Minimal-Energy Clusters

N. J. A. Sloane, R. H. Hardin, T. S. Duff and J. H. Conway

A library of putatively minimal-energy clusters of n equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid, in 3 and 4 dimensions.

Reference:
Minimal-Energy Clusters of Hard Spheres, N. J. A. Sloane, R. H. Hardin, T. S. Duff and J. H. Conway, Discrete Computational Geom., 14 (1995), pp. 237-259. [Note: Fig. 11 , Fig. 12 and "photos" of the putatively optimal clusters of 4 to 10 and 13 to 20 spheres are in separate files]
Go to library of 3-d clusters, library of 4-d clusters
If you use any of these arrangements, please acknowledge this source.
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Last modified May 30 1997.

Summary of results:


dim  npts    moment

  3    4    6.0000000
  3    5    9.3333333
  3    6   12.0000000
  3    7   16.6832816
  3    8   21.1566742
  3    9   25.8989795
  3   10   31.8278963
  3   11   37.8346427
  3   12   42.8162588
  3   13   47.7012169
  3   14   54.8783069
  3   15   62.1070790
  3   16   69.7925795
  3   17   78.1282260
  3   18   86.3012403
  3   19   95.1283745
  3   20  105.0434486
  3   21  114.2222222
  3   22  122.4848485
  3   23  131.7681159
  3   24  141.2777778
  3   25  151.6266667
  3   26  161.3333333
  3   27  172.8888889
  3   28  183.7619048
  3   29  193.4559387
  3   30  205.7135802
  3   31  217.3094385
  3   32  229.3750000
  3   33  240.8305275
  3   34  253.1764706
  3   35  265.6380952
  3   36  277.8518519
  3   37  289.7297297
  3   38  300.0000000
  3   39  317.5384615
  3   40  334.3000000
  3   41  349.2032520
  3   42  364.0952381
  3   43  378.5116279
  3   44  392.6060606
  3   45  408.7111111
  3   46  422.9565217
  3   47  440.0000000
  3   48  456.4166667
  3   49  473.1700680
  3   50  489.3333333
  3   51  504.8714597
  3   52  520.6495727
  3   53  537.0398323
  3   54  553.6378601
  3   55  568.8969697
  3   56  586.1904762
  3   57  603.1578947
  3   58  619.7471264
  3   59  634.6666667
  3   60  656.0814815
  3   61  676.8597450
  3   62  697.9139785
  3   63  717.1428571
  3   64  735.8333333
  3   65  757.0666667
  3   66  778.0942761
  3   67  799.2968491
  3   68  820.2360203
  3   69  840.7149759
  3   70  861.9492064
  3   71  882.2159624
  3   72  902.6666667
  3   73  924.1095890
  3   74  944.1621622
  3   75  964.1600000
  3   76  986.3157895
  3   77 1008.1038961
  3   78 1031.0978158
  3   79 1052.7594937
  3   80 1073.3333333
  3   81 1094.4032922
  3   82 1115.7398374
  3   83 1137.3333333
  3   84 1161.0476190
  3   85 1185.3333333
  3   86 1210.3720930
  3   87 1235.0191571
  3   88 1260.1969697
  3   89 1285.7078652
  3   90 1309.6711284
  3   91 1334.9368293
  3   92 1360.2940080
  3   93 1385.5592027
  3   94 1411.6933350
  3   95 1436.7937622
  3   96 1461.1296297
  3   97 1485.0624921
  3   98 1508.8052406
  3   99 1532.2299539
  4    5    8.0000000
  4    6   11.0000000
  4    7   13.7142857
  4    8   16.0000000
  4    9   20.6075097
  4   10   24.0000000
  4   11   27.8080494
  4   12   31.5497035
  4   13   36.0832338
  4   14   40.2135514
  4   15   44.5974257
  4   16   49.5232044
  4   17   55.1043519
  4   18   60.6925987
  4   19   65.9112022
  4   20   69.9411255
  4   21   76.8885860
  4   22   82.8375729
  4   23   87.4782609
  4   24   91.8333333
  4   25   96.0000000
  4   26  103.6923077
  4   27  111.1111111
  4   28  118.2462885
  4   29  124.7058825
  4   30  131.1804110
  4   31  138.6265097
  4   32  145.0449556

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