Spherical Codes

Nice arrangements of points on a sphere in various dimensions

N. J. A. Sloane

Information Sciences Research
AT&T Shannon Lab
Florham Park, NJ 07932-0971

Email address: njasloane@gmail.com Home page: NeilSloane.com/

Based on joint work with R. H. Hardin, W. D. Smith and Others

 A library of putatively optimal spherical codes, together with other arrangements which may not be optimal but are especially interesting for some reason.

 Keywords: points on a sphere, spherical codes, spherical designs, spherical coverings, 4-dimensional arrangements, 5-dimensional arrangements, 6-dimensional arrangements, 7-dimensional arrangements, 8-dimensional arrangements, 9-dimensional arrangements, 10-dimensional arrangements, 11-dimensional arrangements, 12-dimensional arrangements, 13-dimensional arrangements, 14-dimensional arrangements, 15-dimensional arrangements, 16-dimensional arrangements, 17-dimensional arrangements, 18-dimensional arrangements, 19-dimensional arrangements, 20-dimensional arrangements, 21-dimensional arrangements, 22-dimensional arrangements, 23-dimensional arrangements, 24-dimensional arrangements, higher-dimensional arrangements, Archimedian solids, Barnes-Wall lattice, Coxeter-Todd lattice, cube, cuboctahedron, dodecahedron, E_8 lattice, Gosset lattice, hypercube, hypersphere, icosahedron, Leech lattice, Nordstrom-Robinson code, octahedron, generalized octahedron, orthoplex, Platonic solids, rhombic dodecahedron, simplex, 600-cell, snub cube, improved snub cube, tetrahedron, truncated cube, truncated dodecahedron, truncated icosahedron, truncated octahedron, truncated tetrahedron, 24-cell, twisted cube

 The Problem Place n points on a sphere in d dimensions so as to maximize the minimal distance (or equivalently the minimal angle) between them.

This file has two parts

  1. Part 1. Tables of putatively optimal packings in 3, 4 and 5 dimensions with n = 4, ..., 130 points.
  2. Part 2. Tables of other good (though not always optimal) packings in dimensions 3 through 24 with various numbers of points.

Remarks


Part 1, Dimension 3 through 5, With Up To 130 Points

Summary of results:

Packings of points on a sphere

Arrangements for which no other source is listed are copyright R. H. Hardin, N. J. A. Sloane & W. D. Smith, 1994-1996.


dim npts min separation (degrees)

  3    4  109.4712206
  3    5   90.0000000
  3    6   90.0000000
  3    7   77.8695421
  3    8   74.8584922
  3    9   70.5287794
  3   10   66.1468220
  3   11   63.4349488
  3   12   63.4349488
  3   13   57.1367031
  3   14   55.6705700
  3   15   53.6578501
  3   16   52.2443957
  3   17   51.0903285
  3   18   49.5566548
  3   19   47.6919141
  3   20   47.4310362
  3   21   45.6132231	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
  3   22   44.7401612
  3   23   43.7099642
  3   24   43.6907671
  3   25   41.6344612
  3   26   41.0376616
  3   27   40.6776007	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
  3   28   39.3551436
  3   29   38.7136512
  3   30   38.5971159
  3   31   37.7098291
  3   32   37.4752140
  3   33   36.2545530	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
  3   34   35.8077844
  3   35   35.3198076
  3   36   35.1897322
  3   37   34.4224080
  3   38   34.2506607
  3   39   33.4890466
  3   40   33.1583563
  3   41   32.7290944
  3   42   32.5063863
  3   43   32.0906244
  3   44   31.9834230
  3   45   31.3230814
  3   46   30.9591635
  3   47   30.7818159
  3   48   30.7627855
  3   49   29.9235851
  3   50   29.7529564
  3   51   29.3684069
  3   52   29.1947579
  3   53   28.8138972
  3   54   28.7169205	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
  3   55   28.2627914
  3   56   28.1480466
  3   57   27.8266759
  3   58   27.5564159
  3   59   27.3949757
  3   60   27.1928300
  3   61   26.8732779
  3   62   26.6839970
  3   63   26.4869225
  3   64   26.2350433
  3   65   26.0698299
  3   66   25.9474437	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
  3   67   25.6839813
  3   68   25.4638245
  3   69   25.3336364	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3   70   25.1709200
  3   71   24.9879381
  3   72   24.9264861
  3   73   24.5537792	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3   74   24.4209398	D.A. Kottwitz, private communication
  3   75   24.3017225
  3   76   24.1281944
  3   77   24.0012837
  3   78   23.9310254
  3   79   23.6239917
  3   80   23.5530672	Jim Buddenhagen (jbuddenh(AT)earthlink.net) private communication
  3   81   23.3476377	D.A. Kottwitz, private communication
  3   82   23.1946074
  3   83   23.0829976
  3   84   23.0517306
  3   85   22.7791621
  3   86   22.6743694	ref D.A. Kottwitz (Acta Cryst., A47, 158-165(1991)) thanks to Jim Buddenhagen (jbuddenh(AT)earthlink.net)
  3   87   22.5466574
  3   88   22.4678810
  3   89   22.3166023
  3   90   22.1540232
  3   91   22.0517963
  3   92   22.0275815
  3   93   21.8103801
  3   94   21.7237135	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3   95   21.5945501
  3   96   21.5206099
  3   97   21.4006197
  3   98   21.3710607
  3   99   21.1359674	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  100   21.0312020
  3  101   20.9286834
  3  102   20.8556887	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  103   20.7382700	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  104   20.6566210
  3  105   20.5388524
  3  106   20.4394089
  3  107   20.3612035	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  108   20.3044447	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  109   20.1493196	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  110   20.1113276	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  111   19.9824769	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  112   19.8913044	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  113   19.8056013
  3  114   19.7450093
  3  115   19.6239931	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  116   19.5497969	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  117   19.4612911
  3  118   19.3893497	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  119   19.3257514	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  120   19.3240201
  3  121   19.1357298	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  122   19.0700369	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  123   19.0063891	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  124   18.9539116	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  125   18.8448151	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  126   18.7815856
  3  127   18.6900568	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  128   18.6349726
  3  129   18.5634726	Jim Buddenhagen (jbuddenh(AT)earthlink.net) and D.A. Kottwitz, private communication
  3  130   18.5103522
  4    5  104.4775121
  4    6   90.0000000
  4    7   90.0000000
  4    8   90.0000000
  4    9   80.6761150
  4   10   80.4059318
  4   11   76.6790498
  4   12   75.5224878
  4   13   72.1036706
  4   14   71.3661859
  4   15   69.4519870
  4   16   67.1930021
  4   17   65.6531563
  4   18   64.9872827
  4   19   64.2618766
  4   20   64.2618766
  4   21   61.8760316
  4   22   60.1398863
  4   23   60.0000000
  4   24   60.0000000
  4   25   57.4988826
  4   26   57.2625923
  4   27   56.1106132
  4   28   55.4351398
  4   29   55.0299245
  4   30   54.2511897
  4   31   53.7886221
  4   32   53.4360130
  4   33   53.0759811
  4   34   52.4011334
  4   35   51.5724903
  4   36   51.0745199
  4   37   50.9226685
  4   38   50.1535207
  4   39   49.6895967
  4   40   49.4208547
  4   41   49.0594727
  4   42   48.8204491
  4   43   48.4963928
  4   44   48.0251062
  4   45   47.3961868
  4   46   46.9743371
  4   47   46.7328543
  4   48   46.3832527
  4   49   46.0990591
  4   50   46.0742073
  4   51   45.4625355
  4   52   45.2812081
  4   53   44.9542243
  4   54   44.7883187
  4   55   44.3025299
  4   56   44.0114102
  4   57   43.7495816
  4   58   43.5341667
  4   59   43.3907742
  4   60   43.2167495
  4   61   42.9862214
  4   62   42.7870922
  4   63   42.3770430
  4   64   42.3062196
  4   65   42.0566173
  4   66   41.7550028
  4   67   41.4987735
  4   68   41.3146370
  4   69   41.1813742
  4   70   41.0851193
  4   71   40.8538202
  4   72   40.7098566
  4   73   40.6070686
  4   74   40.5754396
  4   75   40.4865078
  4   76   40.4418535
  4   77   40.0749825
  4   78   39.7568300
  4   79   39.3704881
  4   80   39.1321954
  4   81   38.9404678
  4   82   38.7616584
  4   83   38.6411550
  4   84   38.5216197
  4   85   38.3971687
  4   86   38.2834796
  4   87   38.2316167
  4   88   38.1934424
  4   89   37.9525738
  4   90   37.7779155
  4   91   37.6840939
  4   92   37.6073513
  4   93   37.5288195
  4   94   37.4811292
  4   95   37.1570600
  4   96   37.0622214
  4   97   36.9191853
  4   98   36.7966891
  4   99   36.6983263
  4  100   36.5636318
  4  101   36.4956325
  4  102   36.4186254
  4  103   36.3807276
  4  104   36.2615258
  4  105   36.1859494
  4  106   36.1333759
  4  107   36.0959131
  4  108   36.0509369
  4  109   36.0309659
  4  110   36.0076295
  4  111   36.0013484
  4  112   36.0000058
  4  113   36.0000000
  4  114   36.0000000
  4  115   36.0000000
  4  116   36.0000000
  4  117   36.0000000
  4  118   36.0000000
  4  119   36.0000000
  4  120   36.0000000
  4  121   34.4465422
  4  122   34.2716955
  4  123   34.1743519
  4  124   33.9853008
  4  125   33.8729785
  4  126   33.7679953
  4  127   33.6685430
  4  128   33.5726406
  4  129   33.5033828
  4  130   33.4075174
  4  600   19.8433177
  5    6  101.5369590
  5    7   90.0000000
  5    8   90.0000000
  5    9   90.0000000
  5   10   90.0000000
  5   11   82.3654810
  5   12   81.1451619
  5   13   79.2070847
  5   14   78.4630410
  5   15   78.4630410
  5   16   78.4630410
  5   17   74.3074162
  5   18   74.0080831
  5   19   73.0329080
  5   20   72.5792898
  5   21   71.6441994
  5   22   69.2068599
  5   23   68.2984004
  5   24   68.0230856
  5   25   67.6897899
  5   26   67.0305252
  5   27   66.3175208
  5   28   65.9100901
  5   29   65.7301670
  5   30   65.6046967
  5   31   64.2839257
  5   32   63.7779969
  5   33   62.6125998
  5   34   61.9343521
  5   35   61.4165697
  5   36   61.0530896
  5   37   60.4074688
  5   38   60.2482021
  5   39   60.0017727
  5   40   60.0000000
  5   41   59.0029024
  5   42   58.7856545
  5   43   58.3513041
  5   44   58.1661569
  5   45   57.6514221
  5   46   57.4326036
  5   47   57.1800994
  5   48   56.9676480
  5   49   56.8441582
  5   50   56.8441582
  5   51   56.3071628
  5   52   55.9343732
  5   53   55.4667168
  5   54   55.1747097
  5   55   54.9279561
  5   56   54.7453222
  5   57   54.7356103
  5   58   54.4462372
  5   59   54.0539784
  5   60   53.6051457
  5   61   53.3979587
  5   62   53.2010567
  5   63   52.8751126
  5   64   52.7108774
  5   65   52.5214037
  5   66   52.4752830
  5   67   52.2729583
  5   68   52.0286818
  5   69   51.9176291
  5   70   51.8973507
  5   71   51.8317602
  5   72   51.8317602
  5   73   51.8272924
  5   74   51.8272924
  5   75   51.3428135
  5   76   50.8459281
  5   77   50.4692075
  5   78   50.1905273
  5   79   49.9276863
  5   80   49.6870598
  5   81   49.5416542
  5   82   49.3488655
  5   83   49.0496962
  5   84   48.8819546
  5   85   48.7570741
  5   86   48.6290153
  5   87   48.5000839
  5   88   48.3577476
  5   89   48.2141763
  5   90   48.1896851
  5   91   47.9008875
  5   92   47.7716828
  5   93   47.6364239
  5   94   47.5270717
  5   95   47.4198451
  5   96   47.3535100
  5   97   47.2151568
  5   98   47.1352000
  5   99   46.9889294
  5  100   46.9094506
  5  101   46.7465353
  5  102   46.5719302
  5  103   46.4564803
  5  104   46.3923772
  5  105   46.3118627
  5  106   46.1395418
  5  107   46.0632232
  5  108   45.9430913
  5  109   45.8583450
  5  110   45.7248275
  5  111   45.6276320
  5  112   45.5373551
  5  113   45.4294817
  5  114   45.3304185
  5  115   45.2457325
  5  116   45.1419275
  5  117   45.0181994
  5  118   44.9252672
  5  119   44.8177246
  5  120   44.7301784
  5  121   44.6404435
  5  122   44.5817814
  5  123   44.4453109
  5  124   44.3486865
  5  125   44.2250403
  5  126   44.1807177
  5  127   44.0598726
  5  128   43.9739131
  5  129   43.8672061
  5  130   43.8024944


Part 2, Other Nice Arrangements of Points on Spheres

Remarks

Dimension 3

Dimension 4

Dimension 5

Dimension 6

Dimension 7

Dimension 8

Dimension 9

Dimension 10

Dimension 11

Dimension 12

Dimension 13

Dimension 14

Dimension 15

Dimension 16

Dimension 17

Dimension 18

Dimension 19

Dimension 20

Dimension 21

Dimension 22

Dimension 23

Dimension 24

Higher Dimensions



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