Thomson problem
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 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.
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". The solution of each N-electron problem is obtained when the N-electron configuration constrained to the surface of a sphere of unit radius,, yields a global electrostatic potential energy minimum,.The electrostatic interaction energy occurring between each pair of electrons of equal charges is given by Coulomb's Law,
Here, is Coulomb's constant and is the distance between each pair of electrons located at points on the sphere defined by vectors and, respectively.
Simplified units of and are used without loss of generality. Then,
The total electrostatic potential energy of each N-electron configuration may then be expressed as the sum of all pair-wise interactions
The global minimization of 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,, orKnown 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.
- 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.
- For N = 6, electrons reside at vertices of a regular octahedron.
- For N = 12, electrons reside at the vertices of a regular icosahedron.
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
Traditionally, one considers also known as Riesz -kernels. For integrable Riesz kernels see; for non-integrable Riesz kernels, the Poppy-seed bagel theorem holds, see. Notable cases include α = ∞, the Tammes problem ; α = 1, the Thomson problem; α = 0, Whyte's problem.
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.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.
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 is the number of points in a configuration, is the energy, the symmetry type is given in Schönflies notation, and are the positions of the charges. Most symmetry types require the vector sum of the positions to be zero.It is customary to also consider the polyhedron formed by the convex hull of the points. Thus, is the number of vertices where the given number of edges meet, ' is the total number of edges, is the number of triangular faces, is the number of quadrilateral faces, and is the smallest angle subtended by vectors associated with the nearest charge pair. Note that the edge lengths are generally not equal; thus the convex hull is only topologically equivalent to the figure listed in the last column.
| N | Symmetry | Equivalent polyhedron | ||||||||||||
| 2 | 0.500000000 | 0 | – | – | – | – | – | – | 1 | – | – | 180.000° | digon | |
| 3 | 1.732050808 | 0 | – | – | – | – | – | – | 3 | 1 | – | 120.000° | triangle | |
| 4 | 3.674234614 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 6 | 4 | 0 | 109.471° | tetrahedron | |
| 5 | 6.474691495 | 0 | 2 | 3 | 0 | 0 | 0 | 0 | 9 | 6 | 0 | 90.000° | triangular dipyramid | |
| 6 | 9.985281374 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 12 | 8 | 0 | 90.000° | octahedron | |
| 7 | 14.452977414 | 0 | 0 | 5 | 2 | 0 | 0 | 0 | 15 | 10 | 0 | 72.000° | pentagonal dipyramid | |
| 8 | 19.675287861 | 0 | 0 | 8 | 0 | 0 | 0 | 0 | 16 | 8 | 2 | 71.694° | square antiprism | |
| 9 | 25.759986531 | 0 | 0 | 3 | 6 | 0 | 0 | 0 | 21 | 14 | 0 | 69.190° | triaugmented triangular prism | |
| 10 | 32.716949460 | 0 | 0 | 2 | 8 | 0 | 0 | 0 | 24 | 16 | 0 | 64.996° | gyroelongated square dipyramid | |
| 11 | 40.596450510 | 0.013219635 | 0 | 2 | 8 | 1 | 0 | 0 | 27 | 18 | 0 | 58.540° | edge-contracted icosahedron | |
| 12 | 49.165253058 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 30 | 20 | 0 | 63.435° | icosahedron | |
| 13 | 58.853230612 | 0.008820367 | 0 | 1 | 10 | 2 | 0 | 0 | 33 | 22 | 0 | 52.317° | - | |
| 14 | 69.306363297 | 0 | 0 | 0 | 12 | 2 | 0 | 0 | 36 | 24 | 0 | 52.866° | gyroelongated hexagonal dipyramid | |
| 15 | 80.670244114 | 0 | 0 | 0 | 12 | 3 | 0 | 0 | 39 | 26 | 0 | 49.225° | - | |
| 16 | 92.911655302 | 0 | 0 | 0 | 12 | 4 | 0 | 0 | 42 | 28 | 0 | 48.936° | - | |
| 17 | 106.050404829 | 0 | 0 | 0 | 12 | 5 | 0 | 0 | 45 | 30 | 0 | 50.108° | double-gyroelongated pentagonal dipyramid | |
| 18 | 120.084467447 | 0 | 0 | 2 | 8 | 8 | 0 | 0 | 48 | 32 | 0 | 47.534° | - | |
| 19 | 135.089467557 | 0.000135163 | 0 | 0 | 14 | 5 | 0 | 0 | 50 | 32 | 1 | 44.910° | - | |
| 20 | 150.881568334 | 0 | 0 | 0 | 12 | 8 | 0 | 0 | 54 | 36 | 0 | 46.093° | - | |
| 21 | 167.641622399 | 0.001406124 | 0 | 1 | 10 | 10 | 0 | 0 | 57 | 38 | 0 | 44.321° | - | |
| 22 | 185.287536149 | 0 | 0 | 0 | 12 | 10 | 0 | 0 | 60 | 40 | 0 | 43.302° | - | |
| 23 | 203.930190663 | 0 | 0 | 0 | 12 | 11 | 0 | 0 | 63 | 42 | 0 | 41.481° | - | |
| 24 | 223.347074052 | 0 | 0 | 0 | 24 | 0 | 0 | 0 | 60 | 32 | 6 | 42.065° | snub cube | |
| 25 | 243.812760299 | 0.001021305 | 0 | 0 | 14 | 11 | 0 | 0 | 68 | 44 | 1 | 39.610° | - | |
| 26 | 265.133326317 | 0.001919065 | 0 | 0 | 12 | 14 | 0 | 0 | 72 | 48 | 0 | 38.842° | - | |
| 27 | 287.302615033 | 0 | 0 | 0 | 12 | 15 | 0 | 0 | 75 | 50 | 0 | 39.940° | - | |
| 28 | 310.491542358 | 0 | 0 | 0 | 12 | 16 | 0 | 0 | 78 | 52 | 0 | 37.824° | - | |
| 29 | 334.634439920 | 0 | 0 | 0 | 12 | 17 | 0 | 0 | 81 | 54 | 0 | 36.391° | - | |
| 30 | 359.603945904 | 0 | 0 | 0 | 12 | 18 | 0 | 0 | 84 | 56 | 0 | 36.942° | - | |
| 31 | 385.530838063 | 0.003204712 | 0 | 0 | 12 | 19 | 0 | 0 | 87 | 58 | 0 | 36.373° | - | |
| 32 | 412.261274651 | 0 | 0 | 0 | 12 | 20 | 0 | 0 | 90 | 60 | 0 | 37.377° | pentakis dodecahedron | |
| 33 | 440.204057448 | 0.004356481 | 0 | 0 | 15 | 17 | 1 | 0 | 92 | 60 | 1 | 33.700° | - | |
| 34 | 468.904853281 | 0 | 0 | 0 | 12 | 22 | 0 | 0 | 96 | 64 | 0 | 33.273° | - | |
| 35 | 498.569872491 | 0.000419208 | 0 | 0 | 12 | 23 | 0 | 0 | 99 | 66 | 0 | 33.100° | - | |
| 36 | 529.122408375 | 0 | 0 | 0 | 12 | 24 | 0 | 0 | 102 | 68 | 0 | 33.229° | - | |
| 37 | 560.618887731 | 0 | 0 | 0 | 12 | 25 | 0 | 0 | 105 | 70 | 0 | 32.332° | - | |
| 38 | 593.038503566 | 0 | 0 | 0 | 12 | 26 | 0 | 0 | 108 | 72 | 0 | 33.236° | - | |
| 39 | 626.389009017 | 0 | 0 | 0 | 12 | 27 | 0 | 0 | 111 | 74 | 0 | 32.053° | - | |
| 40 | 660.675278835 | 0 | 0 | 0 | 12 | 28 | 0 | 0 | 114 | 76 | 0 | 31.916° | - | |
| 41 | 695.916744342 | 0 | 0 | 0 | 12 | 29 | 0 | 0 | 117 | 78 | 0 | 31.528° | - | |
| 42 | 732.078107544 | 0 | 0 | 0 | 12 | 30 | 0 | 0 | 120 | 80 | 0 | 31.245° | - | |
| 43 | 769.190846459 | 0.000399668 | 0 | 0 | 12 | 31 | 0 | 0 | 123 | 82 | 0 | 30.867° | - | |
| 44 | 807.174263085 | 0 | 0 | 0 | 24 | 20 | 0 | 0 | 120 | 72 | 6 | 31.258° | - | |
| 45 | 846.188401061 | 0 | 0 | 0 | 12 | 33 | 0 | 0 | 129 | 86 | 0 | 30.207° | - | |
| 46 | 886.167113639 | 0 | 0 | 0 | 12 | 34 | 0 | 0 | 132 | 88 | 0 | 29.790° | - | |
| 47 | 927.059270680 | 0.002482914 | 0 | 0 | 14 | 33 | 0 | 0 | 134 | 88 | 1 | 28.787° | - | |
| 48 | 968.713455344 | 0 | 0 | 0 | 24 | 24 | 0 | 0 | 132 | 80 | 6 | 29.690° | - | |
| 49 | 1011.557182654 | 0.001529341 | 0 | 0 | 12 | 37 | 0 | 0 | 141 | 94 | 0 | 28.387° | - | |
| 50 | 1055.182314726 | 0 | 0 | 0 | 12 | 38 | 0 | 0 | 144 | 96 | 0 | 29.231° | - | |
| 51 | 1099.819290319 | 0 | 0 | 0 | 12 | 39 | 0 | 0 | 147 | 98 | 0 | 28.165° | - | |
| 52 | 1145.418964319 | 0.000457327 | 0 | 0 | 12 | 40 | 0 | 0 | 150 | 100 | 0 | 27.670° | - | |
| 53 | 1191.922290416 | 0.000278469 | 0 | 0 | 18 | 35 | 0 | 0 | 150 | 96 | 3 | 27.137° | - | |
| 54 | 1239.361474729 | 0.000137870 | 0 | 0 | 12 | 42 | 0 | 0 | 156 | 104 | 0 | 27.030° | - | |
| 55 | 1287.772720783 | 0.000391696 | 0 | 0 | 12 | 43 | 0 | 0 | 159 | 106 | 0 | 26.615° | - | |
| 56 | 1337.094945276 | 0 | 0 | 0 | 12 | 44 | 0 | 0 | 162 | 108 | 0 | 26.683° | - | |
| 57 | 1387.383229253 | 0 | 0 | 0 | 12 | 45 | 0 | 0 | 165 | 110 | 0 | 26.702° | - | |
| 58 | 1438.618250640 | 0 | 0 | 0 | 12 | 46 | 0 | 0 | 168 | 112 | 0 | 26.155° | - | |
| 59 | 1490.773335279 | 0.000154286 | 0 | 0 | 14 | 43 | 2 | 0 | 171 | 114 | 0 | 26.170° | - | |
| 60 | 1543.830400976 | 0 | 0 | 0 | 12 | 48 | 0 | 0 | 174 | 116 | 0 | 25.958° | - | |
| 61 | 1597.941830199 | 0.001091717 | 0 | 0 | 12 | 49 | 0 | 0 | 177 | 118 | 0 | 25.392° | - | |
| 62 | 1652.909409898 | 0 | 0 | 0 | 12 | 50 | 0 | 0 | 180 | 120 | 0 | 25.880° | - | |
| 63 | 1708.879681503 | 0 | 0 | 0 | 12 | 51 | 0 | 0 | 183 | 122 | 0 | 25.257° | - | |
| 64 | 1765.802577927 | 0 | 0 | 0 | 12 | 52 | 0 | 0 | 186 | 124 | 0 | 24.920° | - | |
| 65 | 1823.667960264 | 0.000399515 | 0 | 0 | 12 | 53 | 0 | 0 | 189 | 126 | 0 | 24.527° | - | |
| 66 | 1882.441525304 | 0.000776245 | 0 | 0 | 12 | 54 | 0 | 0 | 192 | 128 | 0 | 24.765° | - | |
| 67 | 1942.122700406 | 0 | 0 | 0 | 12 | 55 | 0 | 0 | 195 | 130 | 0 | 24.727° | - | |
| 68 | 2002.874701749 | 0 | 0 | 0 | 12 | 56 | 0 | 0 | 198 | 132 | 0 | 24.433° | - | |
| 69 | 2064.533483235 | 0 | 0 | 0 | 12 | 57 | 0 | 0 | 201 | 134 | 0 | 24.137° | - | |
| 70 | 2127.100901551 | 0 | 0 | 0 | 12 | 50 | 0 | 0 | 200 | 128 | 4 | 24.291° | - | |
| 71 | 2190.649906425 | 0.001256769 | 0 | 0 | 14 | 55 | 2 | 0 | 207 | 138 | 0 | 23.803° | - | |
| 72 | 2255.001190975 | 0 | 0 | 0 | 12 | 60 | 0 | 0 | 210 | 140 | 0 | 24.492° | geodesic sphere 2,1 | |
| 73 | 2320.633883745 | 0.001572959 | 0 | 0 | 12 | 61 | 0 | 0 | 213 | 142 | 0 | 22.810° | - | |
| 74 | 2387.072981838 | 0.000641539 | 0 | 0 | 12 | 62 | 0 | 0 | 216 | 144 | 0 | 22.966° | - | |
| 75 | 2454.369689040 | 0 | 0 | 0 | 12 | 63 | 0 | 0 | 219 | 146 | 0 | 22.736° | - | |
| 76 | 2522.674871841 | 0.000943474 | 0 | 0 | 12 | 64 | 0 | 0 | 222 | 148 | 0 | 22.886° | - | |
| 77 | 2591.850152354 | 0 | 0 | 0 | 12 | 65 | 0 | 0 | 225 | 150 | 0 | 23.286° | - | |
| 78 | 2662.046474566 | 0 | 0 | 0 | 12 | 66 | 0 | 0 | 228 | 152 | 0 | 23.426° | - | |
| 79 | 2733.248357479 | 0.000702921 | 0 | 0 | 12 | 63 | 1 | 0 | 230 | 152 | 1 | 22.636° | - | |
| 80 | 2805.355875981 | 0 | 0 | 0 | 16 | 64 | 0 | 0 | 232 | 152 | 2 | 22.778° | - | |
| 81 | 2878.522829664 | 0.000194289 | 0 | 0 | 12 | 69 | 0 | 0 | 237 | 158 | 0 | 21.892° | - | |
| 82 | 2952.569675286 | 0 | 0 | 0 | 12 | 70 | 0 | 0 | 240 | 160 | 0 | 22.206° | - | |
| 83 | 3027.528488921 | 0.000339815 | 0 | 0 | 14 | 67 | 2 | 0 | 243 | 162 | 0 | 21.646° | - | |
| 84 | 3103.465124431 | 0.000401973 | 0 | 0 | 12 | 72 | 0 | 0 | 246 | 164 | 0 | 21.513° | - | |
| 85 | 3180.361442939 | 0.000416581 | 0 | 0 | 12 | 73 | 0 | 0 | 249 | 166 | 0 | 21.498° | - | |
| 86 | 3258.211605713 | 0.001378932 | 0 | 0 | 12 | 74 | 0 | 0 | 252 | 168 | 0 | 21.522° | - | |
| 87 | 3337.000750014 | 0.000754863 | 0 | 0 | 12 | 75 | 0 | 0 | 255 | 170 | 0 | 21.456° | - | |
| 88 | 3416.720196758 | 0 | 0 | 0 | 12 | 76 | 0 | 0 | 258 | 172 | 0 | 21.486° | - | |
| 89 | 3497.439018625 | 0.000070891 | 0 | 0 | 12 | 77 | 0 | 0 | 261 | 174 | 0 | 21.182° | - | |
| 90 | 3579.091222723 | 0 | 0 | 0 | 12 | 78 | 0 | 0 | 264 | 176 | 0 | 21.230° | - | |
| 91 | 3661.713699320 | 0.000033221 | 0 | 0 | 12 | 79 | 0 | 0 | 267 | 178 | 0 | 21.105° | - | |
| 92 | 3745.291636241 | 0 | 0 | 0 | 12 | 80 | 0 | 0 | 270 | 180 | 0 | 21.026° | - | |
| 93 | 3829.844338421 | 0.000213246 | 0 | 0 | 12 | 81 | 0 | 0 | 273 | 182 | 0 | 20.751° | - | |
| 94 | 3915.309269620 | 0 | 0 | 0 | 12 | 82 | 0 | 0 | 276 | 184 | 0 | 20.952° | - | |
| 95 | 4001.771675565 | 0.000116638 | 0 | 0 | 12 | 83 | 0 | 0 | 279 | 186 | 0 | 20.711° | - | |
| 96 | 4089.154010060 | 0.000036310 | 0 | 0 | 12 | 84 | 0 | 0 | 282 | 188 | 0 | 20.687° | - | |
| 97 | 4177.533599622 | 0.000096437 | 0 | 0 | 12 | 85 | 0 | 0 | 285 | 190 | 0 | 20.450° | - | |
| 98 | 4266.822464156 | 0.000112916 | 0 | 0 | 12 | 86 | 0 | 0 | 288 | 192 | 0 | 20.422° | - | |
| 99 | 4357.139163132 | 0.000156508 | 0 | 0 | 12 | 87 | 0 | 0 | 291 | 194 | 0 | 20.284° | - | |
| 100 | 4448.350634331 | 0 | 0 | 0 | 12 | 88 | 0 | 0 | 294 | 196 | 0 | 20.297° | - | |
| 101 | 4540.590051694 | 0 | 0 | 0 | 12 | 89 | 0 | 0 | 297 | 198 | 0 | 20.011° | - | |
| 102 | 4633.736565899 | 0 | 0 | 0 | 12 | 90 | 0 | 0 | 300 | 200 | 0 | 20.040° | - | |
| 103 | 4727.836616833 | 0.000201245 | 0 | 0 | 12 | 91 | 0 | 0 | 303 | 202 | 0 | 19.907° | - | |
| 104 | 4822.876522746 | 0 | 0 | 0 | 12 | 92 | 0 | 0 | 306 | 204 | 0 | 19.957° | - | |
| 105 | 4919.000637616 | 0 | 0 | 0 | 12 | 93 | 0 | 0 | 309 | 206 | 0 | 19.842° | - | |
| 106 | 5015.984595705 | 0 | 0 | 0 | 12 | 94 | 0 | 0 | 312 | 208 | 0 | 19.658° | - | |
| 107 | 5113.953547724 | 0.000064137 | 0 | 0 | 12 | 95 | 0 | 0 | 315 | 210 | 0 | 19.327° | - | |
| 108 | 5212.813507831 | 0.000432525 | 0 | 0 | 12 | 96 | 0 | 0 | 318 | 212 | 0 | 19.327° | - | |
| 109 | 5312.735079920 | 0.000647299 | 0 | 0 | 14 | 93 | 2 | 0 | 321 | 214 | 0 | 19.103° | - | |
| 110 | 5413.549294192 | 0 | 0 | 0 | 12 | 98 | 0 | 0 | 324 | 216 | 0 | 19.476° | - | |
| 111 | 5515.293214587 | 0 | 0 | 0 | 12 | 99 | 0 | 0 | 327 | 218 | 0 | 19.255° | - | |
| 112 | 5618.044882327 | 0 | 0 | 0 | 12 | 100 | 0 | 0 | 330 | 220 | 0 | 19.351° | - | |
| 113 | 5721.824978027 | 0 | 0 | 0 | 12 | 101 | 0 | 0 | 333 | 222 | 0 | 18.978° | - | |
| 114 | 5826.521572163 | 0.000149772 | 0 | 0 | 12 | 102 | 0 | 0 | 336 | 224 | 0 | 18.836° | - | |
| 115 | 5932.181285777 | 0.000049972 | 0 | 0 | 12 | 103 | 0 | 0 | 339 | 226 | 0 | 18.458° | - | |
| 116 | 6038.815593579 | 0.000259726 | 0 | 0 | 12 | 104 | 0 | 0 | 342 | 228 | 0 | 18.386° | - | |
| 117 | 6146.342446579 | 0.000127609 | 0 | 0 | 12 | 105 | 0 | 0 | 345 | 230 | 0 | 18.566° | - | |
| 118 | 6254.877027790 | 0.000332475 | 0 | 0 | 12 | 106 | 0 | 0 | 348 | 232 | 0 | 18.455° | - | |
| 119 | 6364.347317479 | 0.000685590 | 0 | 0 | 12 | 107 | 0 | 0 | 351 | 234 | 0 | 18.336° | - | |
| 120 | 6474.756324980 | 0.001373062 | 0 | 0 | 12 | 108 | 0 | 0 | 354 | 236 | 0 | 18.418° | - | |
| 121 | 6586.121949584 | 0.000838863 | 0 | 0 | 12 | 109 | 0 | 0 | 357 | 238 | 0 | 18.199° | - | |
| 122 | 6698.374499261 | 0 | 0 | 0 | 12 | 110 | 0 | 0 | 360 | 240 | 0 | 18.612° | geodesic sphere 2,2 | |
| 123 | 6811.827228174 | 0.001939754 | 0 | 0 | 14 | 107 | 2 | 0 | 363 | 242 | 0 | 17.840° | - | |
| 124 | 6926.169974193 | 0 | 0 | 0 | 12 | 112 | 0 | 0 | 366 | 244 | 0 | 18.111° | - | |
| 125 | 7041.473264023 | 0.000088274 | 0 | 0 | 12 | 113 | 0 | 0 | 369 | 246 | 0 | 17.867° | - | |
| 126 | 7157.669224867 | 0 | 0 | 2 | 16 | 100 | 8 | 0 | 372 | 248 | 0 | 17.920° | - | |
| 127 | 7274.819504675 | 0 | 0 | 0 | 12 | 115 | 0 | 0 | 375 | 250 | 0 | 17.877° | - | |
| 128 | 7393.007443068 | 0.000054132 | 0 | 0 | 12 | 116 | 0 | 0 | 378 | 252 | 0 | 17.814° | - | |
| 129 | 7512.107319268 | 0.000030099 | 0 | 0 | 12 | 117 | 0 | 0 | 381 | 254 | 0 | 17.743° | - | |
| 130 | 7632.167378912 | 0.000025622 | 0 | 0 | 12 | 118 | 0 | 0 | 384 | 256 | 0 | 17.683° | - | |
| 131 | 7753.205166941 | 0.000305133 | 0 | 0 | 12 | 119 | 0 | 0 | 387 | 258 | 0 | 17.511° | - | |
| 132 | 7875.045342797 | 0 | 0 | 0 | 12 | 120 | 0 | 0 | 390 | 260 | 0 | 17.958° | geodesic sphere 3,1 | |
| 133 | 7998.179212898 | 0.000591438 | 0 | 0 | 12 | 121 | 0 | 0 | 393 | 262 | 0 | 17.133° | - | |
| 134 | 8122.089721194 | 0.000470268 | 0 | 0 | 12 | 122 | 0 | 0 | 396 | 264 | 0 | 17.214° | - | |
| 135 | 8246.909486992 | 0 | 0 | 0 | 12 | 123 | 0 | 0 | 399 | 266 | 0 | 17.431° | - | |
| 136 | 8372.743302539 | 0 | 0 | 0 | 12 | 124 | 0 | 0 | 402 | 268 | 0 | 17.485° | - | |
| 137 | 8499.534494782 | 0 | 0 | 0 | 12 | 125 | 0 | 0 | 405 | 270 | 0 | 17.560° | - | |
| 138 | 8627.406389880 | 0.000473576 | 0 | 0 | 12 | 126 | 0 | 0 | 408 | 272 | 0 | 16.924° | - | |
| 139 | 8756.227056057 | 0.000404228 | 0 | 0 | 12 | 127 | 0 | 0 | 411 | 274 | 0 | 16.673° | - | |
| 140 | 8885.980609041 | 0.000630351 | 0 | 0 | 13 | 126 | 1 | 0 | 414 | 276 | 0 | 16.773° | - | |
| 141 | 9016.615349190 | 0.000376365 | 0 | 0 | 14 | 126 | 0 | 1 | 417 | 278 | 0 | 16.962° | - | |
| 142 | 9148.271579993 | 0.000550138 | 0 | 0 | 12 | 130 | 0 | 0 | 420 | 280 | 0 | 16.840° | - | |
| 143 | 9280.839851192 | 0.000255449 | 0 | 0 | 12 | 131 | 0 | 0 | 423 | 282 | 0 | 16.782° | - | |
| 144 | 9414.371794460 | 0 | 0 | 0 | 12 | 132 | 0 | 0 | 426 | 284 | 0 | 16.953° | - | |
| 145 | 9548.928837232 | 0.000094938 | 0 | 0 | 12 | 133 | 0 | 0 | 429 | 286 | 0 | 16.841° | - | |
| 146 | 9684.381825575 | 0 | 0 | 0 | 12 | 134 | 0 | 0 | 432 | 288 | 0 | 16.905° | - | |
| 147 | 9820.932378373 | 0.000636651 | 0 | 0 | 12 | 135 | 0 | 0 | 435 | 290 | 0 | 16.458° | - | |
| 148 | 9958.406004270 | 0.000203701 | 0 | 0 | 12 | 136 | 0 | 0 | 438 | 292 | 0 | 16.627° | - | |
| 149 | 10096.859907397 | 0.000638186 | 0 | 0 | 14 | 133 | 2 | 0 | 441 | 294 | 0 | 16.344° | - | |
| 150 | 10236.196436701 | 0 | 0 | 0 | 12 | 138 | 0 | 0 | 444 | 296 | 0 | 16.405° | - | |
| 151 | 10376.571469275 | 0.000153836 | 0 | 0 | 12 | 139 | 0 | 0 | 447 | 298 | 0 | 16.163° | - | |
| 152 | 10517.867592878 | 0 | 0 | 0 | 12 | 140 | 0 | 0 | 450 | 300 | 0 | 16.117° | - | |
| 153 | 10660.082748237 | 0 | 0 | 0 | 12 | 141 | 0 | 0 | 453 | 302 | 0 | 16.390° | - | |
| 154 | 10803.372421141 | 0.000735800 | 0 | 0 | 12 | 142 | 0 | 0 | 456 | 304 | 0 | 16.078° | - | |
| 155 | 10947.574692279 | 0.000603670 | 0 | 0 | 12 | 143 | 0 | 0 | 459 | 306 | 0 | 15.990° | - | |
| 156 | 11092.798311456 | 0.000508534 | 0 | 0 | 12 | 144 | 0 | 0 | 462 | 308 | 0 | 15.822° | - | |
| 157 | 11238.903041156 | 0.000357679 | 0 | 0 | 12 | 145 | 0 | 0 | 465 | 310 | 0 | 15.948° | - | |
| 158 | 11385.990186197 | 0.000921918 | 0 | 0 | 12 | 146 | 0 | 0 | 468 | 312 | 0 | 15.987° | - | |
| 159 | 11534.023960956 | 0.000381457 | 0 | 0 | 12 | 147 | 0 | 0 | 471 | 314 | 0 | 15.960° | - | |
| 160 | 11683.054805549 | 0 | 0 | 0 | 12 | 148 | 0 | 0 | 474 | 316 | 0 | 15.961° | - | |
| 161 | 11833.084739465 | 0.000056447 | 0 | 0 | 12 | 149 | 0 | 0 | 477 | 318 | 0 | 15.810° | - | |
| 162 | 11984.050335814 | 0 | 0 | 0 | 12 | 150 | 0 | 0 | 480 | 320 | 0 | 15.813° | - | |
| 163 | 12136.013053220 | 0.000120798 | 0 | 0 | 12 | 151 | 0 | 0 | 483 | 322 | 0 | 15.675° | - | |
| 164 | 12288.930105320 | 0 | 0 | 0 | 12 | 152 | 0 | 0 | 486 | 324 | 0 | 15.655° | - | |
| 165 | 12442.804451373 | 0.000091119 | 0 | 0 | 12 | 153 | 0 | 0 | 489 | 326 | 0 | 15.651° | - | |
| 166 | 12597.649071323 | 0 | 0 | 0 | 16 | 146 | 4 | 0 | 492 | 328 | 0 | 15.607° | - | |
| 167 | 12753.469429750 | 0.000097382 | 0 | 0 | 12 | 155 | 0 | 0 | 495 | 330 | 0 | 15.600° | - | |
| 168 | 12910.212672268 | 0 | 0 | 0 | 12 | 156 | 0 | 0 | 498 | 332 | 0 | 15.655° | - | |
| 169 | 13068.006451127 | 0.000068102 | 0 | 0 | 13 | 155 | 1 | 0 | 501 | 334 | 0 | 15.537° | - | |
| 170 | 13226.681078541 | 0 | 0 | 0 | 12 | 158 | 0 | 0 | 504 | 336 | 0 | 15.569° | - | |
| 171 | 13386.355930717 | 0 | 0 | 0 | 12 | 159 | 0 | 0 | 507 | 338 | 0 | 15.497° | - | |
| 172 | 13547.018108787 | 0.000547291 | 0 | 0 | 14 | 156 | 2 | 0 | 510 | 340 | 0 | 15.292° | - | |
| 173 | 13708.635243034 | 0.000286544 | 0 | 0 | 12 | 161 | 0 | 0 | 513 | 342 | 0 | 15.225° | - | |
| 174 | 13871.187092292 | 0 | 0 | 0 | 12 | 162 | 0 | 0 | 516 | 344 | 0 | 15.366° | - | |
| 175 | 14034.781306929 | 0.000026686 | 0 | 0 | 12 | 163 | 0 | 0 | 519 | 346 | 0 | 15.252° | - | |
| 176 | 14199.354775632 | 0.000283978 | 0 | 0 | 12 | 164 | 0 | 0 | 522 | 348 | 0 | 15.101° | - | |
| 177 | 14364.837545298 | 0 | 0 | 0 | 12 | 165 | 0 | 0 | 525 | 350 | 0 | 15.269° | - | |
| 178 | 14531.309552587 | 0 | 0 | 0 | 12 | 166 | 0 | 0 | 528 | 352 | 0 | 15.145° | - | |
| 179 | 14698.754594220 | 0.000125113 | 0 | 0 | 13 | 165 | 1 | 0 | 531 | 354 | 0 | 14.968° | - | |
| 180 | 14867.099927525 | 0 | 0 | 0 | 12 | 168 | 0 | 0 | 534 | 356 | 0 | 15.067° | - | |
| 181 | 15036.467239769 | 0.000304193 | 0 | 0 | 12 | 169 | 0 | 0 | 537 | 358 | 0 | 15.002° | - | |
| 182 | 15206.730610906 | 0 | 0 | 0 | 12 | 170 | 0 | 0 | 540 | 360 | 0 | 15.155° | - | |
| 183 | 15378.166571028 | 0.000467899 | 0 | 0 | 12 | 171 | 0 | 0 | 543 | 362 | 0 | 14.747° | - | |
| 184 | 15550.421450311 | 0 | 0 | 0 | 12 | 172 | 0 | 0 | 546 | 364 | 0 | 14.932° | - | |
| 185 | 15723.720074072 | 0.000389762 | 0 | 0 | 12 | 173 | 0 | 0 | 549 | 366 | 0 | 14.775° | - | |
| 186 | 15897.897437048 | 0.000389762 | 0 | 0 | 12 | 174 | 0 | 0 | 552 | 368 | 0 | 14.739° | - | |
| 187 | 16072.975186320 | 0 | 0 | 0 | 12 | 175 | 0 | 0 | 555 | 370 | 0 | 14.848° | - | |
| 188 | 16249.222678879 | 0 | 0 | 0 | 12 | 176 | 0 | 0 | 558 | 372 | 0 | 14.740° | - | |
| 189 | 16426.371938862 | 0.000020732 | 0 | 0 | 12 | 177 | 0 | 0 | 561 | 374 | 0 | 14.671° | - | |
| 190 | 16604.428338501 | 0.000586804 | 0 | 0 | 12 | 178 | 0 | 0 | 564 | 376 | 0 | 14.501° | - | |
| 191 | 16783.452219362 | 0.001129202 | 0 | 0 | 13 | 177 | 1 | 0 | 567 | 378 | 0 | 14.195° | - | |
| 192 | 16963.338386460 | 0 | 0 | 0 | 12 | 180 | 0 | 0 | 570 | 380 | 0 | 14.819° | geodesic sphere 3,2 | |
| 193 | 17144.564740880 | 0.000985192 | 0 | 0 | 12 | 181 | 0 | 0 | 573 | 382 | 0 | 14.144° | - | |
| 194 | 17326.616136471 | 0.000322358 | 0 | 0 | 12 | 182 | 0 | 0 | 576 | 384 | 0 | 14.350° | - | |
| 195 | 17509.489303930 | 0 | 0 | 0 | 12 | 183 | 0 | 0 | 579 | 386 | 0 | 14.375° | - | |
| 196 | 17693.460548082 | 0.000315907 | 0 | 0 | 12 | 184 | 0 | 0 | 582 | 388 | 0 | 14.251° | - | |
| 197 | 17878.340162571 | 0 | 0 | 0 | 12 | 185 | 0 | 0 | 585 | 390 | 0 | 14.147° | - | |
| 198 | 18064.262177195 | 0.000011149 | 0 | 0 | 12 | 186 | 0 | 0 | 588 | 392 | 0 | 14.237° | - | |
| 199 | 18251.082495640 | 0.000534779 | 0 | 0 | 12 | 187 | 0 | 0 | 591 | 394 | 0 | 14.153° | - | |
| 200 | 18438.842717530 | 0 | 0 | 0 | 12 | 188 | 0 | 0 | 594 | 396 | 0 | 14.222° | - | |
| 201 | 18627.591226244 | 0.001048859 | 0 | 0 | 13 | 187 | 1 | 0 | 597 | 398 | 0 | 13.830° | - | |
| 202 | 18817.204718262 | 0 | 0 | 0 | 12 | 190 | 0 | 0 | 600 | 400 | 0 | 14.189° | - | |
| 203 | 19007.981204580 | 0.000600343 | 0 | 0 | 12 | 191 | 0 | 0 | 603 | 402 | 0 | 13.977° | - | |
| 204 | 19199.540775603 | 0 | 0 | 0 | 12 | 192 | 0 | 0 | 606 | 404 | 0 | 14.291° | - | |
| 212 | 20768.053085964 | 0 | 0 | 0 | 12 | 200 | 0 | 0 | 630 | 420 | 0 | 14.118° | geodesic sphere 4,1 | |
| 214 | 21169.910410375 | 0 | 0 | 0 | 12 | 202 | 0 | 0 | 636 | 424 | 0 | 13.771° | - | |
| 216 | 21575.596377869 | 0 | 0 | 0 | 12 | 204 | 0 | 0 | 642 | 428 | 0 | 13.735° | - | |
| 217 | 21779.856080418 | 0 | 0 | 0 | 12 | 205 | 0 | 0 | 645 | 430 | 0 | 13.902° | - | |
| 232 | 24961.252318934 | 0 | 0 | 0 | 12 | 220 | 0 | 0 | 690 | 460 | 0 | 13.260° | - | |
| 255 | 30264.424251281 | 0 | 0 | 0 | 12 | 243 | 0 | 0 | 759 | 506 | 0 | 12.565° | - | |
| 256 | 30506.687515847 | 0 | 0 | 0 | 12 | 244 | 0 | 0 | 762 | 508 | 0 | 12.572° | - | |
| 257 | 30749.941417346 | 0 | 0 | 0 | 12 | 245 | 0 | 0 | 765 | 510 | 0 | 12.672° | - | |
| 272 | 34515.193292681 | 0 | 0 | 0 | 12 | 260 | 0 | 0 | 810 | 540 | 0 | 12.335° | geodesic sphere 3,3 | |
| 282 | 37147.294418462 | 0 | 0 | 0 | 12 | 270 | 0 | 0 | 840 | 560 | 0 | 12.166° | geodesic sphere 4,2 | |
| 292 | 39877.008012909 | 0 | 0 | 0 | 12 | 280 | 0 | 0 | 870 | 580 | 0 | 11.857° | - | |
| 306 | 43862.569780797 | 0 | 0 | 0 | 12 | 294 | 0 | 0 | 912 | 608 | 0 | 11.628° | - | |
| 312 | 45629.313804002 | 0.000306163 | 0 | 0 | 12 | 300 | 0 | 0 | 930 | 620 | 0 | 11.299° | - | |
| 315 | 46525.825643432 | 0 | 0 | 0 | 12 | 303 | 0 | 0 | 939 | 626 | 0 | 11.337° | - | |
| 317 | 47128.310344520 | 0 | 0 | 0 | 12 | 305 | 0 | 0 | 945 | 630 | 0 | 11.423° | - | |
| 318 | 47431.056020043 | 0 | 0 | 0 | 12 | 306 | 0 | 0 | 948 | 632 | 0 | 11.219° | - | |
| 334 | 52407.728127822 | 0 | 0 | 0 | 12 | 322 | 0 | 0 | 996 | 664 | 0 | 11.058° | - | |
| 348 | 56967.472454334 | 0 | 0 | 0 | 12 | 336 | 0 | 0 | 1038 | 692 | 0 | 10.721° | - | |
| 357 | 59999.922939598 | 0 | 0 | 0 | 12 | 345 | 0 | 0 | 1065 | 710 | 0 | 10.728° | - | |
| 358 | 60341.830924588 | 0 | 0 | 0 | 12 | 346 | 0 | 0 | 1068 | 712 | 0 | 10.647° | - | |
| 372 | 65230.027122557 | 0 | 0 | 0 | 12 | 360 | 0 | 0 | 1110 | 740 | 0 | 10.531° | geodesic sphere 4,3 | |
| 382 | 68839.426839215 | 0 | 0 | 0 | 12 | 370 | 0 | 0 | 1140 | 760 | 0 | 10.379° | - | |
| 390 | 71797.035335953 | 0 | 0 | 0 | 12 | 378 | 0 | 0 | 1164 | 776 | 0 | 10.222° | - | |
| 392 | 72546.258370889 | 0 | 0 | 0 | 12 | 380 | 0 | 0 | 1170 | 780 | 0 | 10.278° | - | |
| 400 | 75582.448512213 | 0 | 0 | 0 | 12 | 388 | 0 | 0 | 1194 | 796 | 0 | 10.068° | - | |
| 402 | 76351.192432673 | 0 | 0 | 0 | 12 | 390 | 0 | 0 | 1200 | 800 | 0 | 10.099° | - | |
| 432 | 88353.709681956 | 0 | 0 | 0 | 24 | 396 | 12 | 0 | 1290 | 860 | 0 | 9.556° | - | |
| 448 | 95115.546986209 | 0 | 0 | 0 | 24 | 412 | 12 | 0 | 1338 | 892 | 0 | 9.322° | - | |
| 460 | 100351.763108673 | 0 | 0 | 0 | 24 | 424 | 12 | 0 | 1374 | 916 | 0 | 9.297° | - | |
| 468 | 103920.871715127 | 0 | 0 | 0 | 24 | 432 | 12 | 0 | 1398 | 932 | 0 | 9.120° | - | |
| 470 | 104822.886324279 | 0 | 0 | 0 | 24 | 434 | 12 | 0 | 1404 | 936 | 0 | 9.059° | - |
According to a conjecture, if, 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:.