clust.3.4 contains the coords of a putatively minimal-energy cluster of 4 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.5 contains the coords of a putatively minimal-energy cluster of 5 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.6 contains the coords of a putatively minimal-energy cluster of 6 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.7 contains the coords of a putatively minimal-energy cluster of 7 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.8 contains the coords of a putatively minimal-energy cluster of 8 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.9 contains the coords of a putatively minimal-energy cluster of 9 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.10 contains the coords of a putatively minimal-energy cluster of 10 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.11 contains the coords of a putatively minimal-energy cluster of 11 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.12 contains the coords of a putatively minimal-energy cluster of 12 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.13 contains the coords of a putatively minimal-energy cluster of 13 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.14 contains the coords of a putatively minimal-energy cluster of 14 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.15 contains the coords of a putatively minimal-energy cluster of 15 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

Added April 26 2021: Miranda Holmes-Cerfon and Yair Shenfeld have informed us that there is a typo in one of our coordinates. They kindly provided an alternative set of coordinates for 15 points with the same second moment as our points, and whose graph looks essentially the same as ours. Here are their coordinates:

COORDINATES (N=15) 0 0.0000 0.0000 2.0000 -0.0000 -0.0000 -2.0000 0.0044 -0.0000 1.0000 1.3814 -1.0448 -0.9970 1.3836 1.0448 1.0000 1.4456 0.9541 -0.9968 1.4477 -0.9541 1.0000 -0.5246 -1.6507 -1.0011 -0.5224 1.6507 1.0000 -0.4177 1.6809 -1.0009 -0.4155 -1.6809 1.0000 -1.7312 -0.0556 -1.0038 -1.7290 0.0556 -0.0947 -2.1984 -1.6629 0.0899 -2.1986 1.6629For more information | Reference | Return to my home page

clust.3.16 contains the coords of a putatively minimal-energy cluster of 16 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.17 contains the coords of a putatively minimal-energy cluster of 17 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.18 contains the coords of a putatively minimal-energy cluster of 18 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.19 contains the coords of a putatively minimal-energy cluster of 19 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.20 contains the coords of a putatively minimal-energy cluster of 20 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.21 contains the coords of a putatively minimal-energy cluster of 21 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.22 contains the coords of a putatively minimal-energy cluster of 22 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.23 contains the coords of a putatively minimal-energy cluster of 23 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.24 contains the coords of a putatively minimal-energy cluster of 24 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.25 contains the coords of a putatively minimal-energy cluster of 25 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.26 contains the coords of a putatively minimal-energy cluster of 26 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.27 contains the coords of a putatively minimal-energy cluster of 27 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.28 contains the coords of a putatively minimal-energy cluster of 28 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.29 contains the coords of a putatively minimal-energy cluster of 29 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.30 contains the coords of a putatively minimal-energy cluster of 30 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.31 contains the coords of a putatively minimal-energy cluster of 31 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.32 contains the coords of a putatively minimal-energy cluster of 32 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.33 contains the coords of a putatively minimal-energy cluster of 33 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.34 contains the coords of a putatively minimal-energy cluster of 34 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.35 contains the coords of a putatively minimal-energy cluster of 35 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.36 contains the coords of a putatively minimal-energy cluster of 36 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.37 contains the coords of a putatively minimal-energy cluster of 37 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.38 contains the coords of a putatively minimal-energy cluster of 38 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.39 contains the coords of a putatively minimal-energy cluster of 39 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.40 contains the coords of a putatively minimal-energy cluster of 40 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.41 contains the coords of a putatively minimal-energy cluster of 41 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.42 contains the coords of a putatively minimal-energy cluster of 42 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.43 contains the coords of a putatively minimal-energy cluster of 43 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.44 contains the coords of a putatively minimal-energy cluster of 44 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.45 contains the coords of a putatively minimal-energy cluster of 45 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.46 contains the coords of a putatively minimal-energy cluster of 46 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.47 contains the coords of a putatively minimal-energy cluster of 47 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.48 contains the coords of a putatively minimal-energy cluster of 48 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.49 contains the coords of a putatively minimal-energy cluster of 49 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.50 contains the coords of a putatively minimal-energy cluster of 50 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.51 contains the coords of a putatively minimal-energy cluster of 51 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.52 contains the coords of a putatively minimal-energy cluster of 52 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.53 contains the coords of a putatively minimal-energy cluster of 53 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.54 contains the coords of a putatively minimal-energy cluster of 54 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.55 contains the coords of a putatively minimal-energy cluster of 55 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.56 contains the coords of a putatively minimal-energy cluster of 56 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.57 contains the coords of a putatively minimal-energy cluster of 57 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.58 contains the coords of a putatively minimal-energy cluster of 58 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.59 contains the coords of a putatively minimal-energy cluster of 59 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.60 contains the coords of a putatively minimal-energy cluster of 60 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.61 contains the coords of a putatively minimal-energy cluster of 61 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.62 contains the coords of a putatively minimal-energy cluster of 62 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.63 contains the coords of a putatively minimal-energy cluster of 63 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.64 contains the coords of a putatively minimal-energy cluster of 64 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.65 contains the coords of a putatively minimal-energy cluster of 65 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.66 contains the coords of a putatively minimal-energy cluster of 66 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.67 contains the coords of a putatively minimal-energy cluster of 67 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.68 contains the coords of a putatively minimal-energy cluster of 68 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.69 contains the coords of a putatively minimal-energy cluster of 69 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.70 contains the coords of a putatively minimal-energy cluster of 70 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.71 contains the coords of a putatively minimal-energy cluster of 71 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.72 contains the coords of a putatively minimal-energy cluster of 72 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.73 contains the coords of a putatively minimal-energy cluster of 73 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.74 contains the coords of a putatively minimal-energy cluster of 74 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.75 contains the coords of a putatively minimal-energy cluster of 75 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.76 contains the coords of a putatively minimal-energy cluster of 76 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.77 contains the coords of a putatively minimal-energy cluster of 77 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.78 contains the coords of a putatively minimal-energy cluster of 78 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.79 contains the coords of a putatively minimal-energy cluster of 79 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.80 contains the coords of a putatively minimal-energy cluster of 80 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.81 contains the coords of a putatively minimal-energy cluster of 81 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.82 contains the coords of a putatively minimal-energy cluster of 82 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.83 contains the coords of a putatively minimal-energy cluster of 83 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.84 contains the coords of a putatively minimal-energy cluster of 84 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.85 contains the coords of a putatively minimal-energy cluster of 85 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.86 contains the coords of a putatively minimal-energy cluster of 86 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.87 contains the coords of a putatively minimal-energy cluster of 87 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.88 contains the coords of a putatively minimal-energy cluster of 88 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.89 contains the coords of a putatively minimal-energy cluster of 89 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.90 contains the coords of a putatively minimal-energy cluster of 90 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.91 contains the coords of a putatively minimal-energy cluster of 91 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.92 contains the coords of a putatively minimal-energy cluster of 92 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.93 contains the coords of a putatively minimal-energy cluster of 93 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.94 contains the coords of a putatively minimal-energy cluster of 94 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.95 contains the coords of a putatively minimal-energy cluster of 95 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.96 contains the coords of a putatively minimal-energy cluster of 96 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.97 contains the coords of a putatively minimal-energy cluster of 97 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.98 contains the coords of a putatively minimal-energy cluster of 98 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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clust.3.99 contains the coords of a putatively minimal-energy cluster of 99 equal nonoverlapping hard spheres, in the sense of minimizing the second moment about the centroid.

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