Harmonic energy

How should we arrange N points x1,…,xn on the unit sphere Sn-1 in n dimensions so as to minimize harmonic energy? Harmonic energy is the sum of 1/|xi - xj|n – 2 over 1 ≤ i < jN. When n=3, it agrees with Coulomb energy, and the optimization problem is the Thomson problem; in higher dimensions it is a natural generalization.

The table below shows the best energy known for 3 ≤ n ≤ 32 and n+1 ≤ N ≤ 64, as well as some larger cases. (See also this list of highlights from the larger table.) It includes each configuration's harmonic energy, the cosine of its minimal angle, its strength as a spherical design, and the number of nonzero distances between points in the configuration. Please let me know if you can improve on any of these records.

This data is from the paper Experimental study of energy-minimizing point configurations on spheres by Brandon Ballinger, Grigoriy Blekherman, Henry Cohn, Noah Giansiracusa, Elizabeth Kelly, and Achill Schürmann (Experiment. Math. 18 (2009), 257-283), together with our 2021 update to the data (which includes coordinates for all these configurations). If you cite this data, please refer to the publication or the 2021 data set, not this web page.

N n Harmonic energy Cosine Design Distances
4 3 3.6742346142 -0.3333333333 2 1
5 3 6.4746914947 0 1 3
6 3 9.9852813743 0 3 2
7 3 14.4529774143 0.3090169944 1 4
8 3 19.6752878613 0.3140893679 1 4
9 3 25.7599865313 0.3552741671 1 6
10 3 32.7169494602 0.4226874241 1 7
11 3 40.5964505082 0.5219097665 0 20
12 3 49.1652530577 0.4472135955 5 3
13 3 58.8532306118 0.6112932938 0 28
14 3 69.3063632967 0.6036799145 1 9
15 3 80.6702441143 0.6530918658 1 21
16 3 92.9116553026 0.6568987372 2 12
17 3 106.0504048287 0.6413415741 1 14
18 3 120.0844674475 0.6751471685 1 18
19 3 135.0894675567 0.7082202859 0 52
20 3 150.8815683338 0.6934858853 1 26
21 3 167.6416223993 0.7154427531 0 69
22 3 185.2875361494 0.7277487399 2 17
23 3 203.9301906629 0.7491741098 1 48
24 3 223.3470740519 0.7423816826 3 16
25 3 243.8127602988 0.7703966339 0 160
26 3 265.1333263174 0.7788766002 0 169
27 3 287.3026150331 0.7667138941 1 30
28 3 310.4915423583 0.7899009402 2 35
29 3 334.6344399205 0.8049835832 1 75
30 3 359.6039459038 0.7992402421 1 120
31 3 385.5308380633 0.8051721384 0 94
32 3 412.2612746506 0.7946544723 5 11
33 3 440.2040574477 0.8319551724 0 281
34 3 468.9048532814 0.8360617694 1 153
35 3 498.5698724907 0.8377158534 0 306
36 3 529.1224083755 0.8364850862 1 171
37 3 560.6188877311 0.8449591948 1 52
38 3 593.0385035665 0.8364153906 1 46
39 3 626.3890090169 0.8475580099 1 79
40 3 660.6752788347 0.8488208416 2 46
41 3 695.9167443419 0.8523862772 1 89
42 3 732.0781075437 0.8549594600 1 63
43 3 769.1908464592 0.8583632445 0 254
44 3 807.1742630847 0.8548429936 3 30
45 3 846.1884010611 0.8642115377 1 176
46 3 886.1671136392 0.8678498575 2 92
47 3 927.0592706798 0.8764131754 0 561
48 3 968.7134553438 0.8687209451 3 56
49 3 1011.5571826536 0.8797596437 0 392
50 3 1055.1823147263 0.8770506050 1 75
51 3 1099.8192903189 0.8815885680 1 225
52 3 1145.4189643193 0.8856378701 0 442
53 3 1191.9222904163 0.8899189550 0 374
54 3 1239.3614747292 0.8907719041 0 729
55 3 1287.7727207828 0.8940363852 0 756
56 3 1337.0949452757 0.8935056059 1 406
57 3 1387.3832292529 0.8933522496 1 280
58 3 1438.6182506405 0.8976030101 1 435
59 3 1490.7733352787 0.8974873659 0 870
60 3 1543.8304009764 0.8991180318 1 310
61 3 1597.9418301990 0.9033979359 0 1830
62 3 1652.9094098984 0.8997110311 1 205
63 3 1708.8796815033 0.9044048513 1 341
64 3 1765.8025779274 0.9068967423 1 528
65 3 1823.6679602639 0.9097675820 0 1056
66 3 1882.4415253043 0.9080350212 0 1089
67 3 1942.1227004056 0.9083091438 1 238
68 3 2002.8747017488 0.9104459843 1 595
69 3 2064.5334832348 0.9125737218 1 408
70 3 2127.1009015507 0.9114694689 1 330
71 3 2190.6499064258 0.9149414646 0 1260
72 3 2255.0011909750 0.9100212740 5 52
73 3 2320.6338837454 0.9217928766 0 1332
74 3 2387.0729818384 0.9207372414 0 1369
75 3 2454.3696890397 0.9222924735 1 481
76 3 2522.6748718414 0.9212824416 0 1444
77 3 2591.8501523539 0.9185419982 1 312
78 3 2662.0464745664 0.9175718514 3 138
79 3 2733.2483574788 0.9229675322 0 1585
80 3 2805.3558759813 0.9220094296 1 235
81 3 2878.5228296642 0.9278895083 0 1640
82 3 2952.5696752865 0.9258313530 1 861
83 3 3027.5284889211 0.9294789034 0 1722
84 3 3103.4651244308 0.9303364549 0 1764
85 3 3180.3614429387 0.9304330355 0 1806
86 3 3258.2116057129 0.9302790222 0 1849
87 3 3337.0007500145 0.9306956628 0 1892
88 3 3416.7201967584 0.9305098191 1 990
89 3 3497.4390186247 0.9324360728 0 1980
90 3 3579.0912227229 0.9321324624 1 690
91 3 3661.7136993201 0.9329242179 0 2070
92 3 3745.2916362408 0.9334187892 1 1081
93 3 3829.8443384215 0.9351306131 0 2162
94 3 3915.3092696204 0.9338811839 1 1128
95 3 4001.7716755651 0.9353747914 0 2256
96 3 4089.1540100556 0.9355268075 0 2304
97 3 4177.5335996222 0.9369797135 0 2352
98 3 4266.8224641563 0.9371505031 0 2401
99 3 4357.1391631319 0.9379829133 0 2450
100 3 4448.3506343313 0.9379084810 2 425
101 3 4540.5900516944 0.9396260298 1 867
102 3 4633.7365658987 0.9394538986 1 884
103 3 4727.8366168331 0.9402463343 0 2652
104 3 4822.8765227487 0.9399513995 1 476
105 3 4919.0006376158 0.9406331580 1 936
106 3 5015.9845957048 0.9417157417 1 1431
107 3 5113.9535477139 0.9436459335 0 2862
108 3 5212.8135078307 0.9436471516 0 2916
109 3 5312.7350799203 0.9449342228 0 2970
110 3 5413.5492941925 0.9427836735 1 532
111 3 5515.2932145867 0.9440591732 1 1045
112 3 5618.0448823266 0.9435035366 1 650
113 3 5721.8249780271 0.9456461172 1 1083
114 3 5826.5215721628 0.9464480252 0 3249
115 3 5932.1812857773 0.9485543497 0 2185
116 3 6038.8155935786 0.9489532645 0 3364
117 3 6146.3424465787 0.9479567861 0 3422
118 3 6254.8770277897 0.9485744881 0 3481
119 3 6364.3473174793 0.9492254668 0 3540
120 3 6474.7563249797 0.9487782964 0 3656
121 3 6586.1219495841 0.9499791770 0 2420
122 3 6698.3744992606 0.9477035863 5 86
123 3 6811.8272281741 0.9519151951 0 1978
124 3 6926.1699741935 0.9504564887 1 1952
125 3 7041.4732640233 0.9517727113 0 3906
126 3 7157.6692248670 0.9514845350 1 1022
127 3 7274.8195046755 0.9517159196 1 832
128 3 7393.0074430681 0.9520523919 0 4096
129 3 7512.1073192684 0.9524345811 0 4160
130 3 7632.1673789125 0.9527494489 0 4225
131 3 7753.2051669406 0.9536575863 0 4290
132 3 7875.0453427971 0.9512817237 5 161
133 3 7998.1792128985 0.9556239094 0 2926
134 3 8122.0897211943 0.9552059818 0 4489
135 3 8246.9094869920 0.9540763198 1 1541
136 3 8372.7433025387 0.9537937181 2 782
137 3 8499.5344947816 0.9534004808 1 966
138 3 8627.4063898802 0.9566911025 0 4761
139 3 8756.2270569531 0.9579553071 0 4830
140 3 8885.9806090407 0.9574566488 0 9730
141 3 9016.6153491899 0.9564988889 0 2584
142 3 9148.2715799935 0.9571184808 0 5041
143 3 9280.8398511923 0.9574082980 0 5112
144 3 9414.3717944599 0.9565421742 1 2628
145 3 9548.9288372319 0.9571134262 0 5320
146 3 9684.3818255750 0.9567898087 1 2701
147 3 9820.9323783732 0.9590275006 0 5402
148 3 9958.4060042700 0.9581858538 0 5476
149 3 10096.8599073972 0.9595880434 0 11026
150 3 10236.1964367007 0.9592914473 2 950
151 3 10376.5714692746 0.9604729959 0 5700
152 3 10517.8675928783 0.9606978958 1 2925
153 3 10660.0827482366 0.9593639253 1 1976
154 3 10803.3724211411 0.9608861369 0 5929
155 3 10947.5746922786 0.9613106263 0 6006
156 3 11092.7983114564 0.9621152149 0 6084
157 3 11238.9030411562 0.9615102578 0 6162
158 3 11385.9901861972 0.9613242729 0 6241
159 3 11534.0239609559 0.9614095963 0 6320
160 3 11683.0548055495 0.9614481462 1 3240
161 3 11833.0847394647 0.9621716456 0 6480
162 3 11984.0503358137 0.9621570102 1 2214
163 3 12136.0130532204 0.9628081850 0 6642
164 3 12288.9301053199 0.9629034392 1 3403
165 3 12442.8044513732 0.9629209699 0 6806
166 3 12597.6490713234 0.9631295215 1 1792
167 3 12753.4694297496 0.9631645475 0 6972
168 3 12910.2126722677 0.9629058365 1 2380
169 3 13068.0064511273 0.9634553919 0 7189
170 3 13226.6810785406 0.9633085442 1 1892
171 3 13386.3559307172 0.9636442280 1 2465
172 3 13547.0181087873 0.9645922491 0 3822
173 3 13708.6352430344 0.9649036678 0 7563
174 3 13871.1870922922 0.9642522357 1 3828
175 3 14034.7813069294 0.9647803677 0 7656
176 3 14199.3547756318 0.9654658384 0 15400
177 3 14364.8375452983 0.9646995172 1 1602
178 3 14531.3095525878 0.9652691876 1 4005
179 3 14698.7545942201 0.9660698799 0 15931
180 3 14867.0999275255 0.9656217910 1 4095
181 3 15036.4672397689 0.9659180197 0 8190
182 3 15206.7306109061 0.9652218245 1 1693
183 3 15378.1665710284 0.9670610603 0 16653
184 3 15550.4214503113 0.9662330266 2 1426
185 3 15723.7200740718 0.9669340617 0 8556
186 3 15897.8974370479 0.9670947451 0 17205
187 3 16072.9751863203 0.9666091107 1 1786
188 3 16249.2226788788 0.9670921506 1 4465
189 3 16426.3719388643 0.9673978737 0 8930
190 3 16604.4283385006 0.9681431761 0 5985
191 3 16783.4522193625 0.9694682339 0 18145
192 3 16963.3383864605 0.9667396655 5 330
193 3 17144.5647408799 0.9696829895 0 9312
194 3 17326.6161364711 0.9687980330 0 18721
195 3 17509.4893039301 0.9686914370 1 3201
196 3 17693.4605480823 0.9692268548 0 9604
197 3 17878.3401625712 0.9696733428 1 1980
198 3 18064.2621771951 0.9692849568 0 9801
199 3 18251.0824956400 0.9696460124 0 19701
200 3 18438.8427175305 0.9693521000 1 5050
201 3 18627.5912262442 0.9710086372 0 20100
202 3 18817.2047182617 0.9694918449 1 2081
203 3 19007.9812045802 0.9703933126 0 10402
204 3 19199.5407756026 0.9690554219 3 905
205 3 19392.3691523878 0.9706742585 0 20910
206 3 19585.9558565493 0.9705583907 0 10609
207 3 19780.6569093142 0.9704749433 0 10712
208 3 19976.2032612031 0.9709213718 0 10816
209 3 20172.7546806609 0.9709412482 0 10920
210 3 20370.2516151293 0.9711126112 0 11025
211 3 20568.7406027764 0.9722939290 0 22155
212 3 20768.0530859643 0.9697964114 5 398
213 3 20968.6120254912 0.9718312886 0 22578
214 3 21169.9104103755 0.9712558030 2 1926
215 3 21372.3487893407 0.9721119402 0 23005
216 3 21575.5963778695 0.9714023771 1 3924
217 3 21779.8560804182 0.9707090039 1 2398
218 3 21985.2639489209 0.9721240672 0 11881
219 3 22191.4854748141 0.9724498093 0 11990
220 3 22398.6556025313 0.9719516400 1 6105
221 3 22606.8815472595 0.9725158408 0 12210
222 3 22816.0255702494 0.9723613177 1 4144
223 3 23026.1658811605 0.9724158546 0 12432
224 3 23237.2448734867 0.9719510270 1 6328
225 3 23449.4364606675 0.9735245893 0 25200
226 3 23662.5111226545 0.9728657687 0 8475
227 3 23876.5768937175 0.9729538382 0 12882
228 3 24091.5789858671 0.9732740102 0 12996
229 3 24307.5993132899 0.9737882948 0 26106
230 3 24524.4853509449 0.9733010038 0 13225
231 3 24742.3824947679 0.9737188144 0 13340
232 3 24961.2523189337 0.9733376336 2 2262
233 3 25181.0573995305 0.9730643287 1 4563
234 3 25401.9317866406 0.9738374508 0 13689
235 3 25623.7631442005 0.9740874155 0 27495
236 3 25846.5005631521 0.9737210790 0 13924
237 3 26070.3670154586 0.9742620443 0 27966
238 3 26295.1260479035 0.9744395952 0 14161
239 3 26520.8741174416 0.9749126094 0 28441
240 3 26747.5082050910 0.9744840201 0 14400
241 3 26975.1902839780 0.9749260763 0 9640
242 3 27203.7992855535 0.9747953491 1 4921
243 3 27433.3673523758 0.9749932216 0 14762
244 3 27663.9051121809 0.9746224810 1 7503
245 3 27895.5407453915 0.9753656991 0 29890
246 3 28128.0514643189 0.9750579324 1 7626
247 3 28361.5327772629 0.9755471667 0 15252
248 3 28596.0521029476 0.9754319356 0 15376
249 3 28831.4739098513 0.9754963005 0 15500
250 3 29067.8776739124 0.9759827986 0 15625
251 3 29305.2338616249 0.9763183796 0 31375
252 3 29543.5228681249 0.9756973941 0 15876
253 3 29782.9173643951 0.9759493700 0 31878
254 3 30023.2228618391 0.9769055563 0 32131
255 3 30264.4242512812 0.9760491686 1 5461
256 3 30506.6875158469 0.9760225377 2 2752
5 4 4 -0.25 2 1
6 4 6.5 0 2 2
6 4 6.5 0 1 3
7 4 9.5 0 1 3
8 4 13 0 3 2
9 4 17.5 0.3090169944 1 4
10 4 22.5 0.1666666667 2 2
10 4 22.5 0.3090169944 3 3
11 4 28.2089565051 0.3296355176 0 34
12 4 34.4236445005 0.2548430485 1 5
13 4 41.6 0.3443007135 3 3
14 4 49.2622993678 0.3392126370 0 16
15 4 57.6166048309 0.3535533906 2 5
16 4 66.9498389934 0.4365245956 0 12
17 4 76.9811118697 0.4765073390 0 24
18 4 87.6161773020 0.4848021156 1 45
19 4 98.9529044726 0.4879747558 0 90
20 4 110.7992682643 0.4416879463 1 9
21 4 123.7366293094 0.5204915200 0 17
22 4 137.5161594413 0.5444604211 1 12
23 4 152.0419944701 0.5752443331 0 39
24 4 167 0.5 5 4
25 4 182.9883825868 0.6016707589 0 58
26 4 199.6924939403 0.5644947525 1 15
27 4 217.1541244039 0.6481541788 0 62
28 4 235.3992121309 0.6423783518 1 56
29 4 254.3753344598 0.6269476772 0 211
30 4 274.1861686291 0.6301434697 0 225
31 4 294.7866181201 0.6512424198 0 81
32 4 315.9918325489 0.6280508544 0 66
33 4 337.7920338789 0.6101294879 0 31
34 4 360.5186081802 0.6281714878 0 78
35 4 384.5441132811 0.6887145573 0 86
36 4 409.0693613224 0.6742667622 0 92
37 4 434.1948255532 0.6741025715 0 95
38 4 460.2828047691 0.6886165096 0 40
39 4 487.2547831586 0.6971716692 0 170
40 4 514.9021531846 0.6943205077 0 259
41 4 543.1601453439 0.6814748790 0 107
42 4 572.1560656817 0.6867371660 1 47
43 4 601.9333550375 0.6783527890 0 130
44 4 632.7293557487 0.7084875563 1 68
45 4 664.4751760014 0.7160700425 2 26
46 4 697.2617498943 0.7351897122 1 91
47 4 730.7532359606 0.7399342714 0 196
48 4 764.5870129871 0.7071067812 7 12
49 4 799.6990134405 0.7261343969 0 601
50 4 835.1223665180 0.7172829458 3 19
51 4 871.9796452526 0.7522980782 0 654
52 4 909.1858614099 0.7367267390 0 676
53 4 947.1456571861 0.7347746613 0 247
54 4 985.8785009744 0.7371357685 1 148
55 4 1025.7628733353 0.7482817650 0 757
56 4 1066.6185698313 0.7617569674 0 784
57 4 1108.1678551609 0.7727492135 0 813
58 4 1150.4335736277 0.7677838334 0 841
59 4 1193.3774567569 0.7655330179 0 871
60 4 1236.9130894215 0.7608480757 1 301
61 4 1281.3804443518 0.7627885047 0 521
62 4 1326.5889513885 0.7624471611 0 159
63 4 1373.0883288822 0.7520181626 1 117
64 4 1420.5853243619 0.7783083439 0 1096
65 4 1468.7140042218 0.7841991627 0 1137
66 4 1517.5954748339 0.7794524727 0 1091
67 4 1567.3162668630 0.7885516871 0 2211
68 4 1617.7464131665 0.7733479716 0 233
69 4 1668.8990867939 0.7799037045 0 414
70 4 1721.2129351193 0.7823928708 0 1227
71 4 1774.1239965549 0.7826206815 0 2485
72 4 1827.9306630377 0.7932013508 0 1296
73 4 1882.3796215592 0.7841898530 0 669
74 4 1937.6551068168 0.7905259369 0 748
75 4 1993.5338546768 0.7920300760 0 822
76 4 2050.2197263079 0.7897404853 1 741
77 4 2107.8978840999 0.7852946880 0 610
78 4 2166.0631727030 0.7827489200 1 40
79 4 2226.5142597252 0.8358663558 0 344
80 4 2287.4050587028 0.8131433317 0 206
81 4 2349.1287171527 0.8097204415 0 217
82 4 2411.7796137465 0.8301071668 0 1813
83 4 2474.9383165382 0.8265423611 0 954
84 4 2538.6935982669 0.8245295563 0 380
85 4 2603.8288480969 0.8299954204 0 650
86 4 2669.2946723658 0.8193134882 0 1851
87 4 2735.6421700051 0.8209108973 0 1893
88 4 2802.5166786256 0.8037838643 0 990
89 4 2870.5051007666 0.8186730952 0 429
90 4 2939.7808998581 0.8228657965 0 2027
91 4 3009.5538897899 0.8251410366 0 1059
92 4 3080.1456050203 0.8263158150 0 2116
93 4 3151.3414337247 0.8254958580 1 377
94 4 3223.5098474403 0.8181678366 0 337
95 4 3296.8013698614 0.8249908499 0 772
96 4 3370.9382529401 0.8370773311 0 2304
97 4 3445.6037533871 0.8302871490 0 2353
98 4 3520.7687660554 0.8359639780 0 2403
99 4 3597.1986621937 0.8296772474 0 427
100 4 3674.9697079388 0.8331254016 0 2502
101 4 3753.1657957683 0.8284429684 0 441
102 4 3832.0965209456 0.8364325593 0 1328
103 4 3912.0616266711 0.8383398677 0 909
104 4 3992.8656900646 0.8302344930 0 1078
105 4 4074.1709009041 0.8325935383 0 590
106 4 4156.8297234717 0.8386643417 0 2811
107 4 4240.1626541036 0.8428101010 0 2863
108 4 4324.0100589909 0.8399220195 1 124
109 4 4408.8746255222 0.8389820527 0 631
110 4 4494.7800785232 0.8402398102 0 3027
111 4 4581.3889262609 0.8389754423 0 3084
112 4 4668.8130814016 0.8430920222 0 3138
113 4 4756.8265393942 0.8315616574 0 333
114 4 4845.7989632950 0.8363646500 0 469
115 4 4935.9681504579 0.8469092938 0 3475
116 4 5026.3674918008 0.8373051436 0 395
117 4 5117.9234521644 0.8428969983 0 1938
118 4 5210.1704662379 0.8471289311 0 750
119 4 5302.6162620633 0.8309296975 0 133
120 4 5395 0.8090169944 11 8
121 4 5491.9336100885 0.8681409040 0 446
122 4 5589.2787034083 0.8714855262 0 3931
123 4 5687.0850705340 0.8676754315 0 3786
124 4 5785.6226415451 0.8686937729 0 3846
125 4 5884.9584686256 0.8690039620 0 3907
126 4 5984.7137685374 0.8641339210 0 843
127 4 6085.6740488876 0.8708587665 0 4033
128 4 6187.0713830866 0.8658650541 0 4098
6 5 4.0343576523 -0.2 2 1
7 5 6.1903560341 0 1 3
8 5 8.6989057922 0 1 3
9 5 11.5589519870 0 1 3
10 5 14.7671356238 0 3 2
11 5 19.0272173821 0.3008186797 0 14
12 5 23.6035593670 0.1666666667 1 4
13 5 28.6727458019 0.2955872642 0 20
14 5 34.2864718550 0.2845091753 0 22
15 5 40.2649665922 0.3031887743 0 15
16 5 46.5161831818 0.2 3 2
17 5 54.0569375493 0.3370104941 0 8
18 5 61.9684878601 0.3730704946 0 16
19 5 70.2463973255 0.3360427021 0 21
20 5 78.8184068742 0.3115805327 1 7
21 5 87.9576942620 0.3162277661 2 5
22 5 98.5439193226 0.4463738782 0 16
23 5 109.5192985890 0.4555063136 0 70
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