Numerical results for densest packing of n equal spheres in a larger sphere with Radius=1 Author: Hugo Pfoertner http://www.pfoertner.org/ URL of this file: http://www.randomwalk.de/sequences/a084827.txt History of changes: Jun 28 2005 Improved: n=29, new symmetric config for n=37 Jun 26 2005 Extended until n=45, improved: n=30, 41, 42 Jun 23 2005 Extended until n=43, new symmetric config for n=42 Jun 21 2005 Improved: n=35 Jun 15 2005 Improved: n=29, extended until n=42 Jun 13 2005 Improved: n=24, 40 Jun 12 2005 Double prec. pp: n=34, 36, 37, Jun 11 2005 Extended until n=41 Jun 8 2005 Extended until n=40, double precision pp: n=33 Jun 5 2005 Double precision postprocessing: n=24, 26 ... 32 May 30 2005 Improved: n=26, 29, May 23 2005 Improved: n=36 May 20 2005 New: n=34 ... 38 May 19 2005 Improved: n=24, new: n=31, 32, 33 May 13 2005 n=28, 29, 30 May 12 2005 n=27 May 9 2005 Extended until n=26 May 4 2005 Extended until n=23, list of coordinates updated Apr 24 2005 Nearly exact values from Tammes solutions, extended: n=19, 20, Jun 10 2003 n=18 added, density records indicated Jun 9 2003 Initial version n Radius Required Volume Density D-Records 1 1 1 1 2 0.5 8 0.25 < 3 0.4641016 10.00370331 0.299888942 < 4 0.4494897 11.01135507 0.363261377 < 5 0.4142135 14.07107417 0.355339059 6 0.4142135 14.07107417 0.426406871 < 7 0.3859136 17.39921876 0.402315038 8 0.3780248 18.51139394 0.432166266 < 9 0.3660254 20.39230548 0.441342937 < 10 0.3530494 22.72445973 0.440054466 11 0.3445764 24.44237551 0.450038090 < 12 0.3445764 24.44237551 0.490950644 < 13 0.3333333 27.00000000 0.481481481 14 0.3235046 29.53647195 0.473990259 15 0.3183048 31.00776305 0.483749827 16 0.3109759 33.25215527 0.481171818 17 0.3056940 35.00573944 0.485634649 18 0.3012965 36.56097255 0.492328260 < 19 0.2953323 38.82103923 0.489425332 20 0.2878907 41.90994520 0.477213699 21 0.2868327 42.37542012 0.495570308 < 22 0.2793425 45.87636012 0.479549815 23 0.2756706 47.73408977 0.481835940 24 0.2713413 50.05555671 0.479467248 25 0.2711917 50.13844025 0.498619420 < 26 0.2668496 52.62599639 0.494052403 27 0.2622319 55.45534801 0.486878200 28 0.2602994 56.69966577 0.493830071 29 0.2579251 58.27995090 0.497598223 30 0.2553300 60.07509473 0.499374993 < 31 0.2531141 61.66673912 0.502702112 < 32 0.2507763 63.40748516 0.504672278 < 33 0.2487090 65.00181691 0.507678117 < 34 0.2470391 66.32891766 0.512596937 < 35 0.2448156 68.15264812 0.513553045 < 36 0.2422138 70.37256006 0.511563029 37 0.2406218 71.77861699 0.515473849 < 38 0.2405147 71.87454768 0.528698979 < 39 0.2363671 75.72493678 0.515021889 40 0.2349756 77.07822383 0.518953318 41 0.2326007 79.46336424 0.515961039 42 0.2318415 80.24656773 0.523386871 43 0.2292318 83.01860015 0.517956216 44 0.2279659 84.40930698 0.521269533 45 0.2267412 85.78447105 0.524570467 First Line: n Radius n subsequent lines: Coordinates of sphere centers # x y z Phi Theta R 1 1.0000000 1 0.000000 0.000000 0.000000 0.00000 0.00000 0.000000 2 0.4999999 2 0.000002 -0.000017 0.500000 276.43427 0.00200 0.500000 1 0.000000 0.000000 -0.500000 0.00000 180.00000 0.500000 3 0.4641016 3 0.5358984 0.0000000 0.0000000 0.00000 90.00000 0.5358984 1 -0.2679491 0.4641017 0.0000080 119.99998 89.99914 0.5358984 2 -0.2679491 -0.4641017 0.0000000 240.00000 90.00000 0.5358984 4 0.4494897 1 0.0000000 0.0000000 0.5505102 0.00000 0.00000 0.5505102 2 0.5190260 0.0000000 -0.1835034 0.00000 109.47121 0.5505102 3 -0.2595130 0.4494897 -0.1835034 120.00000 109.47121 0.5505102 4 -0.2595130 -0.4494897 -0.1835034 240.00000 109.47121 0.5505102 5 0.4142135 1 0.0000000 0.0000000 0.5857865 0.00000 0.00000 0.5857865 2 0.5857865 0.0000080 0.0000000 0.00078 90.00000 0.5857865 3 -0.2928934 0.5073059 0.0000000 120.00001 90.00000 0.5857865 4 -0.2928934 -0.5073059 0.0000000 240.00000 90.00000 0.5857865 5 0.0000000 0.0000000 -0.5857865 0.00000 180.00000 0.5857865 6 0.4142135 1 0.0000000 0.0000000 0.5857865 0.00000 0.00000 0.5857865 2 0.5857865 0.0000000 0.0000000 0.00000 90.00000 0.5857865 3 0.0000000 0.5857865 0.0000000 90.00000 90.00000 0.5857865 4 -0.5857865 0.0000000 0.0000000 180.00000 90.00000 0.5857865 5 0.0000000 -0.5857865 0.0000000 270.00000 90.00000 0.5857865 6 0.0000000 0.0000000 -0.5857865 0.00000 180.00000 0.5857865 7 0.3859136 1 0.0000000 0.0000000 0.6140864 206.83449 0.00000 0.6140864 2 0.6003749 0.0000001 0.1290431 0.00001 77.86954 0.6140864 3 -0.3001876 0.5199398 0.1290431 120.00001 77.86954 0.6140864 4 -0.3001873 -0.5199400 0.1290430 240.00002 77.86954 0.6140864 5 0.2228073 0.3859136 -0.4225278 60.00000 133.47667 0.6140864 6 -0.4456146 -0.0000001 -0.4225279 180.00002 133.47667 0.6140865 7 0.2228074 -0.3859135 -0.4225279 300.00000 133.47667 0.6140864 8 0.3780248 1 0.5346079 -0.0000001 0.3178798 0.00000 59.26413 0.6219752 2 0.0000000 0.5346079 0.3178798 90.00000 59.26413 0.6219752 3 -0.5346079 0.0000001 0.3178797 179.99998 59.26413 0.6219752 4 -0.0000001 -0.5346079 0.3178798 270.00000 59.26413 0.6219752 5 0.3780249 0.3780249 -0.3178797 45.00000 120.73586 0.6219752 6 -0.3780248 0.3780248 -0.3178798 135.00000 120.73587 0.6219751 7 -0.3780248 -0.3780248 -0.3178798 225.00000 120.73587 0.6219751 8 0.3780249 -0.3780248 -0.3178798 315.00000 120.73587 0.6219752 9 0.3660254 2 0.2113249 0.3660254 0.4725368 59.99999 41.81031 0.6339746 1 -0.4226497 -0.0000001 0.4725368 180.00002 41.81031 0.6339746 3 0.2113249 -0.3660254 0.4725368 300.00000 41.81031 0.6339746 6 0.6339746 0.0000000 0.0000000 0.00000 90.00000 0.6339746 4 -0.3169873 0.5490382 0.0000000 120.00000 90.00000 0.6339747 5 -0.3169873 -0.5490382 0.0000000 240.00000 90.00000 0.6339747 7 0.2113249 0.3660254 -0.4725368 59.99999 138.18968 0.6339746 8 -0.4226497 0.0000000 -0.4725368 180.00002 138.18968 0.6339746 9 0.2113249 -0.3660254 -0.4725367 300.00000 138.18968 0.6339746 10 0.3530494 10 -0.3301475 0.0000000 0.5563701 180.00002 30.68470 0.6469505 6 0.5421265 0.0000000 0.3530494 0.00000 56.92659 0.6469505 2 0.0822932 0.5358442 0.3530495 81.26891 56.92658 0.6469507 5 0.0822932 -0.5358442 0.3530494 278.73108 56.92659 0.6469506 3 -0.5126716 -0.3946047 0.0000000 217.58557 90.00000 0.6469505 8 -0.5126716 0.3946047 0.0000000 142.41443 90.00000 0.6469505 9 0.0822932 0.5358442 -0.3530494 81.26891 123.07340 0.6469506 1 0.0822932 -0.5358442 -0.3530495 278.73108 123.07340 0.6469507 4 0.5421264 0.0000000 -0.3530495 360.00000 123.07342 0.6469505 7 -0.3301475 0.0000000 -0.5563701 180.00002 149.31529 0.6469505 11 0.3445764 1 -0.0000001 -0.0000001 0.6554234 236.61548 0.00001 0.6554234 2 0.5862287 -0.0000001 0.2931142 0.00000 63.43496 0.6554236 3 0.1811545 0.5575366 0.2931141 72.00001 63.43496 0.6554235 4 -0.4742690 0.3445765 0.2931143 144.00000 63.43495 0.6554236 5 -0.4742690 -0.3445766 0.2931141 216.00000 63.43497 0.6554235 6 0.1811548 -0.5575367 0.2931142 288.00000 63.43496 0.6554236 7 0.4742689 0.3445765 -0.2931143 36.00000 116.56504 0.6554235 8 -0.1811549 0.5575365 -0.2931144 108.00002 116.56506 0.6554236 9 -0.5862287 -0.0000001 -0.2931144 180.00002 116.56506 0.6554236 10 -0.1811547 -0.5575366 -0.2931144 251.99998 116.56506 0.6554236 11 0.4742692 -0.3445764 -0.2931143 324.00003 116.56504 0.6554236 12 0.3445764 1 0.0000000 0.0000000 0.6554234 0.00000 0.00000 0.6554234 2 0.5862284 0.0000000 0.2931141 0.00000 63.43496 0.6554232 3 0.1811547 0.5575365 0.2931141 71.99998 63.43496 0.6554234 4 -0.4742688 0.3445770 0.2931141 143.99994 63.43497 0.6554236 5 -0.4742690 -0.3445763 0.2931142 215.99998 63.43496 0.6554234 6 0.1811547 -0.5575366 0.2931142 288.00000 63.43496 0.6554235 7 0.4742689 0.3445764 -0.2931145 35.99998 116.56507 0.6554235 8 -0.1811543 0.5575365 -0.2931144 107.99997 116.56506 0.6554234 9 -0.5862283 0.0000000 -0.2931144 180.00000 116.56507 0.6554233 10 -0.1811545 -0.5575365 -0.2931141 252.00000 116.56503 0.6554233 11 0.4742689 -0.3445768 -0.2931142 323.99997 116.56503 0.6554236 12 0.0000000 0.0000000 -0.6554235 0.00000 180.00000 0.6554235 13 0.3333333 1 0.0000000 0.0000000 0.0000000 0.00000 0.00000 0.0000000 2 0.0000080 0.0000000 0.6666667 0.00000 0.00069 0.6666667 3 0.5962848 0.0000000 0.2981424 0.00000 63.43495 0.6666667 4 0.1842622 0.5671005 0.2981423 71.99999 63.43496 0.6666666 5 -0.4824046 0.3504876 0.2981423 143.99998 63.43496 0.6666667 6 -0.4824045 -0.3504874 0.2981424 216.00000 63.43494 0.6666666 7 0.1842622 -0.5671005 0.2981424 288.00000 63.43495 0.6666667 8 0.4824046 0.3504874 -0.2981423 36.00000 116.56504 0.6666667 9 -0.1842619 0.5671005 -0.2981426 107.99998 116.56507 0.6666667 10 -0.5962847 0.0000001 -0.2981426 179.99998 116.56507 0.6666667 11 -0.1842622 -0.5671005 -0.2981423 251.99998 116.56504 0.6666666 12 0.4824046 -0.3504875 -0.2981423 323.99997 116.56504 0.6666667 13 0.0000001 0.0000000 -0.6666667 0.00000 179.99998 0.6666667 14 0.3235046 1 0.0000080 0.0000000 0.0000000 0.00000 90.00000 0.0000080 2 0.0000000 0.0000000 0.6764953 0.00000 0.00000 0.6764953 3 0.5682342 0.0000000 0.3670910 0.00000 57.13670 0.6764953 4 0.0000000 0.5682343 0.3670909 90.00000 57.13671 0.6764954 5 -0.5682341 0.0000000 0.3670911 180.00000 57.13669 0.6764953 6 0.0000000 -0.5682341 0.3670912 270.00000 57.13668 0.6764953 7 0.4764085 0.4764087 -0.0609559 45.00001 95.16966 0.6764953 8 -0.4764087 0.4764085 -0.0609557 135.00000 95.16965 0.6764953 9 -0.4764085 -0.4764087 -0.0609555 225.00002 95.16963 0.6764953 10 0.4764086 -0.4764086 -0.0609556 315.00000 95.16965 0.6764953 11 0.4575047 0.0000000 -0.4983326 0.00000 137.44586 0.6764953 12 0.0000000 0.4575046 -0.4983327 90.00000 137.44588 0.6764953 13 -0.4575047 0.0000000 -0.4983326 180.00000 137.44586 0.6764953 14 0.0000000 -0.4575048 -0.4983324 270.00000 137.44585 0.6764953 15 0.3183048 1 0.0000080 0.0000000 0.0000000 0.00000 90.00000 0.0000080 2 0.0000001 0.0000000 0.6816952 24.06850 0.00001 0.6816952 3 0.6697254 0.0000000 0.1271859 360.00000 79.24714 0.6816952 4 0.3183048 0.4643217 0.3844421 55.56833 55.67058 0.6816952 5 -0.3183049 -0.4643217 0.3844420 235.56831 55.67059 0.6816952 6 -0.3183048 0.4643216 0.3844422 124.43167 55.67057 0.6816952 7 0.3183047 -0.4643217 0.3844422 304.43164 55.67057 0.6816952 8 -0.6697254 0.0000000 0.1271858 179.99998 79.24715 0.6816952 9 -0.0000001 0.6697254 -0.1271858 90.00001 100.75284 0.6816952 10 0.0000001 -0.6697253 -0.1271859 270.00000 100.75285 0.6816951 11 0.4643217 0.3183049 -0.3844420 34.43167 124.32941 0.6816952 12 -0.4643217 -0.3183048 -0.3844421 214.43166 124.32941 0.6816952 13 -0.4643216 0.3183047 -0.3844422 145.56833 124.32942 0.6816951 14 0.4643217 -0.3183047 -0.3844422 325.56833 124.32942 0.6816952 15 0.0000000 -0.0000001 -0.6816952 239.86902 179.99998 0.6816952 16 0.3109759 1 0.0000080 0.0000000 0.0000000 0.00000 90.00000 0.0000080 2 -0.0000001 0.0000000 0.6890241 195.76761 0.00000 0.6890241 4 0.3305080 0.4458628 0.4083198 53.45129 53.65784 0.6890241 5 -0.2908655 0.4726801 0.4083197 121.60621 53.65785 0.6890241 6 -0.2556123 -0.4926374 0.4083197 242.57678 53.65785 0.6890242 7 0.3551506 -0.5128281 0.2926255 304.70389 64.86861 0.6890242 8 -0.6366764 -0.0232513 0.2624056 182.09151 67.61449 0.6890240 3 0.6739066 -0.0000001 0.1435411 0.00000 77.97577 0.6890241 9 0.0295019 0.6835765 -0.0812838 87.52875 96.77493 0.6890242 10 -0.0508470 -0.6702458 -0.1514575 265.66168 102.69813 0.6890242 11 -0.5250139 0.4146193 -0.1649408 141.70078 103.85013 0.6890242 12 -0.5453902 -0.3153247 -0.2790592 210.03493 113.89153 0.6890241 13 0.4820739 -0.3878021 -0.3032630 321.18521 116.11243 0.6890241 14 0.5051261 0.2242271 -0.4114899 23.93659 126.67011 0.6890240 15 -0.0778928 0.3445934 -0.5915592 102.73723 149.15376 0.6890241 16 -0.0342330 -0.2745675 -0.6310269 262.89304 156.32344 0.6890241 17 0.3056940 1 0.0000080 0.0000000 0.0000000 0.00000 90.00000 0.0000080 2 0.0000000 0.4323165 0.5432893 90.00000 38.51063 0.6943060 3 -0.4323165 0.0000000 0.5432893 180.00000 38.51063 0.6943060 4 0.0000000 -0.4323165 0.5432893 270.00000 38.51063 0.6943060 5 0.4323165 0.0000000 0.5432893 0.00000 38.51063 0.6943060 6 0.4771202 -0.4771202 0.1636261 315.00000 76.36897 0.6943060 7 -0.4771202 0.4771202 0.1636261 135.00000 76.36897 0.6943060 8 0.4771202 0.4771202 0.1636261 45.00000 76.36897 0.6943060 9 -0.4771202 -0.4771202 0.1636261 225.00000 76.36897 0.6943060 10 0.6747499 0.0000000 -0.1636261 0.00000 103.63103 0.6943061 11 0.0000000 0.6747499 -0.1636261 90.00000 103.63103 0.6943061 12 0.0000000 -0.6747499 -0.1636261 270.00000 103.63103 0.6943061 13 -0.6747499 0.0000000 -0.1636261 180.00000 103.63103 0.6943061 14 0.3056940 0.3056940 -0.5432893 45.00000 141.48935 0.6943061 15 -0.3056940 0.3056940 -0.5432893 135.00000 141.48935 0.6943061 16 -0.3056940 -0.3056940 -0.5432893 225.00000 141.48935 0.6943061 17 0.3056940 -0.3056940 -0.5432893 315.00000 141.48935 0.6943061 18 0.3012965 1 0.0000080 0.0000000 0.0000000 0.00000 90.00000 0.0000080 2 0.0000000 0.0000000 0.6987034 0.00000 0.00000 0.6987034 3 0.5436871 0.0000000 0.4388517 0.00000 51.09033 0.6987035 4 0.1142872 0.5315393 0.4388518 77.86547 51.09032 0.6987035 5 -0.5436870 0.0000000 0.4388517 180.00000 51.09032 0.6987033 6 -0.1142873 -0.5315394 0.4388516 257.86548 51.09034 0.6987034 7 -0.4256611 0.5269113 0.1713580 128.93272 75.80330 0.6987034 8 0.4256611 -0.5269113 0.1713579 308.93271 75.80331 0.6987034 9 0.5431392 0.4387708 0.0258161 38.93274 87.88251 0.6987035 10 -0.5431393 -0.4387707 0.0258160 218.93272 87.88252 0.6987034 11 0.0319622 0.6706459 -0.1933881 87.27142 106.06819 0.6987034 12 -0.0319623 -0.6706458 -0.1933882 267.27139 106.06821 0.6987033 13 -0.6623801 0.1097269 -0.1933883 170.59404 106.06821 0.6987034 14 0.6623802 -0.1097268 -0.1933881 350.59406 106.06819 0.6987034 15 0.3567692 0.2882132 -0.5271009 38.93274 138.97284 0.6987034 16 -0.3567691 -0.2882131 -0.5271010 218.93274 138.97285 0.6987034 18 0.2432059 -0.3010561 -0.5817238 308.93274 146.36427 0.6987034 17 -0.2432057 0.3010562 -0.5817239 128.93271 146.36429 0.6987035 19 0.2953323 19 0.0000000 0.0000000 0.0000000 0.00000 0.00000 0.0000000 1 0.2953323 -0.0000002 0.6397933 359.99994 24.77832 0.7046677 2 -0.2953324 0.0000003 0.6397932 179.99992 24.77833 0.7046677 3 -0.0000005 -0.4930299 0.5034661 269.99994 44.39996 0.7046677 4 0.0000005 0.4930300 0.5034660 89.99994 44.39998 0.7046677 5 -0.5277357 -0.3879746 0.2598603 216.32214 68.36021 0.7046677 6 0.5277357 0.3879745 0.2598604 36.32214 68.36021 0.7046677 7 0.6436692 -0.1991695 0.2063446 342.80643 72.97281 0.7046677 8 -0.6436691 0.1991697 0.2063444 162.80643 72.97283 0.7046677 9 0.2982059 -0.6382889 0.0147355 295.04178 88.80178 0.7046677 10 -0.2982057 0.6382890 0.0147353 115.04176 88.80180 0.7046677 11 -0.2503494 -0.6263235 -0.2039624 248.21278 106.82471 0.7046677 12 0.2503496 0.6263233 -0.2039626 68.21276 106.82473 0.7046677 13 -0.6333655 -0.1496861 -0.2701828 193.29700 112.54560 0.7046677 14 0.6333655 0.1496862 -0.2701828 13.29700 112.54560 0.7046677 15 0.3806336 -0.3380702 -0.4872198 318.38922 133.74245 0.7046677 16 -0.3806335 0.3380701 -0.4872199 138.38922 133.74246 0.7046676 17 -0.1803857 -0.2338423 -0.6397932 232.35345 155.22165 0.7046677 18 0.1803858 0.2338420 -0.6397933 52.35339 155.22166 0.7046677 20 0.2878907 1 0.0000081 0.0000000 0.0000000 0.00000 90.00000 0.0000081 2 0.0000000 0.0000000 0.7121092 0.00000 0.00000 0.7121092 7 0.2785169 0.4534128 0.4732279 58.43893 48.35265 0.7121091 5 -0.2971401 0.4414325 0.4732279 123.94555 48.35265 0.7121092 6 0.2785169 -0.4534126 0.4732281 301.56110 48.35262 0.7121091 4 -0.2971401 -0.4414324 0.4732280 236.05444 48.35263 0.7121092 3 0.6146899 0.0000000 0.3595216 0.00000 59.67736 0.7121091 8 -0.6347889 0.0000000 0.3227113 180.00000 63.05231 0.7121092 10 -0.0533362 0.7096384 0.0258469 94.29825 87.91992 0.7121092 9 -0.0533360 -0.7096384 0.0258471 265.70178 87.91990 0.7121092 12 -0.5580707 0.4405154 -0.0400352 141.71405 93.22290 0.7121092 11 -0.5580706 -0.4405155 -0.0400350 218.28595 93.22289 0.7121092 14 0.4831951 0.5195354 -0.0608687 47.07557 94.90343 0.7121092 13 0.4831952 -0.5195353 -0.0608684 312.92444 94.90340 0.7121091 15 0.6796374 0.0000000 -0.2125853 0.00000 107.36927 0.7121091 16 -0.5822240 0.0000000 -0.4100180 180.00000 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