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Sphere packing

This table gives the best packing densities known for congruent spheres in Euclidean spaces of dimensions 1 through 48 and 56, 64, and 72, along with the best upper bounds known for the optimal packing density (the columns are explained below the table). It was originally based on Tables I.1(a) and I.1(b) in [10] and the Nebe-Sloane Catalogue of Lattices. Please let me know of any improvements that should be incorporated.

A plot of the data is available here. The data is archived here, along with explicit coordinates in up to 38 dimensions.

n Center density # Density Upper bound Ratio References
1 2−1 1 1
2 2−1·3−1/2 0.9068996821171089… 1 [26, 27, 12]
3 2−5/2 0.7404804896930610… 1 [13, 14]
4 2−3 0.6168502750680849… 0.6361073321551329 1.032 [15, 7]
5 2−7/2 0.4652576133092586… 0.5126451306253027 1.102 [16, 7]
6 2−3·3−1/2 0.3729475455820649… 0.4103032818801865 1.101 [16, 7]
7 2−4 0.2952978731457125… 0.3211471056675559 1.088 [16, 7]
8 2−4 0.2536695079010480… 1 [16, 30]
9 2−9/2 0.1457748758081711… 0.1911204152968963 1.312 [5, 7]
10 2−7·5 40 0.0996157828077088… 0.1434100871082547 1.440 [3, 7]
11 2−8·32 72 0.0662380270098011… 0.1067252934567631 1.612 [20, 7]
12 3−3 0.0494541766242440… 0.0797117710668987 1.612 [11, 7]
13 2−8·32 72 0.0320142921603497… 0.0601644380983860 1.880 [20, 7]
14 2−4·3−1/2 0.0216240960824471… 0.0450612211935181 2.084 [18, 7]
15 2−9/2 0.0168575706567626… 0.0337564432797899 2.003 [2, 7]
16 2−4 0.0147081643974308… 0.0249944093845237 1.700 [2, 7]
17 2−4 0.0088113191823211… 0.0184640903350649 2.096 [18, 1]
18 2−18·39 512 0.0061678981253312… 0.0134853404450862 2.187 [4, 1]
19 2−7/2 0.0041208062797686… 0.0098179551395438 2.383 [18, 1]
20 2−31·710 4 0.0033945814107126… 0.0071270536033763 2.100 [28, 1]
21 2−5/2 0.0024658847115024… 0.0051596603948176 2.093 [18, 1]
22 2−23·3−21/2·1111 3 0.0024510340441211… 0.0037259419689206 1.521 [9, 1]
23 2−1 0.0019053281934260… 0.0026842798864291 1.409 [19, 1]
24 1 0.0019295743094039… 1 [19, 6]
25 2−1/2 0.0006772120097731… 0.0013841907222857 2.044 [8, 1]
26 3−1/2 0.0002692200504338… 0.0009910238892216 3.682 [8, 1]
27 2−1/2 16384 0.0001575943907278… 0.0007082297958617 4.495 [29, 1]
28 1 8192 0.0001046381049248… 0.0005052542161057 4.829 [29, 1]
29 2−1/2 32768 0.0000341446469074… 0.0003598581852089 10.54 [29, 1]
30 1 8192 0.0000219153534478… 0.0002559028743732 11.68 [29, 1]
31 2−47/2·315 0.0000118377651859… 0.0001817083813917 15.35 [24, 1]
32 2−24·316 0.0000110407493088… 0.0001288432887595 11.67 [24, 1]
33 2−25·333/2 0.0000041406882896… 0.0000912356039023 22.04 Elkies (see [10, p. xx]), [1]
34 2−25·333/2 0.0000017669738891… 0.0000645221967438 36.52 Elkies (see [10, p. xx]), [1]
35 23/2 0.0000009461904151… 0.0000455743843107 48.17 [10, p. 234], [1]
36 218·3−10 0.0000006161466094… 0.0000321530553313 52.19 [17, 1]
37 25/2 0.0000003213562007… 0.0000226586900106 70.51 [10, p. xxxvii], [1]
38 23 0.0000001835874319… 0.0000159506499105 86.89 [10, p. xxxvii], [1]
39 2−41/2·316·7−1/2 0.0000001004160423… 0.0000112168687009 111.8 [8, Cor. 8], [1]
40 2−45/2·317 0.0000000784800488… 0.0000078801051697 100.5 [8, Cor. 8], [1]
41 2−43/2·317 0.0000000610716131… 0.0000055306464395 90.57 [8, Cor. 8], [1]
42 2−22·318 0.0000000498110957… 0.0000038780970907 77.86 [8, Cor. 8], [1]
43 2−45/2·319 0.0000000401571902… 0.0000027169074727 67.66 [8, Cor. 8], [1]
44 2−43·3−24·1722 4 0.0000000364088490… 0.0000019017703144 52.24 [9, 1]
45 2−44·3−24·1745/2 4 0.0000000278915432… 0.0000013300905665 47.69 [9, 1]
46 3−93/2·1323 3 0.0000000286095747… 0.0000009295151556 32.49 [9, 1]
47 2−70·3−24·547/2·747/2 2 0.0000000221448620… 0.0000006490757338 29.32 [9, 1]
48 2−24·324 0.0000000231782953… 0.0000004529067791 19.55 [20, 1]
56 2-28·528·29-4 0.0000000000535530… 0.0000000248393710 463.9 [25, 1]
64 316 0.0000000000013260… 0.0000000013129981 990.2 [21, 22, 1]
72 236 0.0000000000001458… 0.0000000000673580 461.8 [23, 1]
n Number of dimensions.
Center density Number of spheres per unit volume in space, assuming unit radius.
# Number of spheres per unit cell in an underlying (Bravais) lattice; omitted if 1.
Density Packing density, i.e., πn/2/Γ(n/2+1) times the center density.
Upper bound Best upper bound known for the optimal packing density, rounded up.
Ratio Ratio of the upper bound to the known density, rounded up.

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