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: Jul 22 2006 Improved: n=39, 46 Jul 13 2006 Extended until n=51 Jul 12 2006 Improved: n=50 Jul 10 2006 Improved: n=35 Jul 7 2006 Improved: n=34, 36 with Z2 symmetry Jul 6 2006 Improved: n=47 Jul 3 2006 Improved: n=27, 32 Mar 25 2006 Improved: n=49 Mar 12 2006 Improved: n=46 Mar 11 2006 Improved: n=33, 44, Oct 13 2005 Improved: n=41, not far from Z2 symmetry Oct 9 2005 Extended until n=50. Jul 1 2005 Extended until n=46. Improved: n=35, 43 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 0.2622321 55.45525197 0.486879043 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 0.2507874 63.39903114 0.504739574 < 33 0.2487104 65.00071646 0.507686712 < 34 0.2470423 66.32637786 0.512616565 < 35 0.2448284 68.14195683 0.513633621 < 36 0.2431322 69.57812973 0.517403962 < 37 0.2406218 71.77861699 0.515473849 38 0.2405147 71.87454768 0.528698979 < 39 0.2365522 75.54734627 0.516232560 40 0.2349756 77.07822383 0.518953318 41 0.2326007 79.46336424 0.515961039 42 0.2318415 80.24656773 0.523386871 43 0.2293405 82.90063321 0.518693264 44 0.2281586 84.19559331 0.522592671 45 0.2267412 85.78447105 0.524570467 46 0.2251162 87.65562903 0.524780901 47 0.2234820 89.59258566 0.524596982 48 0.2223540 90.96309057 0.527686556 49 0.2208942 92.77839443 0.528140202 50 0.2192917 94.82733051 0.527274149 51 0.2178880 96.67182148 0.527558074 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 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