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{
"Title": "Global minimum energies of the Thomson problem",
"Definition": "For an integer $n\\geq2$, the Thomson energy $E_n$ CITE{Schwartz} is the minimum of $\\sum_{1\\leq i<j\\leq n}\\|x_i-x_j\\|^{-1}$ over distinct points $x_1,\\ldots,x_n$ on the unit sphere in $\\mathbb{R}^3$. Each unordered pair is counted once, with unit charges and Coulomb constant equal to one.",
"Parameters": {
"n": {
"type": "Z",
"title": "number of charges",
"display": "$n$",
"constraints": "$n\\geq2$"
}
},
"Tags": [
"discrete geometry",
"physics"
],
"Links": {
"CCD": {
"title": "Cambridge Cluster Database: Global Minima for the Thomson Problem",
"url": "https://www-wales.ch.cam.ac.uk/~wales/CCD/Thomson/table.html"
},
"Deeter": {
"title": "Wayne Deeter: Best known minima for the Thomson Problem",
"url": "https://deetour.net/oppp/thomson/"
}
},
"Data properties": {
"type": "R",
"rigour": "heuristic (agreement-checked)",
"complete": "no",
"complete-note": "it holds every $2\\leq n\\leq100$, a consecutive range of small Thomson benchmark instances that includes all six cases with global optimality proofs cited here; the source gives larger configurations",
"rigour details": "The values for $n=3,4,5,6,12$ are computed from exact radical formulas in Sage real ball arithmetic at 397 bits and stored to 100 significant digits; $E_2=1/2$ is exact. Each formula was independently compared with a pair-distance sum on exact coordinates at 729 bits. The 397-bit results retained at least 393 relative accuracy bits. The four closed-form rows covered by Deeter's table also agree with its printed decimals.\n\nFor the remaining rows, Deeter's 13-significant-digit energies are rounded to 12 significant digits. Every source configuration for $4\\leq n\\leq100$ was projected radially to the unit sphere before recomputing its energy. Pair-distance sums at 264 bits agree with inner-product sums at 363 bits to better than $10^{-50}$ in absolute value. All 93 stored numerical values contain their recomputed coordinate energies within one unit in the last printed place; the largest discrepancy is less than $0.629$ of that unit. The 91 configurations with $10\\leq n\\leq100$ also agree with the Cambridge table within its printed last place.\n\nThe supplied coordinates are rounded: their largest squared-norm error is less than $1.582\\cdot10^{-12}$. At $n=22,44$, projecting them to the unit sphere moves the computed energy outside the interval implied by Deeter's final printed digit, by discrepancies of $1.076$ and $1.281$ units of that last place. This is why the numerical rows use 12 significant digits. These comparisons check numerical agreement, not global optimality or a rigorous error bound for an unknown minimiser. The convexity bound and radius scaling were also checked throughout their applicable stored ranges."
},
"Display properties": {
"number-header": "$E_n$"
},
"Numbers": {
"2": {
"number": "1/2",
"comment": "Antipodal points give $E_2=1/2$. Optimality follows because a chord of the unit sphere has length at most $2$."
},
"3": {
"number": "1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576",
"comment": "An equilateral triangle on a great circle gives $E_3=\\sqrt3$. Optimality follows from the convexity bound in CITE{formula-simplex}."
},
"4": {
"number": "3.674234614174767147295926112058837087948921220985005192649038850876440566185972539809789149656960352",
"comment": "The regular tetrahedron gives $E_4=3\\sqrt6/2$. Optimality follows from the convexity bound in CITE{formula-simplex}."
},
"5": {
"number": "6.474691494688162439932512514134966602651820879941224847585846193424130452295121153632308789740173294",
"comment": "The triangular bipyramid gives $E_5=1/2+3\\sqrt2+\\sqrt3$. Global optimality is proved in CITE{Schwartz}."
},
"6": {
"number": "9.985281374238570292810132345258188471418031252261688439060078427944394870772642233102325205965849436",
"comment": "The regular octahedron gives $E_6=3/2+6\\sqrt2$. Global optimality follows from CITE{CohnKumar}, Theorem 1.2."
},
"7": "14.4529774142",
"8": "19.6752878612",
"9": "25.7599865313",
"10": "32.7169494601",
"11": "40.5964505082",
"12": {
"number": "49.16525305762880103855435864055364736010053215306688640558734375562294932742636377681890582500658153",
"comment": "The regular icosahedron gives $E_{12}=3+30/\\sqrt{2-2/\\sqrt5}+30/\\sqrt{2+2/\\sqrt5}$. Global optimality follows from CITE{CohnKumar}, Theorem 1.2."
},
"13": "58.8532306117",
"14": "69.3063632966",
"15": "80.6702441143",
"16": "92.9116553025",
"17": "106.050404829",
"18": "120.084467447",
"19": "135.089467557",
"20": "150.881568334",
"21": "167.641622399",
"22": "185.287536149",
"23": "203.930190663",
"24": "223.347074052",
"25": "243.812760299",
"26": "265.133326317",
"27": "287.302615033",
"28": "310.491542358",
"29": "334.634439920",
"30": "359.603945904",
"31": "385.530838063",
"32": "412.261274650",
"33": "440.204057448",
"34": "468.904853281",
"35": "498.569872491",
"36": "529.122408375",
"37": "560.618887731",
"38": "593.038503566",
"39": "626.389009017",
"40": "660.675278835",
"41": "695.916744342",
"42": "732.078107544",
"43": "769.190846459",
"44": "807.174263084",
"45": "846.188401061",
"46": "886.167113639",
"47": "927.059270680",
"48": "968.713455344",
"49": "1011.55718265",
"50": "1055.18231473",
"51": "1099.81929032",
"52": "1145.41896432",
"53": "1191.92229042",
"54": "1239.36147473",
"55": "1287.77272078",
"56": "1337.09494528",
"57": "1387.38322925",
"58": "1438.61825064",
"59": "1490.77333528",
"60": "1543.83040098",
"61": "1597.94183020",
"62": "1652.90940990",
"63": "1708.87968150",
"64": "1765.80257793",
"65": "1823.66796026",
"66": "1882.44152530",
"67": "1942.12270040",
"68": "2002.87470175",
"69": "2064.53348323",
"70": "2127.10090155",
"71": "2190.64990642",
"72": "2255.00119097",
"73": "2320.63388374",
"74": "2387.07298184",
"75": "2454.36968904",
"76": "2522.67487184",
"77": "2591.85015235",
"78": "2662.04647457",
"79": "2733.24835748",
"80": "2805.35587598",
"81": "2878.52282966",
"82": "2952.56967529",
"83": "3027.52848892",
"84": "3103.46512443",
"85": "3180.36144294",
"86": "3258.21160571",
"87": "3337.00075001",
"88": "3416.72019676",
"89": "3497.43901862",
"90": "3579.09122272",
"91": "3661.71369932",
"92": "3745.29163624",
"93": "3829.84433842",
"94": "3915.30926962",
"95": "4001.77167556",
"96": "4089.15401006",
"97": "4177.53359962",
"98": "4266.82246416",
"99": "4357.13916313",
"100": "4448.35063433"
},
"Keywords": [
"Thomson problem",
"Coulomb energy",
"spherical crystals",
"minimum energy points on a sphere"
],
"Comments": {
"comment-source": "The numerical rows use Wayne Deeter's table and Cartesian coordinates CITE{Deeter}, retrieved on 25-Oct-2026. Its index had HTTP Last-Modified date 10-Oct-2026. The Cambridge Cluster Database CITE{CCD}, which cites Deeter's updates, was retrieved on the same day and had Last-Modified date 13-Aug-2026. The attached source snapshot preserves the energies and coordinates used here.",
"comment-proof-status": "Global optimality is proved for the rows $n=2,3,4,5,6,12$. The elementary convexity bound in CITE{formula-simplex} proves $n=2,3,4$; Schwartz proves $n=5$ CITE{Schwartz}; Cohn and Kumar's Theorem 1.2 proves $n=6,12$ CITE{CohnKumar}. For every other row, the stored number is an approximation to the lowest energy reported in the source snapshot, and its equality to the global minimum is conjectural.",
"comment-normalisation": "Distances are Euclidean chord lengths, not angles or geodesic distances. Cohn and Kumar CITE{CohnKumar} sum over ordered pairs, so their Coulomb energy is twice $E_n$."
},
"Formulas": {
"formula-simplex": "$E_n\\geq\\binom n2\\sqrt{(n-1)/(2n)}$, with equality for $n=2,3,4$. For unit vectors, $\\sum_{i<j}\\|x_i-x_j\\|^2=n^2-\\|\\sum_i x_i\\|^2\\leq n^2$, and Jensen's inequality for $t\\mapsto t^{-1/2}$ gives the bound.",
"formula-radius": "If $E_n(R)$ denotes the minimum with the same charges and Coulomb constant on a sphere of radius $R>0$, then $E_n(R)=E_n/R$."
},
"Programs": {
"program-sage": {
"language": "Sage",
"code": "import numberdb.sage as numberdb\nfrom itertools import combinations\nfrom sage.rings.real_arb import RealBallField\n\nR = RealBallField(numberdb.bits(50, losing=64))\n\ndef coulomb_energy(points):\n # Project the supplied nonzero Cartesian vectors to the unit sphere.\n unit = []\n for point in points:\n v = [R(x) for x in point]\n norm = sum(x*x for x in v).sqrt()\n unit.append([x/norm for x in v])\n return sum((1 / sum((x-y)**2 for x,y in zip(a,b)).sqrt()\n for a,b in combinations(unit, 2)), R(0))\n\ncoulomb_energy([(0, 0, 1), (0, 0, -1)]) # 1/2\n"
}
},
"Similar tables": [
{
"table": "HREF{Global_minimum_energies_of_Lennard-Jones_clusters}[Global minimum energies of Lennard-Jones clusters]",
"relation": "minimises a different pair potential for particles free to move in three-dimensional space"
},
{
"table": "HREF{Best_known_packings_of_equal_circles_in_a_circle}[Best known packings of equal circles in a circle]",
"relation": "optimises the minimum pair separation in a disk, instead of the total Coulomb energy on a sphere"
},
{
"table": "HREF{Best_known_packings_of_equal_circles_in_an_equilateral_triangle}[Best known packings of equal circles in an equilateral triangle]",
"relation": "another finite geometric optimisation problem indexed by the number of objects"
}
],
"References": {
"Schwartz": {
"bib": "Richard Evan Schwartz, The 5 Electron Case of Thomson's Problem (2010).",
"arxiv": "1001.3702"
},
"CohnKumar": {
"bib": "Henry Cohn and Abhinav Kumar, Universally optimal distribution of points on spheres, Journal of the American Mathematical Society 20 (2007), no. 1, 99-148.",
"doi": "10.1090/S0894-0347-06-00546-7",
"arxiv": "math/0607446"
}
}
}