckard Specht [31].
25-Jun-2013: Two improvements for N=507 and 619 by program Packntile [26] dated from June 2012.
25-Jun-2014: Three improvements for N=83, 94, 95 by István Kolossváry. It was my failure to update packomania much too late in June 2018, it is really embarrassing, sorry!
13-Sep-2014: One improvement for N=110 again by István Kolossváry. The method used to get these results are published in [10].
29-Jun-2018: Three improvements for N=83, 96 and 109 by Francesco Bagattini, Fabio Schoen and Luca Tigli [25].
04-Oct-2026: There were very old candidates on my laptop (from July 2022) for N=94–96, 109–120, 123–125, 127, 129, 131, 132, 134–139, 145–152, 154, 155, 157–160, 174–187, 191–194, 197–200 by Xiangjing Lai, Jin-Kao Hao, Renbin Xiao, Fred Glove [30].
05-Oct-2026: A large bundle of new records for N=202, 203, 205, 224, 228, 235–239, 257, 258, 265, 271, 281, 290, 291, 293, 295, 297, 298, 301, 305–309, 326, 328–330, 336, 338–343, 367, 377, 383, 396, 408–411, 417–427, 448, 451, 456–459, 461–463, 501–506, 508–529, 551, 553–557, 559–563, 568–576 by Yuan Gao (Harvard University) [32].
06-Oct-2026: Many new records for N=102, 110, 205, 206, 208, 225, 232, 233, 259, 260, 263–265, 293–296, 300–305, 329, 331, 333, 335–337, 366, 368–371, 373, 375, 376, 378–382, 397, 411, 448, 450, 452–456 by Lu Peng [33].
08-Oct-2026: A continuation of the last update for N=458–465, 469, 509, 511, 512, 518, 551, 552, 555, 558–562, 568, 571–573, 575–578, 601–629, 667–684, 727–732, 788–792, 794–798, 801–811, 865–885, 937–946 by Lu Peng [33].
17-Oct-2026: A continuation of the last update for N=95, 96, 101, 102, 110, 111, 117, 118, 123–125, 130, 135, 137–139, 149, 150, 157, 181, 184–187, 192–195, 200–204, 206–213, 222–229, 231–240, 257–263, 266–268, 271–274, 282, 283, 289–302, 305–310, 321–334, 336–345, 366–378, 380–384, 396, 398, 399, 406–414, 417, 418, 420–427, 446–452, 454–463, 466–469, 501–506, 508–510, 512–529, 553–557, 563, 568–571, 573, 574, 578, 601–603, 605, 607, 608, 610, 613–617, 619–629, 667, 670, 672, 683, 684, 728, 730, 731, 734, 735, 788–795, 797, 801, 804, 806–811, 865–870, 872–874, 876, 878, 879, 881–883 by Lu Peng [33].
17-Oct-2026: The next update for N=101, 210–212, 297, 366, 417, 421, 422, 459–463, 509, 513, 517, 518, 522–524, 555, 556, 573, 630, 734–737, 740–742, 745–747, 793–795, 808, 1596 by Vidit Patankar [34].
17-Oct-2026: The table has been reorganized.

References

[1]   J. Schaer, On the densest packing of spheres in a cube, Canadian Mathematical Bulletin 9 (1966) 3, 265–270.
[2]   J. Schaer, The densest packing of five spheres in a cube, Canadian Mathematical Bulletin 9 (1966) 3, 271–274.
[3]   J. Schaer, The densest packing of six spheres in a cube, Canadian Mathematical Bulletin 9 (1966) 3, 275–280.
[4]   M. Goldberg, On the densest packing of equal spheres in a cube, Math. Mag. 44 (1971) 4, 199–208.
[5]   Th. Gensane, Dense Packings of Equal Spheres in a Cube, The Electronic Journal of Combinatorics 11 (2004) #R33, 211–227.
[6]   D. Boll, Optimal Packing Of Circles And Spheres.
[7]   H. Pfoertner, Densest Packings of Equal Spheres in a Cube, April 13, 2005.
[8]   A. Joós, On the packing of fourteen congruent Spheres in a cube, Geom. Dedicata 140 (2009) 3, 49–80.
[9]   M. Tatarevic, On limits of dense packing of equal spheres in a cube, The Electronic Journal of Combinatorics 22(1) (2015) #P1.35.
[10]   I. Kolossváry, K.J. Bowers, Global optimization of additive potential energy functions: Predicting binary Lennard-Jones clusters, Phys. Rev. E 82 (2010), 056711.
[24]   , private communication, February–April 2012.
[26]   , Results of Pack'n'tile contest, January–June 2012.
[27]   , private communication, February–April 2012.
[28]   , private communication, June–October 2012.
[29]   István Kolossváry, private communication, June 2014, September 2014.
[30]   F. Bagattini, F. Schoen, L. Tigli, Clustering methods for the optimization of atomic structure, J. Chem. Phys 148 (14) (2018), 144102, and private communication, June 2018.
[31]   , program scu, April 2010–November 2012.
[32]   , private communication, September 2026.
[33]   , private communication, October 2026.
[34]   , private communication, October 2026.

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©  E. Specht    17-Oct-2026