Sphere packings of dimension 3

Introduction

The optimal 3-dimensional (3d) lattice in terms of packing density is the face-centered cubic (fcc) lattice. Therefore, spherical subsets of this lattice are asymptotically optimal as the size tends to infinity. The fcc lattice is not unique in this sense: Infinitely other infinite packings exist with the same density as the fcc. The most well-known of these nonlattice packings is the hexagonal close-packing (hcp). The best known packings at finite sizes are, with a few exceptions (M = 1, 2, 3, 4, 6, 37, 38, 39, 40), not lattice subsets.

Conjectured optimal sphere packings were numerically designed in [Sloane95] and described in detail for M ≤ 32, using exact coordinates or contact graphs. The supplementary website [Sloane-web] gives coordinates for the best found packings up to M ≤ 99. These packings are included in this database, in some cases after translation, scaling, and rotation, under the name c3_*.txt.

Subsets of the fcc lattice were designed by enumerating and testing a finite number of possible centroids inside the fundamental simplex [Conway82] of the lattice. This algorithm [Agrell14] straightforwardly generalizes the algorithm in [Chow95] for optimizing 2-dimensional lattice subsets. The obtained packings, which are provenly the optimal fcc subsets although not necessarily the optimal 3-dimensional packings, are included under the name l3_*.txt.

The most power-efficient 3d packing, in the sense of maximizing γ, is the tetrahedron (M = 4). As the size increases, the CFM and γ of good 3d packings both tend to decrease while the gain CFM increases, although these trends are not monotonic. As M → ∞, the gain asymptotically tends to G → 10log10((5/3)(π2/18)1/3) = 1.349 dB, which equals the gain of the fcc lattice over the cubic lattice [Conway99, pp. 73, 112].

Database

FileNMdEEnLatβEbCFM [dB]γ [dB]G [dB]Comment
BPSK3_232210.25Y0.66666717.781510-2.31065
triangle3_3334.2426460.333333Y1.056643.785586.532130.750803-0.914813
tetrahedron3_4342.8284330.375Y1.333331.56.02061.249390.0570721[Gilbert52]. The tetrahedron, one of the five Platonic solids [Coxeter73], [Agrell11a]. Applied in optical communications in [Dochhan13].
l3_5351.414210.960.48Y1.547950.4134494.94850.8254750.00907843A square pyramid.
doublesimplex3_5354.242648.40.466667N1.547953.617685.070840.947820.131423Two tetrahedra sharing the same base. A subset of the hcp.
biortho3_6361.4142110.5Y1.723310.3868534.771211.114240.610616[Gilbert52]. The octahedron, one of the five Platonic solids [Coxeter73], [Agrell11a]. Used in biorthogonal modulation.
l3_6361.4142110.5Y1.723310.3868534.771211.114240.610616Equivalent to biortho3_6.
l3_7379.89949600.612245Y1.8715721.37243.891660.5931220.357831
c3_73722.383330.595831N1.871570.8489584.009680.7111390.475848Two pentagonal pyramids sharing the same base.
OOK3_83811.51.5Y20.50-3.0103-3.0103[Oliver48]. A regular cube with nonnegative coordinates.
cube3_838230.75Y213.010300[Oliver48], [Gilbert52, p. 517], [Viterbi61]. A regular zero-mean cube, one of the five platonic solids [Coxeter73], [Agrell11a].
l3_8385.65685220.6875Y27.333333.388190.3778860.377886
g3_83822.707110.676777N20.9023693.456460.4461590.446159[Gilbert52]
c3_83822.644580.661146N20.8815283.557940.5476380.547638
l3_93912.72791240.765432Y2.1132839.11762.921850.1508230.360533
c3_93922.877660.719416N2.113280.9078023.191110.4200880.629799Three square pyramids, each sharing two points with another.
l3_103107.07107410.82Y2.2146212.34222.622770.05516380.454171
c3_1031023.182790.795697N2.214620.9581152.753430.1858230.58483[Gilbert52]. Two square pyramids placed base-to-base.
l3_1131115.55632140.884298Y2.3062961.85992.29493-0.09653650.475083
c3_11311663745.630.859878N2.306291082.732.416540.0250780.596698
icosa3_123122.3511450.904508N2.389981.394712.19679-0.03988030.690451The icosahedron, one of the five Platonic solids [Coxeter73], [Agrell11a].
l3_1231216.97062620.909722Y2.3899873.08312.17182-0.06484180.665489
c3_1231223.568020.892005N2.389980.9952742.257240.02057150.750902
l3_133131.414211.846150.923077Y2.466960.4989012.108530.009554490.886828[Gilbert52]. All points with even parity in a 3×3×3 cube. Also, the 12 vertices of the cuboctahedron plus a central point. A “kissing” packing [Agrell09].
c3_133131482.2014756.0.917331N2.46696544464.2.135650.03667230.913946
odd3_143141.414212.142861.07143Y2.538240.562821.46128-0.5139990.500122All points with odd parity in a 3x3x3 cube.
l3_143149.89949970.989796Y2.5382425.4771.80546-0.1698230.844298
c3_1431423.919880.97997N2.538241.029551.84879-0.1264930.887628
l3_1531521.21324721.04889Y2.60459120.8121.55362-0.3095820.832624
c3_1531524.140471.03512N2.604591.059791.61101-0.2521860.890021
l3_1631611.31371411.10156Y2.6666735.251.34082-0.4200910.842525
c3_1631624.362041.09051N2.666671.090511.38462-0.3762930.886324
l3_1731724.04166721.16263Y2.72498164.4051.1065-0.5604760.815767
c3_1731724.595781.14894N2.724981.124361.15792-0.5090520.867191
l3_1831812.72791971.21605Y2.7799547.24310.9114-0.668830.815
c3_1831824.794511.19863N2.779951.149780.974067-0.6061630.877667
l3_193191.414212.526321.26316Y2.831950.5947170.746336-0.7534050.832599
c3_1931925.006761.25169N2.831951.178640.785948-0.7137940.87221
dodeca3_203203.8042328.41641.96353N2.881296.57494-1.16945-2.59419-0.910892The dodecahedron, one of the five Platonic solids [Coxeter73], [Agrell11a]. Possibly useful for ornamentation, but weak as a sphere packing.
l3_2032028.284310621.3275Y2.88129245.7240.530567-0.894170.789128
c3_2032025.252171.31304N2.881291.215240.578123-0.8466140.836684
l3_2132129.698512081.36961Y2.92821275.0260.394929-0.9596460.816523
c3_21321422398.671.35979N2.92821546.1050.426199-0.9283760.847794A subset of the hcp.
l3_2232231.112713701.41529Y2.97295307.2140.25246-1.036260.828757
c3_2232225.567491.39187N2.972951.248480.324916-0.9638020.901212A subset of the hcp.
l3_2332332.526915461.46125Y3.01571341.7660.113674-1.113030.837141
c3_23323463030.671.43226N3.01571669.9740.200688-1.026020.924154A subset of the hcp.
l3_243242.82843121.5Y3.056642.617250-1.168150.863797
c3_24324483390.671.47164N3.05664739.5190.0828864-1.085270.946684A subset of the hcp.
l3_2532535.355319321.5456Y3.0959416.034-0.130059-1.242780.867829
c3_25325503790.671.51627N3.0959816.276-0.0468433-1.159570.951045A subset of the hcp.
l3_2632618.38485351.58284Y3.13363113.819-0.233458-1.293590.892806
c3_2632626.205131.55128N3.133631.32012-0.145995-1.206120.980269A subset of the hcp.
l3_2732738.183823781.631Y3.16993500.117-0.363631-1.373740.885757
c3_27327544668.1.60082N3.16993981.727-0.282521-1.292630.966867A subset of the hcp.
l3_283282.8284313.28571.66071Y3.20492.76362-0.442037-1.404490.92563
c3_28328565145.331.64073N3.20491070.3-0.389462-1.351910.978205A subset of the hcp.
l3_2932941.012228801.71225Y3.23865592.839-0.574752-1.491710.906712
c3_2932926.670891.66772N3.238651.37318-0.460328-1.377281.02114
l3_303308.485281261.75Y3.2712625.6782-0.669468-1.542920.921637
c3_30330606171.411.71428N3.271261257.7-0.579905-1.453351.0112
l3_3133143.840634401.7898Y3.3028694.361-0.767138-1.598920.92974
c3_3133127.009981.7525N3.30281.41496-0.675656-1.507441.02122
SP-QAM3_323322.82843151.875Y3.333333-0.9691-1.760910.829944A cubic subset of fcc. Similar to the 4-dimensional SP-QAM packings.
l3_3233222.62749331.82227Y3.33333186.6-0.845204-1.637020.95384
c3_32332576.594540.1.79199N3.33333118908.-0.772449-1.564261.0266
l3_3333346.66940401.85491Y3.36293800.889-0.922322-1.675740.975517
c3_33333594.643740.1.82447N3.36293127615.-0.850463-1.603891.04738
l3_3433424.041610891.88408Y3.39164214.056-0.990088-1.706591.00339
c3_3433427.446371.86159N3.391641.46367-0.937932-1.654431.05554
l3_3533549.497546881.91347Y3.41952913.968-1.0573-1.738251.02885
c3_3533527.589661.89741N3.419521.47967-1.02071-1.701661.06544
l3_3633650.911750221.9375Y3.44662971.387-1.1115-1.758171.06453
c3_3633627.718111.92953N3.446621.49289-1.0936-1.740271.08244
l3_3733752.325953601.95763Y3.472971028.9-1.1564-1.769991.10689
c3_3733727.830531.95763Y3.472971.50314-1.1564-1.769991.10689A lattice packing, identical to l3_37.
l3_383381.414213.947371.97368Y3.498620.752177-1.19186-1.77351.15621The three points (0,0,1), (1,1,1), and (0,1,2) with all sign combinations and all permutations.
c3_3833827.894741.97368Y3.498621.50435-1.19186-1.77351.15621A lattice packing, identical to l3_38.
l3_3933918.38486882.0355Y3.5236130.17-1.3258-1.876541.10471
c3_3933928.142012.0355Y3.52361.54047-1.3258-1.876541.10471A lattice packing, identical to l3_39.
l3_4034056.568566862.08938Y3.547951256.31-1.43925-1.960071.07148
c3_403408013372.2.08938N3.547952512.62-1.43925-1.960071.07148The best known packing with M = 40 is not unique. Although this packing is not a lattice subset, it has the same En as l3_40.
l3_4134157.982871822.13623Y3.57171340.54-1.53556-2.027411.05328
c3_413418214317.32.12929N3.57172672.36-1.52143-2.013281.06742
l3_4234259.39776982.18197Y3.594881427.59-1.62758-2.091341.03739
c3_423428415292.2.16723N3.594882835.89-1.59814-2.06191.06682
l3_4334360.811282202.22282Y3.617511514.85-1.70814-2.144641.03105
c3_4334328.80262.20065N3.617511.62222-1.6646-2.10111.07459
l3_4434431.112721902.2624Y3.63962401.141-1.78477-2.194811.02683
c3_4434428.922872.23072N3.639621.63439-1.72353-2.133571.08808
l3_4534563.639693162.30025Y3.661241696.33-1.85683-2.241161.02547
c3_45345471420.504614756449.2.27062N3.6612491884241548.-1.80053-2.184851.08178
l3_4634665.053898742.33318Y3.682371787.61-1.91856-2.277881.03279
c3_4634629.194712.29868N3.682371.66463-1.85387-2.213191.09749
l3_4734766.468104642.36849Y3.703061883.85-1.98381-2.31881.03503
c3_4734794.20680.2.34043N3.703063723.05-1.93204-2.267031.08681
l3_4834833.941127662.40104Y3.72331495.258-2.04308-2.354391.04174
c3_4834829.508682.37717N3.723311.70255-1.99969-2.3111.08514
l3_4934969.2965116802.43232Y3.743142080.25-2.09929-2.387531.05009
c3_4934910476.264942831.2.41413N3.7431447187270.-2.0667-2.354941.08268
l3_5035035.355330822.4656Y3.76257546.081-2.15831-2.424071.05425
c3_5035029.786672.44667N3.762571.73404-2.12484-2.390591.08773
l3_5135172.1249129922.4975Y3.781622290.38-2.21414-2.457971.06028
c3_51351689.0952.47486N3.7816215.7067-2.17459-2.418421.09983
l3_5235218.38488552.52959Y3.80029149.988-2.26958-2.492011.06544
c3_5235278.15229.2.50312N3.800292671.55-2.22391-2.446341.11111
l3_5335374.9533143922.56177Y3.818612512.6-2.32448-2.526021.06993
c3_53353691.19542.53321N3.8186115.9212-2.27579-2.477341.11862
l3_5435438.183837782.59122Y3.83659656.485-2.37413-2.555281.0785
c3_5435412369.0922.56314N3.8365964.1354-2.32681-2.507951.12582
l3_553551.414215.236362.61818Y3.854240.905732-2.41909-2.58031.09064
c3_55355330281604.2.5859N3.8542448709.-2.3652-2.526411.14453
l3_5635679.196166782.65912Y3.871572871.88-2.48647-2.62821.07928
c3_5635616873860.2.61692N3.8715712718.4-2.41699-2.558721.14875
l3_5735780.6102175102.69468Y3.888593001.94-2.54415-2.666831.07657
c3_57357342309420.2.64543N3.8885953047.5-2.46405-2.586721.15668
l3_5835882.0244183582.7286Y3.905323133.84-2.59848-2.702511.07622
c3_58358348323508.2.67132N3.9053255225.2-2.50635-2.610391.16835
l3_5935983.4386192082.75898Y3.921763265.2-2.64657-2.732361.08115
c3_59359696.81362.68927N3.9217616.4575-2.53542-2.621211.19229
buckyball3_603602.3511433.94436.14058N3.937935.74656-6.12118-6.1891-2.34138The buckyball, or truncated icosahedron, represents a stable carbon molecule (fullerene) and also a classical soccer ball design.
l3_6036084.8528200742.78806Y3.937933398.4-2.6921-2.760021.08769
c3_6036012039364.92.73367N3.937936664.23-2.60655-2.674481.17324
l3_6136186.267209742.81833Y3.953823536.49-2.739-2.789431.09197
c3_61361699.86462.77402N3.9538216.8385-2.67018-2.72061.1608
l3_6236287.6812218902.84729Y3.969463676.4-2.78341-2.816691.09787
c3_62362372.389436.2.81417N3.9694665405.3-2.73259-2.765871.14869
l3_6336389.0955228222.87503Y3.984853818.12-2.82551-2.841991.10523
c3_63363378.406620.2.8458N3.9848568027.6-2.78114-2.797621.14961
l3_6436490.5097237942.90454Y43965.67-2.86986-2.869861.10954
c3_6436448.6622.52.87435N41103.75-2.82448-2.824481.15492
l3_6536591.9239247682.93112Y4.014914112.67-2.90943-2.893271.11784
c3_6536530.2620.622.91179N4.01491435.147-2.8807-2.864531.14657
l3_6636646.66964572.96465Y4.02961068.26-2.95882-2.92681.11555
c3_66366396.462188.2.94733N4.029676465.6-2.93337-2.901361.141
l3_6736794.7523268962.99577Y4.044064433.83-3.00417-2.956591.11656
c3_67367402.481976.2.98245N4.0440679454.2-2.98482-2.937251.13591
l3_683682.8284324.17653.02206Y4.058313.97152-3.04212-2.979271.12426
c3_683681224.4517860.3.01557N4.05831742158.-3.03279-2.969941.13359
l3_6936997.5807290903.05503Y4.072354762.2-3.08924-3.011391.12209
c3_69369212.18433.04607N4.072351.99463-3.07648-2.998631.13485
l3_7037098.9949302343.0851Y4.086194932.71-3.13178-3.03921.12383
c3_70370212.31363.07839N4.086192.00897-3.12232-3.029741.13329
l3_71371100.409313803.11248Y4.099835102.65-3.17015-3.063091.12909
c3_71371212.42563.10639N4.099832.0205-3.16165-3.054591.13758
l3_7237225.455820373.14352Y4.11328330.15-3.21325-3.091961.12898
c3_72372212.5373.13426N4.113282.03196-3.20044-3.079151.14179
l3_73373103.238338283.17395Y4.126555465.1-3.25509-3.119821.12951
c3_73373212.6593.16476N4.126552.04514-3.24249-3.107221.14211
l3_74374104.652350663.20179Y4.139645647.2-3.29302-3.143991.13336
c3_7437414869868.3.18974N4.1396411251.9-3.27664-3.127621.14974
l3_75375106.066363703.23289Y4.152555838.99-3.335-3.172451.13257
c3_75375212.85553.21387N4.152552.06387-3.30937-3.146821.1582
l3_7637653.740194193.26143Y4.165291507.54-3.37316-3.197321.13502
c3_763767618740.3.24446N4.165292999.39-3.35051-3.174661.15767
l3_773779.89949322.3643.28942Y4.1778651.44-3.41029-3.221351.13797
c3_77377213.09233.27306N4.177862.08915-3.38863-3.19971.15962
l3_78378110.309403943.31969Y4.190276426.64-3.45006-3.248251.13773
c3_783787820106.43.3048N4.190273198.91-3.43054-3.228721.15725
l3_793791.414216.683543.34177Y4.202521.06024-3.47886-3.264361.14795
c3_7937915883168.3.33152N4.2025213193.4-3.46551-3.251011.16129
l3_8038056.5685107973.37406Y4.214621707.87-3.52062-3.293641.14469
c3_80380213.41673.35417N4.214622.12224-3.49493-3.267951.17037
l3_81381114.551446803.40497Y4.226577047.49-3.56022-3.320941.14309
c3_81381162.88646.73.37779N4.2265713982.5-3.52541-3.286131.1779
l3_8238257.9828115513.43575Y4.238371816.89-3.59931-3.347921.14153
c3_82382213.60663.40165N4.238372.14022-3.55598-3.304591.18485
l3_83383117.38477443.46523Y4.250037489.21-3.63641-3.37311.14147
c3_83383213.70283.4257N4.250032.14945-3.58658-3.323271.1913
l3_8438459.397123373.49688Y4.261541929.97-3.6759-3.400831.13857
c3_84384213.8223.4555N4.261542.16228-3.62419-3.349121.19028
l3_85385120.208509683.5272Y4.272937952.08-3.71338-3.426731.13723
c3_85385213.94513.48627N4.272932.17573-3.6627-3.376051.18791
l3_8638660.8112131553.55733Y4.284182047.07-3.75033-3.452251.13599
c3_86386172104092.3.51852N4.2841816197.9-3.70269-3.404621.18362
l3_873871.414217.172413.58621Y4.29531.11322-3.78544-3.476111.13615
c3_87387214.19563.54891N4.29532.20328-3.74003-3.43071.18156
l3_8838831.112735003.6157Y4.30629541.843-3.82101-3.500581.13543
c3_88388176.110897.3.58011N4.3062917168.3-3.77805-3.457611.1784
l3_89389125.865576843.64121Y4.317168907.72-3.85154-3.520161.13936
c3_89389178.114428.3.61154N4.3171617670.3-3.81601-3.484631.17489
l3_9039063.6396148633.66988Y4.32792289.48-3.8856-3.543431.13935
c3_903906841702049.3.63798N4.3279262182.-3.84768-3.505511.17727
l3_91391128.693612243.69665Y4.338539407.8-3.91718-3.564351.14144
c3_9139110374.394686088.3.66741N4.3385360648209.-3.88268-3.529851.17593
l3_923921.414217.434783.71739Y4.349041.13968-3.94147-3.578131.15043
c3_9239210488.406602761.3.69645N4.3490462328336.-3.91694-3.55361.17496
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References

[Agrell14]
E. Agrell, unpublished, 2014.
[Sloane-web]
http://neilsloane.com/cluster (previously http://www2.research.att.com/~njas/cluster/), accessed April 2010.