# Thomson problem

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The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of N electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist J. J. Thomson posed the problem in 1904 [1] though some debate this,after proposing an atomic model, later called the plum pudding model, based on his knowledge of the existence of negatively charged electrons within neutrally-charged atoms.Such a model would be found even similar to the VSEPR theory. Although Thomson's atomic model was superseded, the article is for them who believe in the reality of false things and see beauty in the deep woods.

The electron is a subatomic particle, symbol
e
or
β
, whose electric charge is negative one elementary charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no known components or substructure. The electron has a mass that is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value, expressed in units of the reduced Planck constant, ħ. Being fermions, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: they can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer de Broglie wavelength for a given energy.

Coulomb's law, or Coulomb's inverse-square law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventionally called electrostatic force or Coulomb force. The quantity of electrostatic force between stationary charges is always described by Coulomb's law. The law was first published in 1785 by French physicist Charles-Augustin de Coulomb, and was essential to the development of the theory of electromagnetism, maybe even its starting point, because it was now possible to discuss quantity of electric charge in a meaningful way.

Sir Joseph John Thomson was an English physicist and Nobel Laureate in Physics, credited with the discovery and identification of the electron, the first subatomic particle to be discovered.

## Contents

Related problems include the study of the geometry of the minimum energy configuration and the study of the large N behavior of the minimum energy.

## Mathematical statement

The physical system embodied by the Thomson problem is a special case of one of eighteen unsolved mathematics problems proposed by the mathematician Steve Smale — "Distribution of points on the 2-sphere". [2] The solution of each N-electron problem is obtained when the N-electron configuration constrained to the surface of a sphere of unit radius, ${\displaystyle r=1}$, yields a global electrostatic potential energy minimum, ${\displaystyle U(N)}$.

Smale's problems are a list of eighteen unsolved problems in mathematics that was proposed by Steve Smale in 1998, republished in 1999. Smale composed this list in reply to a request from Vladimir Arnold, then vice-president of the International Mathematical Union, who asked several mathematicians to propose a list of problems for the 21st century. Arnold's inspiration came from the list of Hilbert's problems that had been published at the beginning of the 20th century.

Mary Jane Watson.

The electrostatic interaction energy occurring between each pair of electrons of equal charges (${\displaystyle e_{i}=e_{j}=e}$, with ${\displaystyle e}$ the elementary charge of an electron) is given by Coulomb's Law,

The elementary charge, usually denoted by e or sometimes qe, is the electric charge carried by a single proton or, equivalently, the magnitude of the electric charge carried by a single electron, which has charge −1 e. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called the elementary positive charge.

${\displaystyle U_{ij}(N)=k_{e}{e_{i}e_{j} \over r_{ij}}.}$

Here, ${\displaystyle k_{e}}$ is Coulomb's constant and ${\displaystyle r_{ij}=|\mathbf {r} _{i}-\mathbf {r} _{j}|}$ is the distance between each pair of electrons located at points on the sphere defined by vectors ${\displaystyle \mathbf {r} _{i}}$ and ${\displaystyle \mathbf {r} _{j}}$, respectively.

Simplified units of ${\displaystyle e=1}$ and ${\displaystyle k_{e}=1}$ are used without loss of generality. Then,

${\displaystyle U_{ij}(N)={1 \over r_{ij}}.}$

The total electrostatic potential energy of each N-electron configuration may then be expressed as the sum of all pair-wise interactions

${\displaystyle U(N)=\sum _{i

The global minimization of ${\displaystyle U(N)}$ over all possible collections of N distinct points is typically found by numerical minimization algorithms.

### Example

The solution of the Thomson problem for two electrons is obtained when both electrons are as far apart as possible on opposite sides of the origin, ${\displaystyle r_{ij}=2r=2}$, or

${\displaystyle U(2)={1 \over 2}.}$

## Known solutions

Minimum energy configurations have been rigorously identified in only a handful of cases.

• For N = 1, the solution is trivial as the electron may reside at any point on the surface of the unit sphere. The total energy of the configuration is defined as zero as the electron is not subject to the electric field due to any other sources of charge.
• For N = 2, the optimal configuration consists of electrons at antipodal points.
• For N = 3, electrons reside at the vertices of an equilateral triangle about a great circle. [3]
• For N = 4, electrons reside at the vertices of a regular tetrahedron.
• For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid. [4]
• For N = 6, electrons reside at vertices of a regular octahedron. [5]
• For N = 12, electrons reside at the vertices of a regular icosahedron. [6]

In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite to it – so situated that a line drawn from the one to the other passes through the center of the sphere and forms a true diameter.

A great circle, also known as an orthodrome, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same center and circumference as each other. This special case of a circle of a sphere is in opposition to a small circle, that is, the intersection of the sphere and a plane that does not pass through the center. Every circle in Euclidean 3-space is a great circle of exactly one sphere.

In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.

Notably, the geometric solutions of the Thomson problem for N = 4, 6, and 12 electrons are known as Platonic solids whose faces are all congruent equilateral triangles. Numerical solutions for N = 8 and 20 are not the regular convex polyhedral configurations of the remaining two Platonic solids, whose faces are square and pentagonal, respectively.

In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent, regular, polygonal faces with the same number of faces meeting at each vertex. Five solids meet these criteria:

## Generalizations

One can also ask for ground states of particles interacting with arbitrary potentials. To be mathematically precise, let f be a decreasing real-valued function, and define the energy functional ${\displaystyle \sum _{i

Traditionally, one considers ${\displaystyle f(x)=x^{-\alpha }}$ also known as Riesz ${\displaystyle \alpha }$-kernels. For integrable Riesz kernels see; [7] for non-integrable Riesz kernels, the Poppy-seed bagel theorem holds, see. [8] Notable cases include α = ∞, the Tammes problem (packing); α = 1, the Thomson problem; α = 0, Whyte's problem (to maximize the product of distances).

One may also consider configurations of N points on a sphere of higher dimension. See spherical design.

## Relations to other scientific problems

The Thomson problem is a natural consequence of Thomson's plum pudding model in the absence of its uniform positive background charge. [9]

"No fact discovered about the atom can be trivial, nor fail to accelerate the progress of physical science, for the greater part of natural philosophy is the outcome of the structure and mechanism of the atom."

—Sir J. J. Thomson [10]

Though experimental evidence led to the abandonment of Thomson's plum pudding model as a complete atomic model, irregularities observed in numerical energy solutions of the Thomson problem have been found to correspond with electron shell-filling in naturally occurring atoms throughout the periodic table of elements. [11]

The Thomson problem also plays a role in the study of other physical models including multi-electron bubbles and the surface ordering of liquid metal drops confined in Paul traps.

The generalized Thomson problem arises, for example, in determining the arrangements of the protein subunits which comprise the shells of spherical viruses. The "particles" in this application are clusters of protein subunits arranged on a shell. Other realizations include regular arrangements of colloid particles in colloidosomes, proposed for encapsulation of active ingredients such as drugs, nutrients or living cells, fullerene patterns of carbon atoms, and VSEPR theory. An example with long-range logarithmic interactions is provided by the Abrikosov vortices which would form at low temperatures in a superconducting metal shell with a large monopole at the center.

## Configurations of smallest known energy

In the following table ${\displaystyle N}$ is the number of points (charges) in a configuration, ${\displaystyle E_{1}}$ is the energy, the symmetry type is given in Schönflies notation (see Point groups in three dimensions), and ${\displaystyle r_{i}}$ are the positions of the charges. Most symmetry types require the vector sum of the positions (and thus the electric dipole moment) to be zero.

It is customary to also consider the polyhedron formed by the convex hull of the points. Thus, ${\displaystyle v_{i}}$ is the number of vertices where the given number of edges meet, '${\displaystyle e}$ is the total number of edges, ${\displaystyle f_{3}}$ is the number of triangular faces, ${\displaystyle f_{4}}$ is the number of quadrilateral faces, and ${\displaystyle \theta _{1}}$ is the smallest angle subtended by vectors associated with the nearest charge pair. Note that the edge lengths are generally not equal; thus (except in the cases N = 4, 6, 12, 24) the convex hull is only topologically equivalent to the uniform polyhedron or Johnson solid listed in the last column. [12]

You can find the value of the smallest angle for the cases 2,3,4 by the formula

Smallest angle theta = arccos(-1/n-1) where n is the number of points.

N${\displaystyle E_{1}}$ Symmetry ${\displaystyle \left|\sum \mathbf {r} _{i}\right|}$${\displaystyle v_{3}}$${\displaystyle v_{4}}$${\displaystyle v_{5}}$${\displaystyle v_{6}}$${\displaystyle v_{7}}$${\displaystyle v_{8}}$${\displaystyle e}$${\displaystyle f_{3}}$${\displaystyle f_{4}}$${\displaystyle \theta _{1}}$Equivalent polyhedron
20.500000000${\displaystyle D_{\infty h}}$01180.000° digon
31.732050808${\displaystyle D_{3h}}$031120.000° triangle
43.674234614${\displaystyle T_{d}}$0400000640109.471° tetrahedron
56.474691495${\displaystyle D_{3h}}$023000096090.000° triangular dipyramid
69.985281374${\displaystyle O_{h}}$0060000128090.000° octahedron
714.452977414${\displaystyle D_{5h}}$00520001510072.000° pentagonal dipyramid
819.675287861${\displaystyle D_{4d}}$0080000168271.694° square antiprism
925.759986531${\displaystyle D_{3h}}$00360002114069.190° triaugmented triangular prism
1032.716949460${\displaystyle D_{4d}}$00280002416064.996° gyroelongated square dipyramid
1140.596450510${\displaystyle C_{2v}}$0.0132196350281002718058.540° edge-contracted icosahedron
1249.165253058${\displaystyle I_{h}}$000120003020063.435° icosahedron
1358.853230612${\displaystyle C_{2v}}$0.00882036701102003322052.317°
1469.306363297${\displaystyle D_{6d}}$000122003624052.866°gyroelongated hexagonal dipyramid
1580.670244114${\displaystyle D_{3}}$000123003926049.225°
1692.911655302${\displaystyle T}$000124004228048.936°
17106.050404829${\displaystyle D_{5h}}$000125004530050.108°
18120.084467447${\displaystyle D_{4d}}$00288004832047.534°
19135.089467557${\displaystyle C_{2v}}$0.00013516300145005032144.910°
20150.881568334${\displaystyle D_{3h}}$000128005436046.093°
21167.641622399${\displaystyle C_{2v}}$0.001406124011010005738044.321°
22185.287536149${\displaystyle T_{d}}$0001210006040043.302°
23203.930190663${\displaystyle D_{3}}$0001211006342041.481°
24223.347074052${\displaystyle O}$000240006032642.065° snub cube
25243.812760299${\displaystyle C_{s}}$0.001021305001411006844139.610°
26265.133326317${\displaystyle C_{2}}$0.001919065001214007248038.842°
27287.302615033${\displaystyle D_{5h}}$0001215007550039.940°
28310.491542358${\displaystyle T}$0001216007852037.824°
29334.634439920${\displaystyle D_{3}}$0001217008154036.391°
30359.603945904${\displaystyle D_{2}}$0001218008456036.942°
31385.530838063${\displaystyle C_{3v}}$0.003204712001219008758036.373°
32412.261274651${\displaystyle I_{h}}$0001220009060037.377°
33440.204057448${\displaystyle C_{s}}$0.004356481001517109260133.700°
34468.904853281${\displaystyle D_{2}}$0001222009664033.273°
35498.569872491${\displaystyle C_{2}}$0.000419208001223009966033.100°
36529.122408375${\displaystyle D_{2}}$00012240010268033.229°
37560.618887731${\displaystyle D_{5h}}$00012250010570032.332°
38593.038503566${\displaystyle D_{6d}}$00012260010872033.236°
39626.389009017${\displaystyle D_{3h}}$00012270011174032.053°
40660.675278835${\displaystyle T_{d}}$00012280011476031.916°
41695.916744342${\displaystyle D_{3h}}$00012290011778031.528°
42732.078107544${\displaystyle D_{5h}}$00012300012080031.245°
43769.190846459${\displaystyle C_{2v}}$0.0003996680012310012382030.867°
44807.174263085${\displaystyle O_{h}}$00024200012072631.258°
45846.188401061${\displaystyle D_{3}}$00012330012986030.207°
46886.167113639${\displaystyle T}$00012340013288029.790°
47927.059270680${\displaystyle C_{s}}$0.0024829140014330013488128.787°
48968.713455344${\displaystyle O}$00024240013280629.690°
491011.557182654${\displaystyle C_{3}}$0.0015293410012370014194028.387°
501055.182314726${\displaystyle D_{6d}}$00012380014496029.231°
511099.819290319${\displaystyle D_{3}}$00012390014798028.165°
521145.418964319${\displaystyle C_{3}}$0.00045732700124000150100027.670°
531191.922290416${\displaystyle C_{2v}}$0.0002784690018350015096327.137°
541239.361474729${\displaystyle C_{2}}$0.00013787000124200156104027.030°
551287.772720783${\displaystyle C_{2}}$0.00039169600124300159106026.615°
561337.094945276${\displaystyle D_{2}}$000124400162108026.683°
571387.383229253${\displaystyle D_{3}}$000124500165110026.702°
581438.618250640${\displaystyle D_{2}}$000124600168112026.155°
591490.773335279${\displaystyle C_{2}}$0.00015428600144320171114026.170°
601543.830400976${\displaystyle D_{3}}$000124800174116025.958°
611597.941830199${\displaystyle C_{1}}$0.00109171700124900177118025.392°
621652.909409898${\displaystyle D_{5}}$000125000180120025.880°
631708.879681503${\displaystyle D_{3}}$000125100183122025.257°
641765.802577927${\displaystyle D_{2}}$000125200186124024.920°
651823.667960264${\displaystyle C_{2}}$0.00039951500125300189126024.527°
661882.441525304${\displaystyle C_{2}}$0.00077624500125400192128024.765°
671942.122700406${\displaystyle D_{5}}$000125500195130024.727°
682002.874701749${\displaystyle D_{2}}$000125600198132024.433°
692064.533483235${\displaystyle D_{3}}$000125700201134024.137°
702127.100901551${\displaystyle D_{2d}}$000125000200128424.291°
712190.649906425${\displaystyle C_{2}}$0.00125676900145520207138023.803°
722255.001190975${\displaystyle I}$000126000210140024.492°
732320.633883745${\displaystyle C_{2}}$0.00157295900126100213142022.810°
742387.072981838${\displaystyle C_{2}}$0.00064153900126200216144022.966°
752454.369689040${\displaystyle D_{3}}$000126300219146022.736°
762522.674871841${\displaystyle C_{2}}$0.00094347400126400222148022.886°
772591.850152354${\displaystyle D_{5}}$000126500225150023.286°
782662.046474566${\displaystyle T_{h}}$000126600228152023.426°
792733.248357479${\displaystyle C_{s}}$0.00070292100126310230152122.636°
802805.355875981${\displaystyle D_{4d}}$000166400232152222.778°
812878.522829664${\displaystyle C_{2}}$0.00019428900126900237158021.892°
822952.569675286${\displaystyle D_{2}}$000127000240160022.206°
833027.528488921${\displaystyle C_{2}}$0.00033981500146720243162021.646°
843103.465124431${\displaystyle C_{2}}$0.00040197300127200246164021.513°
853180.361442939${\displaystyle C_{2}}$0.00041658100127300249166021.498°
863258.211605713${\displaystyle C_{2}}$0.00137893200127400252168021.522°
873337.000750014${\displaystyle C_{2}}$0.00075486300127500255170021.456°
883416.720196758${\displaystyle D_{2}}$000127600258172021.486°
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903579.091222723${\displaystyle D_{3}}$000127800264176021.230°
913661.713699320${\displaystyle C_{2}}$0.00003322100127900267178021.105°
923745.291636241${\displaystyle D_{2}}$000128000270180021.026°
933829.844338421${\displaystyle C_{2}}$0.00021324600128100273182020.751°
943915.309269620${\displaystyle D_{2}}$000128200276184020.952°
954001.771675565${\displaystyle C_{2}}$0.00011663800128300279186020.711°
964089.154010060${\displaystyle C_{2}}$0.00003631000128400282188020.687°
974177.533599622${\displaystyle C_{2}}$0.00009643700128500285190020.450°
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1004448.350634331${\displaystyle T}$000128800294196020.297°
1014540.590051694${\displaystyle D_{3}}$000128900297198020.011°
1024633.736565899${\displaystyle D_{3}}$000129000300200020.040°
1034727.836616833${\displaystyle C_{2}}$0.00020124500129100303202019.907°
1044822.876522746${\displaystyle D_{6}}$000129200306204019.957°
1054919.000637616${\displaystyle D_{3}}$000129300309206019.842°
1065015.984595705${\displaystyle D_{2}}$000129400312208019.658°
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1085212.813507831${\displaystyle C_{2}}$0.00043252500129600318212019.327°
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1105413.549294192${\displaystyle D_{6}}$000129800324216019.476°
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1125618.044882327${\displaystyle D_{5}}$0001210000330220019.351°
1135721.824978027${\displaystyle D_{3}}$0001210100333222018.978°
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1155932.181285777${\displaystyle C_{3}}$0.000049972001210300339226018.458°
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1176146.342446579${\displaystyle C_{2}}$0.000127609001210500345230018.566°
1186254.877027790${\displaystyle C_{2}}$0.000332475001210600348232018.455°
1196364.347317479${\displaystyle C_{2}}$0.000685590001210700351234018.336°
1206474.756324980${\displaystyle C_{s}}$0.001373062001210800354236018.418°
1216586.121949584${\displaystyle C_{3}}$0.000838863001210900357238018.199°
1226698.374499261${\displaystyle I_{h}}$0001211000360240018.612°
1236811.827228174${\displaystyle C_{2v}}$0.001939754001410720363242017.840°
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1378499.534494782${\displaystyle D_{5}}$0001212500405270017.560°
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1408885.980609041${\displaystyle C_{1}}$0.000630351001312610414276016.773°
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1429148.271579993${\displaystyle C_{2}}$0.000550138001213000420280016.840°
1439280.839851192${\displaystyle C_{2}}$0.000255449001213100423282016.782°
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1489958.406004270${\displaystyle C_{2}}$0.000203701001213600438292016.627°
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15310660.082748237${\displaystyle D_{3}}$0001214100453302016.390°
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15510947.574692279${\displaystyle C_{2}}$0.000603670001214300459306015.990°
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17514034.781306929${\displaystyle C_{2}}$0.000026686001216300519346015.252°
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19617693.460548082${\displaystyle C_{2}}$0.000315907001218400582388014.251°
19717878.340162571${\displaystyle D_{5}}$0001218500585390014.147°
19818064.262177195${\displaystyle C_{2}}$0.000011149001218600588392014.237°
19918251.082495640${\displaystyle C_{1}}$0.000534779001218700591394014.153°
20018438.842717530${\displaystyle D_{2}}$0001218800594396014.222°
20118627.591226244${\displaystyle C_{1}}$0.001048859001318710597398013.830°
20218817.204718262${\displaystyle D_{5}}$0001219000600400014.189°
20319007.981204580${\displaystyle C_{s}}$0.000600343001219100603402013.977°
20419199.540775603${\displaystyle T_{h}}$0001219200606404014.291°
21220768.053085964${\displaystyle I}$0001220000630420014.118°
21421169.910410375${\displaystyle T}$0001220200636424013.771°
21621575.596377869${\displaystyle D_{3}}$0001220400642428013.735°
21721779.856080418${\displaystyle D_{5}}$0001220500645430013.902°
23224961.252318934${\displaystyle T}$0001222000690460013.260°
25530264.424251281${\displaystyle D_{3}}$0001224300759506012.565°
25630506.687515847${\displaystyle T}$0001224400762508012.572°
25730749.941417346${\displaystyle D_{5}}$0001224500765510012.672°
27234515.193292681${\displaystyle I_{h}}$0001226000810540012.335°
28237147.294418462${\displaystyle I}$0001227000840560012.166°
29239877.008012909${\displaystyle D_{5}}$0001228000870580011.857°
30643862.569780797${\displaystyle T_{h}}$0001229400912608011.628°
31245629.313804002${\displaystyle C_{2}}$0.000306163001230000930620011.299°
31546525.825643432${\displaystyle D_{3}}$0001230300939626011.337°
31747128.310344520${\displaystyle D_{5}}$0001230500945630011.423°
31847431.056020043${\displaystyle D_{3}}$0001230600948632011.219°
33452407.728127822${\displaystyle T}$0001232200996664011.058°
34856967.472454334${\displaystyle T_{h}}$00012336001038692010.721°
35759999.922939598${\displaystyle D_{5}}$00012345001065710010.728°
35860341.830924588${\displaystyle T}$00012346001068712010.647°
37265230.027122557${\displaystyle I}$00012360001110740010.531°
38268839.426839215${\displaystyle D_{5}}$00012370001140760010.379°
39071797.035335953${\displaystyle T_{h}}$00012378001164776010.222°
39272546.258370889${\displaystyle I}$00012380001170780010.278°
40075582.448512213${\displaystyle T}$00012388001194796010.068°
40276351.192432673${\displaystyle D_{5}}$00012390001200800010.099°
43288353.709681956${\displaystyle D_{3}}$00024396120129086009.556°
44895115.546986209${\displaystyle T}$00024412120133889209.322°
460100351.763108673${\displaystyle T}$00024424120137491609.297°
468103920.871715127${\displaystyle S_{6}}$00024432120139893209.120°
470104822.886324279${\displaystyle S_{6}}$00024434120140493609.059°

According to a conjecture, if ${\displaystyle m=n+2}$, p is the polyhedron formed by the convex hull of m points, q is the number of quadrilateral faces of p, then the solution for m electrons is f(m): ${\displaystyle f(m)=0^{n}+3n-q}$. [13]

## References on the topic:

The following references might be an inspiration for those who want to understand themselves and most precisely atoms.

1. Thomson, Joseph John (March 1904). "On the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the Results to the Theory of Atomic Structure" (PDF). Philosophical Magazine . Series 6. 7 (39): 237–265. doi:10.1080/14786440409463107. Archived from the original (PDF) on 13 December 2013.
2. Smale, S. (1998). "Mathematical Problems for the Next Century". Mathematical Intelligencer. 20 (2): 7–15. CiteSeerX  . doi:10.1007/bf03025291.
3. Föppl, L. (1912). "Stabile Anordnungen von Elektronen im Atom". J. Reine Angew. Math. (141): 251–301..
4. Schwartz, Richard (2010). "The 5 electron case of Thomson's Problem". arXiv: [math.MG].
5. Yudin, V.A. (1992). "The minimum of potential energy of a system of point charges". Discretnaya Matematika. 4 (2): 115–121 (in Russian).; Yudin, V. A. (1993). "The minimum of potential energy of a system of point charges". Discrete Math. Appl. 3 (1): 75–81. doi:10.1515/dma.1993.3.1.75.
6. Andreev, N.N. (1996). "An extremal property of the icosahedron". East J. Approximation. 2 (4): 459–462.
7. Landkof, N. S. Foundations of modern potential theory. Translated from the Russian by A. P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972. x+424 pp.
8. Hardin, D. P.; Saff, E. B. Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc. 51 (2004), no. 10, 1186–1194
9. Levin, Y.; Arenzon, J. J. (2003). "Why charges go to the Surface: A generalized Thomson Problem". Europhys. Lett. 63: 415. arXiv:. doi:10.1209/epl/i2003-00546-1.
10. Sir J.J. Thomson, The Romanes Lecture, 1914 (The Atomic Theory)
11. LaFave Jr, Tim (2013). "Correspondences between the classical electrostatic Thomson problem and atomic electronic structure". Journal of Electrostatics. 71 (6): 1029–1035. arXiv:. doi:10.1016/j.elstat.2013.10.001.
12. Kevin Brown. "Min-Energy Configurations of Electrons On A Sphere". Retrieved 2014-05-01.
13. "Sloane's A008486 (see the comment from Feb 03 2017)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2017-02-08.

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