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
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
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