This table shows the conjectural minimal-energy configurations of N points on the unit sphere in Rn, under the harmonic potential function: the potential energy between two points at distance r is 1/rn-2.

In each entry in the table, the column for N links to coordinates for the configuration (given as a plain text file in which each line lists the coordinates for one point).

Skip to dimension: 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

NnMaximal CosineHarmonic Energy
2 3 -1.000000000000 0.500000000000
3 3 -0.500000000000 1.732050807569
4 3 -0.333333333333 3.674234614175
5 3 0.000000000000 6.474691494688
6 3 0.000000000000 9.985281374239
7 3 0.309016994375 14.452977414221
8 3 0.314089367889 19.675287861233
9 3 0.355274167046 25.759986531270
10 3 0.422687424044 32.716949460148
11 3 0.521909766439 40.596450508191
12 3 0.447213595500 49.165253057629
13 3 0.611293293702 58.853230611702
14 3 0.603679914436 69.306363296626
15 3 0.653091865726 80.670244114294
16 3 0.656898737182 92.911655302545
17 3 0.641341574047 106.050404828619
18 3 0.675147168450 120.084467447492
19 3 0.708220285859 135.089467556679
20 3 0.693485885254 150.881568333756
21 3 0.715442753070 167.641622399271
22 3 0.727748739817 185.287536149308
23 3 0.749174109733 203.930190662879
24 3 0.742381682501 223.347074051805
25 3 0.770396633833 243.812760298766
26 3 0.778876600117 265.133326317357
27 3 0.766713894069 287.302615033039
28 3 0.789900940149 310.491542358202
29 3 0.804983583197 334.634439920416
30 3 0.799240242051 359.603945903764
31 3 0.805172138397 385.530838063300
32 3 0.794654472292 412.261274650530
33 3 0.831955172396 440.204057447647
34 3 0.836061769319 468.904853281343
35 3 0.837715853354 498.569872490645
36 3 0.836485086120 529.122408375414
37 3 0.844959194726 560.618887731045
38 3 0.836415390593 593.038503566451
39 3 0.847558009871 626.389009016823
40 3 0.848820841523 660.675278834622
41 3 0.852386277111 695.916744341886
42 3 0.854959459924 732.078107543673
43 3 0.858363244495 769.190846459159
44 3 0.854842993573 807.174263084628
45 3 0.864211537636 846.188401061077
46 3 0.867849857414 886.167113639192
47 3 0.876413175379 927.059270679709
48 3 0.868720945091 968.713455343785
49 3 0.879759643661 1011.557182653571
50 3 0.877050604933 1055.182314726296
51 3 0.881588567998 1099.819290318900
52 3 0.885637870042 1145.418964319278
53 3 0.889918954950 1191.922290416226
54 3 0.890771904098 1239.361474729160
55 3 0.894036385178 1287.772720782709
56 3 0.893505605848 1337.094945275652
57 3 0.893352249506 1387.383229252842
58 3 0.897603010098 1438.618250640397
59 3 0.897487365876 1490.773335278694
60 3 0.899118031717 1543.830400976379
61 3 0.903397935805 1597.941830198988
62 3 0.899711031084 1652.909409898302
63 3 0.904404851252 1708.879681503257
64 3 0.906896742285 1765.802577927304
2 4 -1.000000000000 0.250000000000
3 4 -0.500000000000 1.000000000000
4 4 -0.333333333333 2.250000000000
5 4 -0.250000000000 4.000000000000
6 4 0.000000000000 6.500000000000
7 4 0.000000000000 9.500000000000
8 4 0.000000000000 13.000000000000
9 4 0.309016994375 17.500000000000
10 4 0.309016994375 22.500000000000
11 4 0.329635517510 28.208956505019
12 4 0.254843048489 34.423644500473
13 4 0.344300713493 41.600000000000
14 4 0.339212636909 49.262299367800
15 4 0.353553390593 57.616604830847
16 4 0.436524595599 66.949838993307
17 4 0.476507338983 76.981111869607
18 4 0.484802115534 87.616177301986
19 4 0.487974755791 98.952904472532
20 4 0.441687946265 110.799268264278
21 4 0.520491519904 123.736629309374
22 4 0.544460421056 137.516159441209
23 4 0.575244333066 152.041994470053
24 4 0.500000000000 167.000000000000
25 4 0.601670758813 182.988382586738
26 4 0.564494752408 199.692493940280
27 4 0.648154178782 217.154124403836
28 4 0.642378351798 235.399212130882
29 4 0.626947677100 254.375334459738
30 4 0.630143469666 274.186168629074
31 4 0.651242419771 294.786618120006
32 4 0.628050854399 315.991832548801
33 4 0.610129487846 337.792033878877
34 4 0.628171487764 360.518608180190
35 4 0.688714557286 384.544113281013
36 4 0.674266762107 409.069361322359
37 4 0.674102571487 434.194825553103
38 4 0.688616509599 460.282804769032
39 4 0.697171669146 487.254783158551
40 4 0.694320507661 514.902153184528
41 4 0.681474878908 543.160145343872
42 4 0.686737165958 572.156065681656
43 4 0.678352788984 601.933355037443
44 4 0.708487556292 632.729355748660
45 4 0.716070042432 664.475176001384
46 4 0.735189712110 697.261749894285
47 4 0.739934271308 730.753235960535
48 4 0.707106781187 764.587012987012
49 4 0.726134396894 799.699013440439
50 4 0.717282945743 835.122366517995
51 4 0.752298078106 871.979645252522
52 4 0.736726738991 909.185861409823
53 4 0.734774661242 947.145657186085
54 4 0.737135768471 985.878500974354
55 4 0.748281764916 1025.762873335294
56 4 0.761756967323 1066.618569831205
57 4 0.772749213428 1108.167855160873
58 4 0.767783833352 1150.433573627662
59 4 0.765533017859 1193.377456756878
60 4 0.760848075621 1236.913089421486
61 4 0.762788504606 1281.380444351720
62 4 0.762447161081 1326.588951388450
63 4 0.752018162591 1373.088328882171
64 4 0.778308343893 1420.585324361836
2 5 -1.000000000000 0.125000000000
3 5 -0.500000000000 0.577350269190
4 5 -0.333333333333 1.377837980316
5 5 -0.250000000000 2.529822128135
6 5 -0.200000000000 4.034357652299
7 5 0.000000000000 6.190356034067
8 5 0.000000000000 8.698905792181
9 5 0.000000000000 11.558951986988
10 5 0.000000000000 14.767135623731
11 5 0.300818679628 19.027217382084
12 5 0.166666666667 23.603559366985
13 5 0.295587264172 28.672745801806
14 5 0.284509175241 34.286471854935
15 5 0.303188774230 40.264966592184
16 5 0.200000000000 46.516183181792
17 5 0.337010494050 54.056937549224
18 5 0.373070494561 61.968487860091
19 5 0.336042702021 70.246397325421
20 5 0.311580532614 78.818406874182
21 5 0.316227766017 87.957694261916
22 5 0.446373878136 98.543919322585
23 5 0.455506313529 109.519298588937
24 5 0.467862584804 120.827425571194
25 5 0.441051807815 132.602000441342
26 5 0.480462411294 144.980932266263
27 5 0.507363752093 158.343226510795
28 5 0.437253883022 171.930764783014
29 5 0.455447643940 185.927813923839
30 5 0.416651757771 200.289869187587
31 5 0.496361036874 215.869962585760
32 5 0.447213595500 231.613797549627
33 5 0.524943829647 249.128634241518
34 5 0.553803035977 266.917816022602
35 5 0.572301673217 285.364140056579
36 5 0.517221915341 303.987436006878
37 5 0.577290714957 323.638969804658
38 5 0.565121417506 343.575572139474
39 5 0.586026710139 364.158382756335
40 5 0.562483166415 384.875224615964
41 5 0.565508366009 407.153122333631
42 5 0.570858267272 429.829279281721
43 5 0.615229632398 453.352129730022
44 5 0.610793038760 477.194734377022
45 5 0.611162712943 501.799430222859
46 5 0.608805540750 527.127697241018
47 5 0.603699400460 553.090769078033
48 5 0.630892424935 579.598117250757
49 5 0.634111806255 606.632796424053
50 5 0.638614090665 634.572451445523
51 5 0.619758053763 663.167095352760
52 5 0.645805650988 691.825981395443
53 5 0.656883142162 721.467651157638
54 5 0.641453305278 751.719389355839
55 5 0.631901478194 782.947112773653
56 5 0.633451720742 814.590956316019
57 5 0.625691227845 847.247840734903
58 5 0.640517919930 880.569116898194
59 5 0.642421360413 914.741982180555
60 5 0.649254621146 949.212880775003
61 5 0.643483422705 984.491598520461
62 5 0.656830279115 1020.138570844414
63 5 0.639553499868 1056.739123258765
64 5 0.650119724197 1094.212947739920
2 6 -1.000000000000 0.062500000000
3 6 -0.500000000000 0.333333333333
4 6 -0.333333333333 0.843750000000
5 6 -0.250000000000 1.600000000000
6 6 -0.200000000000 2.604166666667
7 6 -0.166666666667 3.857142857143
8 6 0.000000000000 5.666666666667
9 6 0.000000000000 7.725000000000
10 6 0.000000000000 10.031250000000
11 6 0.000000000000 12.583333333333
12 6 0.000000000000 15.375000000000
13 6 0.288646470266 19.241880030605
14 6 0.166666666667 23.271785714286
15 6 0.309016994375 27.650000000000
16 6 0.242516438155 32.494175418956
17 6 0.214779108265 37.618598408635
18 6 0.292301679513 43.117480869564
19 6 0.341491933840 48.911968458451
20 6 0.293125899724 54.969070321712
21 6 0.325590843692 62.184724246498
22 6 0.268083645603 69.429381171872
23 6 0.359832090163 77.325312952393
24 6 0.349121229819 85.425915810222
25 6 0.328514617855 93.782551052541
26 6 0.272946879203 102.268911266040
27 6 0.250000000000 111.000000000000
28 6 0.406953046585 121.918957447886
29 6 0.415066412819 133.042607194026
30 6 0.446028980307 144.384249499257
31 6 0.447199418614 156.118407392185
32 6 0.472221756237 168.162320660137
33 6 0.386526620583 180.227018861729
34 6 0.409714694096 193.284655524846
35 6 0.381276972183 206.491648443095
36 6 0.368218625554 219.707312464904
37 6 0.467268273058 234.573168536793
38 6 0.385994185960 249.179869536682
39 6 0.460593044438 264.728590521165
40 6 0.433942050419 280.390457157410
41 6 0.403139599070 296.279083732584
42 6 0.400000000000 312.139493519303
43 6 0.415130837064 329.302661833036
44 6 0.408248290464 346.404999999998
45 6 0.481070417457 365.908756585185
46 6 0.519870631495 385.907808454010
47 6 0.507143040973 406.252228956546
48 6 0.488191836317 426.096951401117
49 6 0.547161465348 447.686121561771
50 6 0.539206845336 468.927161528207
51 6 0.551447333954 491.130485727676
52 6 0.538235195272 513.283330491626
53 6 0.484998472069 535.292514795480
54 6 0.495175460255 558.535280965364
55 6 0.552783871197 582.544218205019
56 6 0.559636028179 606.924568107308
57 6 0.564618619342 631.924883286584
58 6 0.544027802930 657.009084765690
59 6 0.570877373509 683.358005020038
60 6 0.551227412454 709.982587363387
61 6 0.573173585707 737.282080691852
62 6 0.602359092457 764.865630714736
63 6 0.623137258876 793.246398298272
64 6 0.582337633177 821.468030352232
2 7 -1.000000000000 0.031250000000
3 7 -0.500000000000 0.192450089730
4 7 -0.333333333333 0.516689242618
5 7 -0.250000000000 1.011928851254
6 7 -0.200000000000 1.680982355125
7 7 -0.166666666667 2.525092729874
8 7 -0.142857142857 3.544893358184
9 7 0.000000000000 5.031216464027
10 7 0.000000000000 6.693229823431
11 7 0.000000000000 8.530300053713
12 7 0.000000000000 10.541184179230
13 7 0.005601325977 12.717855630196
14 7 0.000000000000 15.067992404917
15 7 0.275693724306 18.471182203710
16 7 0.142857142857 21.893707673497
17 7 0.287500772686 25.687926245647
18 7 0.180723283607 29.641398501789
19 7 0.223494397861 34.056556994179
20 7 0.205947526330 38.605157328708
21 7 0.220177134935 43.425486568916
22 7 0.262062800350 48.556738705778
23 7 0.276207735985 53.930100927488
24 7 0.215625121409 59.407048347458
25 7 0.294405648045 65.755713796338
26 7 0.321018479787 72.456930066644
27 7 0.310958047107 79.659692174111
28 7 0.336899123286 87.069526163537
29 7 0.306554419480 94.527558315173
30 7 0.290083431645 102.297104109393
31 7 0.368404570887 110.398544993971
32 7 0.375950611057 118.602915678910
33 7 0.337882816572 126.923952190470
34 7 0.340257348996 136.566259627852
35 7 0.304198826060 145.759997194544
36 7 0.345780256879 156.740889034419
37 7 0.431846391626 168.027878349593
38 7 0.449787858810 179.356468480985
39 7 0.424164211027 190.831537661810
40 7 0.434702420638 202.756130292346
41 7 0.428893269741 214.990632909411
42 7 0.485749314078 227.436385439754
43 7 0.424930614375 240.306216678356
44 7 0.443849045963 253.327160420418
45 7 0.483619791753 266.505470081624
46 7 0.501293312422 279.832386963721
47 7 0.454011314500 294.080360211686
48 7 0.424592196160 308.574953944075
49 7 0.428330571459 323.660516121796
50 7 0.372568843567 338.330257461766
51 7 0.450254540706 354.155610465227
52 7 0.450919508367 370.108973011949
53 7 0.391328041788 386.102750275669
54 7 0.366661966838 401.970660343672
55 7 0.348220748214 418.018413940449
56 7 0.333333333333 434.255147529244
57 7 0.570789672884 454.868427043592
58 7 0.502243091321 475.194970436891
59 7 0.535561271152 496.103118026385
60 7 0.500000000000 516.697030457169
61 7 0.494907887192 537.951263695158
62 7 0.490292271333 558.506024830782
63 7 0.526808587812 580.555229248216
64 7 0.502197371801 601.776879637518
2 8 -1.000000000000 0.015625000000
3 8 -0.500000000000 0.111111111111
4 8 -0.333333333333 0.316406250000
5 8 -0.250000000000 0.640000000000
6 8 -0.200000000000 1.085069444444
7 8 -0.166666666667 1.653061224490
8 8 -0.142857142857 2.344726562500
9 8 -0.125000000000 3.160493827160
10 8 0.000000000000 4.360351562500
11 8 0.000000000000 5.684311224490
12 8 0.000000000000 7.131944444444
13 8 0.000000000000 8.702500000000
14 8 0.009725719110 10.389088793939
15 8 0.004026055652 12.193630460536
16 8 0.000000000000 14.125000000000
17 8 0.265575964016 17.049472818256
18 8 0.125000000000 19.896014986683
19 8 0.271350519895 23.122982878721
20 8 0.166666666667 26.270000000000
21 8 0.193152513486 29.946250246215
22 8 0.189571026100 33.736749659203
23 8 0.210782261417 37.739532360590
24 8 0.228830694665 41.794753994376
25 8 0.236368530074 46.204231632079
26 8 0.252008724214 50.749794834434
27 8 0.255029415992 55.419493371257
28 8 0.227859305776 60.320844918883
29 8 0.234112162158 65.836604909078
30 8 0.253200766480 71.304418290428
31 8 0.259536448358 77.736855882694
32 8 0.270555886239 84.365378350856
33 8 0.294975714443 91.005848672133
34 8 0.270049494153 97.859794998923
35 8 0.355283519635 104.953711966162
36 8 0.322980023510 111.964533396359
37 8 0.351394436427 119.519188483273
38 8 0.300114481078 126.924129681282
39 8 0.291115845451 134.466206509579
40 8 0.351845263029 143.192299139738
41 8 0.303782997407 151.830684561582
42 8 0.354170116673 160.887420170321
43 8 0.342440864559 170.201102211712
44 8 0.348168793774 179.504767445868
45 8 0.348643944755 190.008094535102
46 8 0.369542307074 200.801789071240
47 8 0.425140673125 211.911170106343
48 8 0.409157879186 223.078234410257
49 8 0.402652146739 234.312546065971
50 8 0.413634809765 246.098329132572
51 8 0.399279419225 258.003968555821
52 8 0.364614796437 269.980747868073
53 8 0.381033192539 281.991542921334
54 8 0.378504798416 294.524979228274
55 8 0.403236785243 307.790866928879
56 8 0.415660180627 320.869522725934
57 8 0.377212686452 334.633318570276
58 8 0.405132697135 348.109254849679
59 8 0.392723765521 362.731387434715
60 8 0.390316085818 376.740782183675
61 8 0.418791934084 391.862720255831
62 8 0.424631435158 406.756749438557
63 8 0.397041427466 421.793914210828
64 8 0.418569521580 437.069649099857
2 9 -1.000000000000 0.007812500000
3 9 -0.500000000000 0.064150029910
4 9 -0.333333333333 0.193758465982
5 9 -0.250000000000 0.404771540502
6 9 -0.200000000000 0.700409314635
7 9 -0.166666666667 1.082182598517
8 9 -0.142857142857 1.550890844206
9 9 -0.125000000000 2.106995884774
10 9 -0.111111111111 2.750782875071
11 9 0.000000000000 3.705798642443
12 9 0.000000000000 4.748496359545
13 9 0.000000000000 5.878590871526
14 9 0.000000000000 7.095620345314
15 9 0.012775932517 8.395278193516
16 9 0.007106798058 9.773053588817
17 9 0.003027419967 11.240739459187
18 9 0.000000000000 12.798234561358
19 9 0.257258322189 15.263478529392
20 9 0.111111111111 17.593240554829
21 9 0.258507690180 20.297008313144
22 9 0.166666666667 22.864609275713
23 9 0.229039159821 25.789473313319
24 9 0.199721290558 28.872921813399
25 9 0.212132640324 32.115463810975
26 9 0.206509335328 35.473130679828
27 9 0.198334052430 38.887407932809
28 9 0.214382661790 42.499743925059
29 9 0.240248258060 46.278051044424
30 9 0.217964451139 50.210331998887
31 9 0.187632965309 54.104706033351
32 9 0.174534834998 58.038596738587
33 9 0.292250428896 63.215145366466
34 9 0.259504299054 67.870002395255
35 9 0.233064313715 72.886703301151
36 9 0.341787893824 78.597526548002
37 9 0.308060808616 84.324342475942
38 9 0.270607474968 90.166569901063
39 9 0.279150278307 96.315076336038
40 9 0.295902333299 102.572683170721
41 9 0.316714277680 108.916582690018
42 9 0.274988390151 115.262689467483
43 9 0.288325903594 121.799342807885
44 9 0.320851877867 128.626383702419
45 9 0.298044498268 135.238613163019
46 9 0.331548113690 142.791930850563
47 9 0.313398413468 150.411927840206
48 9 0.298045427728 158.133351940581
49 9 0.353872550919 166.386296788474
50 9 0.330945382063 174.790627959225
51 9 0.356440171314 183.317559618489
52 9 0.372235521082 192.194207475630
53 9 0.374360640294 201.227140680542
54 9 0.351238030531 210.170042989278
55 9 0.339421255174 219.716508083892
56 9 0.353798719996 230.149462252315
57 9 0.380321350807 240.181683317350
58 9 0.364655714255 250.599412588978
59 9 0.361893088614 260.959505082842
60 9 0.455309689300 272.056666460490
61 9 0.350588126155 283.058574976551
62 9 0.363025308952 293.420140775860
63 9 0.379760704482 304.506358210737
64 9 0.388447390625 316.393496743640
2 10 -1.000000000000 0.003906250000
3 10 -0.500000000000 0.037037037037
4 10 -0.333333333333 0.118652343750
5 10 -0.250000000000 0.256000000000
6 10 -0.200000000000 0.452112268519
7 10 -0.166666666667 0.708454810496
8 10 -0.142857142857 1.025817871094
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