Spherical codes Dimension: 3 Number of points: 7 Cosine: 0.210138312731 Minimal polynomial: 3*x^3-9*x^2-3*x+1 Provably optimal References and notes: K. Schütte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96–124, doi:10.1007/BF02054944. Points: -0.3223221167653308,0.3383117161972068,0.8841117778469062 -0.46822436668895356,0.8455398516414839,-0.25657026664215327 -0.9589537361413936,-0.28354188552247034,-0.0034250685869030755 0.6843998237157204,0.6946892202891244,0.22136794824901684 0.8475530755750805,-0.43615898510346285,-0.3023559554510865 -0.17333163675994265,-0.14313187085903079,-0.9744072101757627 0.023241932797164445,-0.8265184890658953,0.5624295509591237 Dimension: 3 Number of points: 8 Cosine: 0.261203874964 Minimal polynomial: 7*x^2+2*x-1 Provably optimal References and notes: K. Schütte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96–124, doi:10.1007/BF02054944. Points: 1,0,0.5946035575013605 -1,0,0.5946035575013605 0,1,0.5946035575013605 0,-1,0.5946035575013605 0.7071067811865476,0.7071067811865476,-0.5946035575013605 0.7071067811865476,-0.7071067811865476,-0.5946035575013605 -0.7071067811865476,0.7071067811865476,-0.5946035575013605 -0.7071067811865476,-0.7071067811865476,-0.5946035575013605 Dimension: 3 Number of points: 9 Cosine: 0.333333333334 Minimal polynomial: 3*x-1 Provably optimal References and notes: K. Schütte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96–124, doi:10.1007/BF02054944. Points: -0.6666666666666666,0,0.7453559924999299 0.3333333333333333,-0.5773502691896257,0.7453559924999299 0.3333333333333333,0.5773502691896257,0.7453559924999299 1,0,0 -0.5,0.8660254037844386,0 -0.5,-0.8660254037844386,0 -0.6666666666666666,0,-0.7453559924999299 0.3333333333333333,-0.5773502691896257,-0.7453559924999299 0.3333333333333333,0.5773502691896257,-0.7453559924999299 Dimension: 3 Number of points: 10 Cosine: 0.404394325217 Minimal polynomial: 7*x^3-4*x^2-2*x+1 Provably optimal References and notes: K. Schütte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96–124, doi:10.1007/BF02054944; L. Danzer, Finite point-sets on S^2 with minimum distance as large as possible, Discrete Math. 60 (1986), 3–66, doi:10.1016/0012-365X(86)90002-6. Points: -0.44228789746574293,0.7786489448207033,0.4450699231412271 -0.10792350346817989,-0.297320788081394,-0.9486584561230731 0.4549578185126939,-0.8662271731151231,-0.20655233702660794 0.8280659508232893,0.09962437313638146,-0.5517080435921746 0.028992238230054937,0.8429169556483159,-0.5372619994034445 -0.6282093932461281,-0.7697099526296717,-0.11357617294203981 -0.8788956836709402,0.25172572059977705,-0.40518704176357084 0.6408793142930791,0.6821317243352519,0.35209375905665896 0.5833166879070142,-0.3367299725897504,0.7391580123146181 -0.49985052385180784,-0.24021275210429904,0.8321341763991862 Dimension: 3 Number of points: 11 Cosine: 0.447213595500 Minimal polynomial: 5*x^2-1 Provably optimal References and notes: K. Schütte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96–124, doi:10.1007/BF02054944; L. Danzer, Finite point-sets on S^2 with minimum distance as large as possible, Discrete Math. 60 (1986), 3–66, doi:10.1016/0012-365X(86)90002-6. Points: -0.8506508083520399,0,-0.5257311121191336 -0.8506508083520399,0,0.5257311121191336 -0.5257311121191336,-0.8506508083520399,0 -0.5257311121191336,0.8506508083520399,0 0,-0.5257311121191336,-0.8506508083520399 0,-0.5257311121191336,0.8506508083520399 0,0.5257311121191336,-0.8506508083520399 0,0.5257311121191336,0.8506508083520399 0.5257311121191336,-0.8506508083520399,0 0.5257311121191336,0.8506508083520399,0 0.8506508083520399,0,-0.5257311121191336 Dimension: 3 Number of points: 12 Cosine: 0.447213595500 Minimal polynomial: 5*x^2-1 Provably optimal References and notes: L. Fejes [Tóth], Über eine Abschätzung des kürzesten Abstandes zweier Punkte eines auf einer Kugelfläche liegenden Punktsystems, Jber. Deutsch. Math.-Verein. 53 (1943), 66–68. Points: -0.8506508083520399,0,-0.5257311121191336 -0.8506508083520399,0,0.5257311121191336 -0.5257311121191336,-0.8506508083520399,0 -0.5257311121191336,0.8506508083520399,0 0,-0.5257311121191336,-0.8506508083520399 0,-0.5257311121191336,0.8506508083520399 0,0.5257311121191336,-0.8506508083520399 0,0.5257311121191336,0.8506508083520399 0.5257311121191336,-0.8506508083520399,0 0.5257311121191336,0.8506508083520399,0 0.8506508083520399,0,-0.5257311121191336 0.8506508083520399,0,0.5257311121191336 Dimension: 3 Number of points: 13 Cosine: 0.542636486830 Minimal polynomial: x^8-24*x^7-12*x^6+8*x^5+38*x^4+24*x^3-12*x^2-8*x+1 Provably optimal References and notes: K. Schütte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96–124, doi:10.1007/BF02054944; O. R. Musin and A. S. Tarasov, The strong thirteen spheres problem, Discrete Comput. Geom. 48 (2012), 128–141, doi:10.1007/s00454-011-9392-2. Points: 0.6373327466340105,-0.7646532440643388,0.0954588204925604 0.03025988760798175,0.997172120093437,0.06879027627733728 0.12549897985538594,-0.6936017924507256,-0.7093423429941271 -0.6819139515787054,-0.7044577859216173,-0.19680597169986438 -0.5357091071008355,-0.01586301635348692,-0.8442535858860148 0.5302559671604951,-0.1638375557676447,0.8318568774799977 0.8380483468867255,-0.07854572720036006,-0.5399125271921307 -0.17008018480008075,-0.7755092375352305,0.6079951917868232 -0.8435014868270659,-0.10115484492077158,0.5275158187861135 0.8356915672618433,0.5037197182189701,0.21882881411012883 0.19997909302455927,0.5919676944861877,-0.7807577159642947 -0.8028513290549335,0.5784842338049598,-0.1441725864157798 -0.1671388548176494,0.49023625753774225,0.8554139436586436 Dimension: 3 Number of points: 14 Cosine: 0.563950300361 Minimal polynomial: 4*x^4-2*x^3+3*x^2-1 Provably optimal References and notes: K. Schütte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96–124, doi:10.1007/BF02054944; O. R. Musin and A. S. Tarasov, The Tammes problem for N=14, Exp. Math. 24 (2015), 460–468, doi:10.1080/10586458.2015.1022842. Points: -0.36384865465579125,0.7520331583614871,0.5496001139283153 -0.18474353795360404,0.19209609512017917,-0.9638303353930942 -0.6808468036148474,0.6570317303814883,-0.3236617606072004 -0.9884728493405397,-0.1214241352998813,0.09043011380768509 0.6828883063204896,0.33481958095005376,-0.6492760655554489 0.18712919330704644,0.9669596260195705,-0.17312350117862757 -0.5533990282166157,-0.5826834918295714,-0.5951717936177784 0.36429375297659555,-0.535784925164908,-0.7617247373613785 -0.4394355584103401,-0.8493051555849686,0.29253571183940075 0.9880277510197355,-0.09482409789669771,0.12169450962537805 0.43704990305689767,-0.3097505655547812,0.8444181247323209 0.4743484275283747,-0.874457473290643,0.10157606365430075 0.5533990282166157,0.5826834918295714,0.5951717936177784 -0.47638993023401704,-0.11739383804089905,0.8713617625083486 Dimension: 3 Number of points: 15 Cosine: 0.592605902926 Minimal polynomial: 13*x^5-x^4+6*x^3+2*x^2-3*x-1 References and notes: K. Schütte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96–124, doi:10.1007/BF02054944; D. A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. Points: -0.4532679310936769,0.4441075078541507,0.7728626683374177 -0.217659192699563,-0.5733771626276939,-0.7898500523582875 0.6430062154659175,-0.5642047032049129,-0.5178958000926964 -0.7899098976193717,-0.6062000708201343,-0.09254095191112921 0.020260153806930907,-0.9993320718934239,-0.030412764637833843 -0.3537740341980906,0.9350728958198743,0.021968437137011198 0.2854321271942969,-0.04066694830566801,0.9575357435003891 -0.9620326286005992,0.2686477532642007,0.04818304861586315 0.7112597311891703,-0.5902014988125672,0.38180071449397357 0.45896903811746215,0.8156731836642221,-0.35217137802581805 0.29628162971904176,0.15274758100994656,-0.9428071766732785 0.36143478871848217,0.7577335340520057,0.543327511614389 -0.5844956697062614,0.2511736230691911,-0.771541718359367 0.9875798807480268,0.14022696819840297,-0.07086872745857055 -0.4863376136173134,-0.4574944647019364,0.7444290028928922 Dimension: 3 Number of points: 16 Cosine: 0.612294616483 Minimal polynomial: 23*x^6+6*x^5+5*x^4+4*x^3-3*x^2-2*x-1 References and notes: K. Schütte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz?, Math. Ann. 123 (1951), 96–124, doi:10.1007/BF02054944. Points: -0.9832469510387222,-0.1771335537840721,-0.04300159760848491 -0.6090637476008695,0.11467374989327107,-0.7847874122613078 0.8242057056450016,0.505340669700561,-0.25556948630228427 0.17295690248844647,-0.15511153944211295,0.9726388436688614 0.06262897381580342,-0.4442297182062034,-0.8937211920398982 0.8430562940428193,-0.3262417595881327,0.42757736069116936 0.20979876607133485,0.4238672195493357,-0.8810908341064845 0.35510921304303694,-0.9346375779189531,0.01870948298986356 -0.6600253360475268,0.024761951133561663,0.7508351360661064 -0.22167021920952393,0.7724368341226924,0.5951501081291289 -0.4589954487734546,-0.8355512184610581,-0.30195585659744445 0.8190665814412473,-0.35296906705005066,-0.45226405215533727 0.6180366941671751,0.5073441025445691,0.6005269405082136 -0.3149901785113359,-0.7635073577729333,0.5637709659675556 -0.7378648816028184,0.6658818978117303,-0.11025749255224053 0.080997632069386,0.9750753674677949,-0.20656091439741717 Dimension: 3 Number of points: 17 Cosine: 0.628094415071 Minimal polynomial: 4*x^10-92*x^9+24*x^8+128*x^7-58*x^6-78*x^5+35*x^4+32*x^3-4*x^2-6*x-1 References and notes: L. Danzer, Finite point-sets on S^2 with minimum distance as large as possible, Discrete Math. 60 (1986), 3–66, doi:10.1016/0012-365X(86)90002-6. Points: 0.8883492390004214,0.0934786383907809,-0.44955241488838427 -0.4944780065487319,0.5640431676647443,0.6613220138860595 0.43744133346593206,-0.6110027881189465,-0.6597883544640819 0.27620405036272144,0.8653251892412606,0.4182387349682044 -0.7718420410987927,-0.6354755167430398,0.020754064966080916 0.20115381803285587,0.1596672673126606,-0.9664592620693913 0.882048069735028,0.251841001945828,0.3982051134975952 -0.394523126729936,0.9107492063204825,-0.12201387487431936 -0.6083493831559222,0.329378181948608,-0.7220948976900742 0.8217864321279917,-0.5589733826083665,0.11052518946243116 0.229758616244978,-0.6410807718909484,0.7322748269419745 0.40318655517513985,0.8063225780505834,-0.4327637945252176 0.22610948736519995,0.19210855639945543,0.954970576657503 -0.9783954125628307,0.20147809754855514,0.04635723119668837 -0.5556607499210141,-0.2880132791622619,0.7799291519256204 0.008816061902035124,-0.9997553435086337,-0.020286206606296736 -0.41391860073978154,-0.5102639667978167,-0.7538581273352287 Dimension: 3 Number of points: 18 Cosine: 0.648695832224 Minimal polynomial: 216*x^11+108*x^10+118*x^9+209*x^8-100*x^7-160*x^6+40*x^5+46*x^4-20*x^3-12*x^2+2*x+1 References and notes: T. Tarnai and Zs. Gáspár, Improved packing of equal circles on a sphere and rigidity of its graph, Math. Proc. Cambridge Philos. Soc. 93 (1983), 191–218, doi:10.1017/S0305004100060485. Points: -0.3272617535826605,0.8830690709136323,0.3362867238499559 0.3985146309148577,0.9134924396503428,-0.08196128139835106 0.8190085284258968,-0.09153229021377211,0.5664334649487694 -0.3593806522347611,0.26011817977729346,0.8962053778842579 -0.09176527973753576,-0.9543120962841817,0.2843722143958207 0.020512402405338512,-0.8422494142797933,-0.5386976568474255 -0.8173674752076884,0.1329096094043358,-0.5605759950270756 0.6572544231563049,-0.3167992499717346,-0.6838529509030573 0.3769923240158556,0.6083286233431908,0.6984361629057465 0.3883209341498003,0.4695786204098325,-0.7929077949894457 -0.6873442982303142,-0.44188325436575343,0.5764520840463742 -0.3209220123720445,0.805421495335585,-0.4983023949636079 -0.14704219527011292,-0.1482831662780089,-0.9779522971028364 -0.9203835244248655,0.29636756388588176,0.2550694710144883 0.12432717004577357,-0.42445075770484003,0.8968747454757529 -0.7341118230493899,-0.6360973826490033,-0.23761302793868172 0.9433638235074043,0.2830549433688822,-0.17304506792142282 0.6798508200922458,-0.7320330436201549,0.04393729017717415 Dimension: 3 Number of points: 19 Cosine: 0.673116887561 Minimal polynomial: 26805627119546756712*x^75-11948562371904894036*x^74-585034737208997137002*x^73+3205538219765481672033*x^72-1811715795208777326138*x^71-32027083460029622026050*x^70+131321687130974923909328*x^69-215430828926691745481942*x^68+143578141387958588992206*x^67-18406761611971518339214*x^66+28337005985031047347118*x^65+236368444777002037271073*x^64-1082229764974994074739344*x^63+1516092906483969195053744*x^62+179570427167174666506496*x^61-3217396144386747746240528*x^60+3537833960550350228385744*x^59+991490736993357330354848*x^58-5706214109060395312919304*x^57+3869375067145621202200948*x^56+3222022425168550972367752*x^55-6451162558970750017599320*x^54+1418757773915326072392384*x^53+4886013800460103250409752*x^52-4187512596952067529671448*x^51-1524646701886268332978696*x^50+4008909869245135167286008*x^49-876772290328159683949020*x^48-2388289625028580376909744*x^47+1571763036057352365864208*x^46+929968188649714429241600*x^45-1244795308423067845031536*x^44-175117573311018995290304*x^43+701512392486712668409464*x^42-56214865908605694262284*x^41-313009485469054026132306*x^40+67705510705054228901124*x^39+114643074677718448810196*x^38-35069626854099241213472*x^37-34325988490554840080772*x^36+12846271813523410492212*x^35+7897687687110894499948*x^34-3748244298939587917724*x^33-995351057402496601890*x^32+1023858388193512945808*x^31-197416589532207246000*x^30-348107335768936904960*x^29+159027821846227712400*x^28+140445584391790929904*x^27-44538947685690751744*x^26-50311525445451632040*x^25+3837870000595205380*x^24+14032664738240095304*x^23+2555747252845771112*x^22-2415894472515188544*x^21-1293572825268210408*x^20-3476911855421016*x^19+210842363175215224*x^18+80264372524640792*x^17+4168369152867604*x^16-7094624800170320*x^15-2960968913050640*x^14-459867970025728*x^13+55487582029808*x^12+46737611704216*x^11+11559923551036*x^10+1215281611558*x^9-143093193863*x^8-87545738138*x^7-19809266850*x^6-2929239600*x^5-308513494*x^4-23230866*x^3-1198926*x^2-38178*x-567 References and notes: D. E. Lazić, V. Šenk, and I. Šeškar, Arrangements of points on a sphere which maximize the least distance, Bull. Appl. Math. (Technical University Budapest) 47 (1987), 7–21. Points: 0.44403583795127244,0.7950973189599577,-0.41310099007094936 -0.2953678757225916,-0.9083957470442618,-0.2959307093611967 0.12327524035214414,-0.5643249587288006,0.8162968553607557 0.4453811171027206,0.803295136400416,0.39541444632659745 -0.04835836985527009,0.39750479140933737,-0.916324947205717 -0.575898219022926,-0.6943311127472562,0.4315617536313649 0.9494536186637963,0.31387122700196457,-0.004762233352508299 -0.8975368363089203,-0.37207413961654195,-0.23661881179902955 0.6758570855100854,0.08901007145445668,-0.7316381668184663 0.38825262766197277,-0.9114874462482251,0.13583273701675336 -0.4780223738829758,-0.027287216687982962,0.8779236970675228 0.39727736691319815,-0.6637059998212834,-0.6337704943745383 0.8167918293391426,-0.31891907689272453,0.48077201449201407 -0.3299647987205419,0.7208914017914655,0.6094578069304291 -0.38394423178079945,-0.3359685765714391,-0.8600650803507911 -0.7608647404557389,0.36674707946741625,-0.5353330051784045 0.2904593615604909,0.21830442373341602,0.9316525843147182 -0.3146747258237049,0.9337751686076307,-0.1704222738363949 -0.924384574929861,0.27702568080332846,0.26224021393980085 Dimension: 3 Number of points: 20 Cosine: 0.676477138130 Minimal polynomial: 21*x^3-9*x^2-5*x+1 References and notes: B. L. van der Waerden, Punkte auf der Kugel. Drei Zusätze, Math. Ann. 125 (1952), 213–222, doi:10.1007/BF01343118. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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Tarnai and Zs. Gáspár, Arrangement of 23 points on a sphere (on a conjecture of R. M. Robinson), Proc. Roy. Soc. London Ser. A 433 (1991), 257–267, doi:10.1098/rspa.1991.0046. 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M. Robinson, Arrangements of 24 points on a sphere, Math. Ann. 144 (1961), 17–48, doi:10.1007/BF01396539. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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References and notes: D. A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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Tarnai and Zs. Gáspár, Improved packing of equal circles on a sphere and rigidity of its graph, Math. Proc. Cambridge Philos. Soc. 93 (1983), 191–218, doi:10.1017/S0305004100060485. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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W. Clare and D. L. Kepert, The closest packing of equal circles on a sphere, Proc. Roy. Soc. London Ser. A 405 (1986), 329–344, doi:10.1098/rspa.1986.0056. 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Strohmajer, Über die Verteilung von Punkten auf der Kugel, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 6 (1963), 49–53. 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Danzer, Finite point-sets on S^2 with minimum distance as large as possible, Discrete Math. 60 (1986), 3–66, doi:10.1016/0012-365X(86)90002-6. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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Székely, Sur le problème de Tammes, Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae 17 (1974), 157–175, http://annalesm.elte.hu/annales17-1974/Annales_1974_T-XVII.pdf. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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W. Clare and D. L. Kepert, The closest packing of equal circles on a sphere, Proc. Roy. Soc. London Ser. A 405 (1986), 329–344, doi:10.1098/rspa.1986.0056. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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Székely, Sur le problème de Tammes, Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae 17 (1974), 157–175, http://annalesm.elte.hu/annales17-1974/Annales_1974_T-XVII.pdf. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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M. Robinson, Finite sets of points on a sphere with each nearest to five others, Math. Ann. 179 (1969), 296–318, doi:10.1007/BF01350775. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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Székely, Sur le problème de Tammes, Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae 17 (1974), 157–175, http://annalesm.elte.hu/annales17-1974/Annales_1974_T-XVII.pdf. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996; J. Buddenhagen and D. A. Kottwitz, Multiplicity and symmetry breaking in (conjectured) densest packings of congruent circles on a circle, undated manuscript. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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A. Kottwitz, 1997. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996; J. Buddenhagen and D. A. Kottwitz, Multiplicity and symmetry breaking in (conjectured) densest packings of congruent circles on a circle, undated manuscript. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996; K. J. Nurmela, Constructing spherical codes by global optimization methods, Research Report A32, Digital Systems Laboratory, Helsinki University of Technology, 1995, https://research.ics.aalto.fi/publications/info/bibdb.HUT-TCS-A32.shtml. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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A. Kottwitz, The densest packing of equal circles on a sphere, Acta Cryst. Sect. A 47 (1991), 158–165, doi:10.1107/S0108767390011370. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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A. Kottwitz, 1997. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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A. Kottwitz, 1997. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996; J. Buddenhagen and D. A. Kottwitz, Multiplicity and symmetry breaking in (conjectured) densest packings of congruent circles on a circle, undated manuscript. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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M. Robinson, Finite sets of points on a sphere with each nearest to five others, Math. Ann. 179 (1969), 296–318, doi:10.1007/BF01350775. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith, and others, Tables of spherical codes, 1994–1996. 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Cosine: 0.948765511183 References and notes: X. Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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0.953609890346 References and notes: X. Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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0.953890527065 References and notes: Henry Cohn, 2024. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121 (see footnote 1). 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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0.961828786663 References and notes: X. Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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0.9595850620431908,0.2641822224902475,-0.09697557436632785 0.7395755840357607,-0.6709501667722427,-0.0534212430261881 Dimension: 3 Number of points: 180 Cosine: 0.962128794086 Minimal polynomial: 448505*x^18+1214430*x^17+1867769*x^16+1054048*x^15-1540076*x^14-3472264*x^13-2273628*x^12+709568*x^11+2115198*x^10+712036*x^9-369522*x^8-221472*x^7+20772*x^6+3416*x^5-6956*x^4+256*x^3+81*x^2-18*x+1 References and notes: T. Tarnai, Packing of 180 equal circles on a sphere, Elem. Math. 38 (1983), 119–122, doi:10.5169/seals-37194. 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0.9904869494561224,0.058592709337725604,-0.12450902525347207 Dimension: 3 Number of points: 181 Cosine: 0.962508969964 References and notes: X. Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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0.965083352280 References and notes: Henry Cohn, 2024. 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0.965251036900 References and notes: X. Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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0.967547223611 References and notes: Henry Cohn, 2024. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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0.970127376655 References and notes: Henry Cohn, 2025. 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0.971893361216 References and notes: Henry Cohn, 2025. 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0.45960529397032784,-0.8823201053970928,0.1013617549498416 Dimension: 3 Number of points: 243 Cosine: 0.971985441053 References and notes: Henry Cohn, 2024. 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0.04690138879925862,-0.9794773146893584,-0.19602155426795395 Dimension: 3 Number of points: 260 Cosine: 0.973743009714 References and notes: Henry Cohn, 2024. 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64*x^26+160*x^25+12800*x^24+10984*x^23+825696*x^22-487414*x^21+19978631*x^20-30112768*x^19+156813818*x^18-155014322*x^17-71661645*x^16+39637008*x^15+39387704*x^14+18787364*x^13-12513954*x^12-6708928*x^11+2521884*x^10+241020*x^9-659698*x^8-42552*x^7+38104*x^6-4494*x^5-1445*x^4+512*x^3+58*x^2-10*x-1 References and notes: T. Tarnai and Zs. Gáspár, Multi-symmetric close packings of equal spheres on the spherical surface, Acta Cryst. Sect. A 43 (1987), 612–616, doi:10.1107/S0108767387098842. 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0.975124578016 References and notes: Henry Cohn, 2024. 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Lai, D. Yue, J.-K. Hao, F. Glover, and Z. Lü, Iterated dynamic neighborhood search for packing equal circles on a sphere, Comput. Oper. Res. 151 (2023), Paper No. 106121, 17 pp., doi:10.1016/j.cor.2022.106121. 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0.976439859800 References and notes: Henry Cohn, 2024. 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Cosine: 0.976500970455 References and notes: Henry Cohn, 2024. 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Dimension: 3 Number of points: 291 Cosine: 0.976554952830 References and notes: Henry Cohn, 2024. 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H. Hardin, N. J. A. Sloane, and W. D. Smith, Tables of spherical codes with icosahedral symmetry, 1994–2000. 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0.977704320343 References and notes: Henry Cohn, 2024. 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0.978826091828 References and notes: Henry Cohn, 2024. 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0.979756802402 References and notes: Henry Cohn, 2025. 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0.980684301407 References and notes: Henry Cohn, 2024. 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0.980732527244 References and notes: Henry Cohn, 2024. 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0.980776958770 References and notes: Henry Cohn, 2024. 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Tarnai and Zs. Gáspár, Multi-symmetric close packings of equal spheres on the spherical surface, Acta Cryst. Sect. A 43 (1987), 612–616, doi:10.1107/S0108767387098842. 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0.982281435205 References and notes: Henry Cohn, 2024. 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0.983049964328 References and notes: Henry Cohn, 2024. 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