Thomson problem explained

The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of 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] 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 behavior of the minimum energy.

Mathematical statement

The electrostatic interaction energy occurring between each pair of electrons of equal charges (

e _{i} = e _{j} = e , with e the elementary charge of an electron) is given by Coulomb's law,U _{ij} (N) = {e _{i} e _{j} \over 4 \pi\epsilon _{0} r _{ij} },where \epsilon _{0} is the electric constant and r _{ij} = | r _{i} - r _{j|} is the distance between each pair of electrons located at points on the sphere defined by vectors r _{i} and r _{j} , respectively.

Simplified units of

e =1 and k _{e=1/4\pi\epsilon} _{0=1} (the Coulomb constant) are used without loss of generality. Then,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 interaction energiesU(N) = \sum _{i} (1)/(r _{ij}) .

The global minimization of

U(N) over all possible configurations of N distinct points is typically found by numerical minimization algorithms.

Thomson's problem is related to the 7th of the eighteen unsolved mathematics problems proposed by the mathematician Steve Smale — "Distribution of points on the 2-sphere".^[2] The main difference is that in Smale's problem the function to minimise is not the electrostatic potential

1 \over r _{ij} but a logarithmic potential given by -log r _{ij} . A second difference is that Smale's question is about the asymptotic behaviour of the total potential when the number N of points goes to infinity, not for concrete values of N.

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,

r _{ij} = 2 r = 2 , orU( 2 ) = { 1 \over 2 }.

Known exact solutions

Mathematically exact minimum energy configurations have been rigorously identified in only a handful of cases.

• For N = 1, the solution is trivial. The single electron may reside at any point on the surface of the unit sphere. The total energy of the configuration is defined as zero because the charge of the electron is subject to no electric field due to other sources of charge.
• For N = 2, the optimal configuration consists of electrons at antipodal points. This represents the first one-dimensional solution.
• For N = 3, electrons reside at the vertices of an equilateral triangle about any great circle.^[3] The great circle is often considered to define an equator about the sphere and the two points perpendicular to the plane are often considered poles to aid in discussions about the electrostatic configurations of many-N electron solutions. Also, this represents the first two-dimensional solution.
• For N = 4, electrons reside at the vertices of a regular tetrahedron. Of interest, this represents the first three-dimensional solution.
• For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid.^[4] Of interest, it is impossible for any N solution with five or more electrons to exhibit global equidistance among all pairs of electrons.
• For N = 6, electrons reside at vertices of a regular octahedron.^[5] The configuration may be imagined as four electrons residing at the corners of a square about the equator and the remaining two residing at the poles.
• For N = 12, electrons reside at the vertices of a regular icosahedron.^[6]

Geometric solutions of the Thomson problem for N = 4, 6, and 12 electrons are 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, the cube and dodecahedron respectively.^[7]

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

$$\backslash sum\_\; f(|x\_i-x\_j|).$$

Traditionally, one considers

f(x) = x ^{-\alpha} also known as Riesz \alpha -kernels. For integrable Riesz kernels see the 1972 work of Landkof.^[8] For non-integrable Riesz kernels, the Poppy-seed bagel theorem holds, see the 2004 work of Hardin and Saff.^[9] Notable cases include:^[10]

• α = ∞, the Tammes problem (packing);
• α = 1, the Thomson problem;
• α = 0, to maximize the product of distances, latterly known as Whyte's problem;
• α = −1 : maximum average distance problem.

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

Solution algorithms

Several algorithms have been applied to this problem. The focus since the millennium has been on local optimization methods applied to the energy function, although random walks have made their appearance:^[10]

• constrained global optimization (Altschuler et al. 1994),
• steepest descent (Claxton and Benson 1966, Erber and Hockney 1991),
• random walk (Weinrach et al. 1990),
• genetic algorithm (Morris et al. 1996)

While the objective is to minimize the global electrostatic potential energy of each N-electron case, several algorithmic starting cases are of interest.

Continuous spherical shell charge

The energy of a continuous spherical shell of charge distributed across its surface is given by

U _{shell} (N) = (N ^{2})/(2) and is, in general, greater than the energy of every Thomson problem solution. Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell. For example, a spherical shell of N =1 represents the uniform distribution of a single electron's charge, - e , across the entire shell.

Randomly distributed point charges

The expected global energy of a system of electrons distributed in a purely random manner across the surface of the sphere is given by

U _{rand} (N) = (N(N -1 ))/(2) and is, in general, greater than the energy of every Thomson problem solution.

Here, N is a discrete variable that counts the number of electrons in the system. As well,

U _{rand} (N) < U _{shell} (N) .

Charge-centered distribution

For every Nth solution of the Thomson problem there is an

(N +1 ) th configuration that includes an electron at the origin of the sphere whose energy is simply the addition of N to the energy of the Nth solution. That is,^[11] U _{0(N+1)=U} _{Thom} (N) + N. Thus, if U _{Thom} (N) is known exactly, then U _{0(N+1)} is known exactly.

In general,

U _{0(N+1)} is greater than U _{Thom} (N +1 ) , but is remarkably closer to each (N +1 ) th Thomson solution than U _{shell} (N +1 ) and U _{rand} (N +1 ) . Therefore, the charge-centered distribution represents a smaller "energy gap" to cross to arrive at a solution of each Thomson problem than algorithms that begin with the other two charge configurations.

Relations to other scientific problems

The Thomson problem is a natural consequence of J. J. Thomson's plum pudding model in the absence of its uniform positive background charge.^[12]

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.^[13]

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 arrangements of protein subunits that 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 Abrikosov vortices that form at low temperatures in a superconducting metal shell with a large monopole at its center.

Configurations of smallest known energy

In the following table

N is the number of points (charges) in a configuration, U _{rm{Thom} } is the energy, the symmetry type is given in Schönflies notation (see Point groups in three dimensions), and 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,

v _{i} is the number of vertices where the given number of edges meet, e is the total number of edges, f _{3} is the number of triangular faces, f _{4} is the number of quadrilateral faces, and \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 = 2, 3, 4, 6, 12, and the geodesic polyhedra, the convex hull is only topologically equivalent to the figure listed in the last column.^[14]

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

According to a conjecture, if

P is the polyhedron formed by the convex hull of the solution configuation to the Thomson Problem for m electrons and q is the number of quadrilateral faces of P , then P has f(m) = \delta _{0,m-2} + 3 (m -2 ) - q edges.^[15]

References

1. Thomson. Joseph John. Joseph John Thomson. 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. . 10.1080/14786440409463107 . Series 6 . 7 . 39. 237–265. March 1904. dead. https://web.archive.org/web/20131213172104/http://www.cond-mat.physik.uni-mainz.de/~oettel/ws10/thomson_PhilMag_7_237_1904.pdf. 13 December 2013.
2. S. . Smale . Mathematical Problems for the Next Century . Mathematical Intelligencer . 1998 . 20 . 2 . 7–15 . 10.1.1.35.4101 . 10.1007/bf03025291 . 1331144.
3. L.. Föppl . Stabile Anordnungen von Elektronen im Atom . J. Reine Angew. Math. . 141 . 1912 . 141 . 251–301. 10.1515/crll.1912.141.251 . 120309200 . .
4. Schwartz. Richard. 1001.3702. The 5 electron case of Thomson's Problem. 2010. math.MG.
5. V.A. . Yudin . The minimum of potential energy of a system of point charges . Discretnaya Matematika . 4 . 2. 1992. 115–121 (in Russian).

   V. A. . Yudin. Discrete Math. Appl.. 3 . 1. 1993. 75–81 . 10.1515/dma.1993.3.1.75 . The minimum of potential energy of a system of point charges. 117117450.

6. N.N. . Andreev. An extremal property of the icosahedron. East J. Approximation. 2 . 4 . 1996. 459–462.,
7. Polyhedra in physics, chemistry and geometry . 2003 . math-ph/0303071 . Atiyah . Michael . Sutcliffe . Paul .
8. 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.
9. Hardin, D. P.; Saff, E. B. Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc. 51 (2004), no. 10, 1186–1194
10. News: Batagelj . Vladimir . Plestenjak . Bor . Optimal arrangements of n points on a sphere and in a circle . 25 June 2018 . https://web.archive.org/web/20180625050324/https://www.fmf.uni-lj.si/~plestenjak/Talks/preddvor.pdf . IMFM/TCS.
11. LaFave Jr. Tim. Discrete transformations in the Thomson Problem. Journal of Electrostatics . 10.1016/j.elstat.2013.11.007. 72 . 1. 39–43. February 2014. 1403.2592. 119309183.
12. Y.. Levin. J. J. . Arenzon . Why charges go to the Surface: A generalized Thomson Problem. Europhys. Lett.. 63. 3. 415. 2003. cond-mat/0302524 . 10.1209/epl/i2003-00546-1 . 2003EL.....63..415L. 18929981.
13. LaFave Jr. Tim. Correspondences between the classical electrostatic Thomson problem and atomic electronic structure. Journal of Electrostatics. 71. 6. 1029–1035. 2013. 10.1016/j.elstat.2013.10.001. 1403.2591. 118480104.
14. Kevin Brown."Min-Energy Configurations of Electrons On A Sphere".Retrieved 2014-05-01.
15. Web site: Sloane's A008486 (see the comment from Feb 03 2017). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2017-02-08.

Notes

• L.L. . Whyte. Unique arrangements of points on a sphere . Amer. Math. Monthly . 1952 . 59 . 9 . 606–611. 10.2307/2306764. 2306764.
• Harvey . Cohn. 1956. Stability configurations of electrons on a sphere. Math. Comput. . 10. 55. 117–120 . 10.1090/S0025-5718-1956-0081133-0 . free.
• Michael . Goldberg . Stability configurations of electrons on a sphere. Math. Comp. . 1969 . 23 . 108 . 785–786 . 10.1090/S0025-5718-69-99642-2 . free .
• T. . Erber. G. M. . Hockney . equilibrium configurations of N equal charges on a sphere. 1991. J. Phys. A: Math. Gen. . 24 . L1369. 10.1088/0305-4470/24/23/008. 23. 1991JPhA...24L1369E. 122561279 .
• J. R. . Morris. D. M. . Deaven. K. M. . Ho. Genetic-algorithm energy minimization for point charges on a sphere . Phys. Rev. B . 53 . 4. 1996. R1740–R1743 . 10.1103/PhysRevB.53.R1740. 9983695. 1996PhRvB..53.1740M. 10.1.1.28.93.
• Book: T.. Erber . G. M.. Hockney. Complex Systems: Equilibrium Configurations of N Equal Charges on a Sphere ( 2 \leq N\leq 112 ) . Advances in Chemical Physics. 98. 495–594. 1997 . 10.1002/9780470141571.ch5. 9780470141571 . .
• E. L. . Altschuler . T. J. . Williams. E. R. . Ratner . R. . Tipton . R. . Stong . F. . Dowla . F. . Wooten. Possible global minimum lattice configurations for Thomson's problem of charges on a sphere. 1997. Phys. Rev. Lett. . 78 . 14 . 2681–2685 . 10.1103/PhysRevLett.78.2681. 1997PhRvL..78.2681A.
• M. . Bowick. A. . Cacciuto. D. R. . Nelson. A. . Travesset . Crystalline order on a sphere and the generalized Thomson Problem. Phys. Rev. Lett. . 89 . 2002 . 249902 . 10.1103/PhysRevLett.89.185502. 12398614. 18. cond-mat/0206144. 2002PhRvL..89r5502B. 20362989.
• P. D.. Dragnev. D. A.. Legg. D. W. . Townsend . Discrete logarithmic energy on the sphere. Pacific J. Math. . 207 . 2002. 2. 345–358. 10.2140/pjm.2002.207.345. free. .
• A. . Katanforoush . M. . Shahshahani . Distributing points on the sphere. I. 2003 . Exper. Math. . 12. 2. 199–209 . 10.1080/10586458.2003.10504492. 7306812 .
• David J. . Wales . Sidika. Ulker. Structure and dynamics of spherical crystals characterized for the Thomson problem . Phys. Rev. B . 74 . 2006. 21. 212101 . 10.1103/PhysRevB.74.212101. 2006PhRvB..74u2101W. 119932997 . Configurations reprinted in Web site: D. J. . Wales . S. . Ulker . The Cambridge cluster database.
• A. . Slosar. R. . Podgornik. On the connected-charges Thomson problem. Europhys. Lett.. 2006. 75. 4. 631. 10.1209/epl/i2006-10146-1. cond-mat/0606765. 2006EL.....75..631S. 119005054.
• Henry . Cohn . Abhinav. Kumar . Universally optimal distribution of points on spheres. J. Amer. Math. Soc. . 20 . 2007. 1. 99–148 . 10.1090/S0894-0347-06-00546-7. math/0607446. 2007JAMS...20...99C. 26614691 .
• D. J. . Wales. H. . McKay . E. L. . Altschuler. Defect motifs for spherical topologies. Phys. Rev. B. 2009 . 79 . 224115 . 22 . 10.1103/PhysRevB.79.224115. 2009PhRvB..79v4115W. . Configurations reproduced in Web site: D. J. . Wales . S. . Ulker . The Cambridge cluster database.
• W. J. M. . Ridgway . A. F. . Cheviakov. 10.1016/j.cpc.2018.03.029. 2018. Comput. Phys. Commun. . An iterative procedure for finding locally and globally optimal arrangements of particles on the unit sphere. 233. 84–109. 2018CoPhC.233...84R . 52097788 .
• Web site: Cris. Cecka. Mark J.. Bowick. Alan A.. Middleton. Thomson Problem @ S.U.. 2009-11-24. 2018-04-09. https://web.archive.org/web/20180409064239/http://thomson.phy.syr.edu/. dead.
• This webpage contains many more electron configurations with the lowest known energy: https://www.hars.us.