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 (

ei=ej=e

, with

e

the elementary charge of an electron) is given by Coulomb's law,

Uij(N)={eiej\over4\pi\epsilon0rij

},where

\epsilon0

is the electric constant and

rij=|ri-rj|

is the distance between each pair of electrons located at points on the sphere defined by vectors

ri

and

rj

, respectively.

Simplified units of

e=1

and

ke=1/4\pi\epsilon0=1

(the Coulomb constant) are used without loss of generality. Then,

Uij(N)={1\overrij

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

U(N)=\sumi

1
rij

.

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\overrij

but a logarithmic potential given by

-logrij.

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,

rij=2r=2

, or

U(2)={1\over2}.

Known exact solutions

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

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

\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]

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]

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

Ushell(N)=

N2
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

Urand(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,

Urand(N)<Ushell(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]

U0(N+1)=UThom(N)+N.

Thus, if

UThom(N)

is known exactly, then

U0(N+1)

is known exactly.

In general,

U0(N+1)

is greater than

UThom(N+1)

, but is remarkably closer to each

(N+1)

th Thomson solution than

Ushell(N+1)

and

Urand(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,

Urm{Thom

} is the energy, the symmetry type is given in Schönflies notation (see Point groups in three dimensions), and

ri

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,

vi

is the number of vertices where the given number of edges meet,

e

is the total number of edges,

f3

is the number of triangular faces,

f4

is the number of quadrilateral faces, and

\theta1

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

Urm{Thom

}
Symmetry

\left

\sum \mathbf_i \right

v3

v4

v5

v6

v7

v8

e

f3

f4

\theta1

Equivalent polyhedron
align=right 2align=right 0.500000000align=center

Dinfty

align=center 0align=center align=center align=center align=center align=center align=center align=right 1align=center align=center align=right 180.000°digon
align=right 3align=right 1.732050808align=center

D3h

align=center 0align=center align=center align=center align=center align=center align=center align=right 3align=right 2align=center align=right 120.000°triangle
align=right 4align=right 3.674234614align=center

Td

align=center 0align=right 4align=right 0align=right 0align=right 0align=right 0align=right 0align=right 6align=right 4align=right 0align=right 109.471°tetrahedron
align=right 5align=right 6.474691495align=center

D3h

align=center 0align=right 2align=right 3align=right 0align=right 0align=right 0align=right 0align=right 9align=right 6align=right 0align=right 90.000°triangular dipyramid
align=right 6align=right 9.985281374align=center

Oh

align=center 0align=right 0align=right 6align=right 0align=right 0align=right 0align=right 0align=right 12align=right 8align=right 0align=right 90.000°octahedron
align=right 7align=right 14.452977414align=center

D5h

align=center 0align=right 0align=right 5align=right 2align=right 0align=right 0align=right 0align=right 15align=right 10align=right 0align=right 72.000°pentagonal dipyramid
align=right 8align=right 19.675287861align=center

D4d

align=center 0align=right 0align=right 8align=right 0align=right 0align=right 0align=right 0align=right 16align=right 8align=right 2align=right 71.694°square antiprism
align=right 9align=right 25.759986531align=center

D3h

align=center 0align=right 0align=right 3align=right 6align=right 0align=right 0align=right 0align=right 21align=right 14align=right 0align=right 69.190°triaugmented triangular prism
align=right 10align=right 32.716949460align=center

D4d

align=center 0align=right 0align=right 2align=right 8align=right 0align=right 0align=right 0align=right 24align=right 16align=right 0align=right 64.996°gyroelongated square dipyramid
align=right 11align=right 40.596450510align=center

C2v

align=center 0.013219635align=right 0align=right 2align=right 8align=right 1align=right 0align=right 0align=right 27align=right 18align=right 0align=right 58.540°edge-contracted icosahedron
align=right 12align=right 49.165253058align=center

Ih

align=center 0align=right 0align=right 0align=right 12align=right 0align=right 0align=right 0align=right 30align=right 20align=right 0align=right 63.435°icosahedron
(geodesic sphere 1,0)
align=right 13align=right 58.853230612align=center

C2v

align=center 0.008820367align=right 0align=right 1align=right 10align=right 2align=right 0align=right 0align=right 33align=right 22align=right 0align=right 52.317°
align=right 14align=right 69.306363297align=center

D6d

align=center 0align=right 0align=right 0align=right 12align=right 2align=right 0align=right 0align=right 36align=right 24align=right 0align=right 52.866°gyroelongated hexagonal dipyramid
align=right 15align=right 80.670244114align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 3align=right 0align=right 0align=right 39align=right 26align=right 0align=right 49.225°
align=right 16align=right 92.911655302align=center

T

align=center 0align=right 0align=right 0align=right 12align=right 4align=right 0align=right 0align=right 42align=right 28align=right 0align=right 48.936°tetrahedrally diminished dodecahedron
align=right 17align=right 106.050404829align=center

D5h

align=center 0align=right 0align=right 0align=right 12align=right 5align=right 0align=right 0align=right 45align=right 30align=right 0align=right 50.108°double-gyroelongated pentagonal dipyramid
align=right 18align=right 120.084467447align=center

D4d

align=center 0align=right 0align=right 2align=right 8align=right 8align=right 0align=right 0align=right 48align=right 32align=right 0align=right 47.534°
align=right 19align=right 135.089467557align=center

C2v

align=center 0.000135163align=right 0align=right 0align=right 14align=right 5align=right 0align=right 0align=right 50align=right 32align=right 1align=right 44.910°
align=right 20align=right 150.881568334align=center

D3h

align=center 0align=right 0align=right 0align=right 12align=right 8align=right 0align=right 0align=right 54align=right 36align=right 0align=right 46.093°
align=right 21align=right 167.641622399align=center

C2v

align=center 0.001406124align=right 0align=right 1align=right 10align=right 10align=right 0align=right 0align=right 57align=right 38align=right 0align=right 44.321°
align=right 22align=right 185.287536149align=center

Td

align=center 0align=right 0align=right 0align=right 12align=right 10align=right 0align=right 0align=right 60align=right 40align=right 0align=right 43.302°
align=right 23align=right 203.930190663align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 11align=right 0align=right 0align=right 63align=right 42align=right 0align=right 41.481°
align=right 24align=right 223.347074052align=center

O

align=center 0align=right 0align=right 0align=right 24align=right 0align=right 0align=right 0align=right 60align=right 32align=right 6align=right 42.065°snub cube
align=right 25align=right 243.812760299align=center

Cs

align=center 0.001021305align=right 0align=right 0align=right 14align=right 11align=right 0align=right 0align=right 68align=right 44align=right 1align=right 39.610°
align=right 26align=right 265.133326317align=center

C2

align=center 0.001919065align=right 0align=right 0align=right 12align=right 14align=right 0align=right 0align=right 72align=right 48align=right 0align=right 38.842°
align=right 27align=right 287.302615033align=center

D5h

align=center 0align=right 0align=right 0align=right 12align=right 15align=right 0align=right 0align=right 75align=right 50align=right 0align=right 39.940°
align=right 28align=right 310.491542358align=center

T

align=center 0align=right 0align=right 0align=right 12align=right 16align=right 0align=right 0align=right 78align=right 52align=right 0align=right 37.824°
align=right 29align=right 334.634439920align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 17align=right 0align=right 0align=right 81align=right 54align=right 0align=right 36.391°
align=right 30align=right 359.603945904align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 18align=right 0align=right 0align=right 84align=right 56align=right 0align=right 36.942°
align=right 31align=right 385.530838063align=center

C3v

align=center 0.003204712align=right 0align=right 0align=right 12align=right 19align=right 0align=right 0align=right 87align=right 58align=right 0align=right 36.373°
align=right 32align=right 412.261274651align=center

Ih

align=center 0align=right 0align=right 0align=right 12align=right 20align=right 0align=right 0align=right 90align=right 60align=right 0align=right 37.377°pentakis dodecahedron
(geodesic sphere 1,1)
align=right 33align=right 440.204057448align=center

Cs

align=center 0.004356481align=right 0align=right 0align=right 15align=right 17align=right 1align=right 0align=right 92align=right 60align=right 1align=right 33.700°
align=right 34align=right 468.904853281align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 22align=right 0align=right 0align=right 96align=right 64align=right 0align=right 33.273°
align=right 35align=right 498.569872491align=center

C2

align=center 0.000419208align=right 0align=right 0align=right 12align=right 23align=right 0align=right 0align=right 99align=right 66align=right 0align=right 33.100°
align=right 36align=right 529.122408375align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 24align=right 0align=right 0align=right 102align=right 68align=right 0align=right 33.229°
align=right 37align=right 560.618887731align=center

D5h

align=center 0align=right 0align=right 0align=right 12align=right 25align=right 0align=right 0align=right 105align=right 70align=right 0align=right 32.332°
align=right 38align=right 593.038503566align=center

D6d

align=center 0align=right 0align=right 0align=right 12align=right 26align=right 0align=right 0align=right 108align=right 72align=right 0align=right 33.236°
align=right 39align=right 626.389009017align=center

D3h

align=center 0align=right 0align=right 0align=right 12align=right 27align=right 0align=right 0align=right 111align=right 74align=right 0align=right 32.053°
align=right 40align=right 660.675278835align=center

Td

align=center 0align=right 0align=right 0align=right 12align=right 28align=right 0align=right 0align=right 114align=right 76align=right 0align=right 31.916°
align=right 41align=right 695.916744342align=center

D3h

align=center 0align=right 0align=right 0align=right 12align=right 29align=right 0align=right 0align=right 117align=right 78align=right 0align=right 31.528°
align=right 42align=right 732.078107544align=center

D5h

align=center 0align=right 0align=right 0align=right 12align=right 30align=right 0align=right 0align=right 120align=right 80align=right 0align=right 31.245°
align=right 43align=right 769.190846459align=center

C2v

align=center 0.000399668align=right 0align=right 0align=right 12align=right 31align=right 0align=right 0align=right 123align=right 82align=right 0align=right 30.867°
align=right 44align=right 807.174263085align=center

Oh

align=center 0align=right 0align=right 0align=right 24align=right 20align=right 0align=right 0align=right 120align=right 72align=right 6align=right 31.258°
align=right 45align=right 846.188401061align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 33align=right 0align=right 0align=right 129align=right 86align=right 0align=right 30.207°
align=right 46align=right 886.167113639align=center

T

align=center 0align=right 0align=right 0align=right 12align=right 34align=right 0align=right 0align=right 132align=right 88align=right 0align=right 29.790°
align=right 47align=right 927.059270680align=center

Cs

align=center 0.002482914align=right 0align=right 0align=right 14align=right 33align=right 0align=right 0align=right 134align=right 88align=right 1align=right 28.787°
align=right 48align=right 968.713455344align=center

O

align=center 0align=right 0align=right 0align=right 24align=right 24align=right 0align=right 0align=right 132align=right 80align=right 6align=right 29.690°
align=right 49align=right 1011.557182654align=center

C3

align=center 0.001529341align=right 0align=right 0align=right 12align=right 37align=right 0align=right 0align=right 141align=right 94align=right 0align=right 28.387°
align=right 50align=right 1055.182314726align=center

D6d

align=center 0align=right 0align=right 0align=right 12align=right 38align=right 0align=right 0align=right 144align=right 96align=right 0align=right 29.231°
align=right 51align=right 1099.819290319align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 39align=right 0align=right 0align=right 147align=right 98align=right 0align=right 28.165°
align=right 52align=right 1145.418964319align=center

C3

align=center 0.000457327align=right 0align=right 0align=right 12align=right 40align=right 0align=right 0align=right 150align=right 100align=right 0align=right 27.670°
align=right 53align=right 1191.922290416align=center

C2v

align=center 0.000278469align=right 0align=right 0align=right 18align=right 35align=right 0align=right 0align=right 150align=right 96align=right 3align=right 27.137°
align=right 54align=right 1239.361474729align=center

C2

align=center 0.000137870align=right 0align=right 0align=right 12align=right 42align=right 0align=right 0align=right 156align=right 104align=right 0align=right 27.030°
align=right 55align=right 1287.772720783align=center

C2

align=center 0.000391696align=right 0align=right 0align=right 12align=right 43align=right 0align=right 0align=right 159align=right 106align=right 0align=right 26.615°
align=right 56align=right 1337.094945276align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 44align=right 0align=right 0align=right 162align=right 108align=right 0align=right 26.683°
align=right 57align=right 1387.383229253align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 45align=right 0align=right 0align=right 165align=right 110align=right 0align=right 26.702°
align=right 58align=right 1438.618250640align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 46align=right 0align=right 0align=right 168align=right 112align=right 0align=right 26.155°
align=right 59align=right 1490.773335279align=center

C2

align=center 0.000154286align=right 0align=right 0align=right 14align=right 43align=right 2align=right 0align=right 171align=right 114align=right 0align=right 26.170°
align=right 60align=right 1543.830400976align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 48align=right 0align=right 0align=right 174align=right 116align=right 0align=right 25.958°
align=right 61align=right 1597.941830199align=center

C1

align=center 0.001091717align=right 0align=right 0align=right 12align=right 49align=right 0align=right 0align=right 177align=right 118align=right 0align=right 25.392°
align=right 62align=right 1652.909409898align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 50align=right 0align=right 0align=right 180align=right 120align=right 0align=right 25.880°
align=right 63align=right 1708.879681503align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 51align=right 0align=right 0align=right 183align=right 122align=right 0align=right 25.257°
align=right 64align=right 1765.802577927align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 52align=right 0align=right 0align=right 186align=right 124align=right 0align=right 24.920°
align=right 65align=right 1823.667960264align=center

C2

align=center 0.000399515align=right 0align=right 0align=right 12align=right 53align=right 0align=right 0align=right 189align=right 126align=right 0align=right 24.527°
align=right 66align=right 1882.441525304align=center

C2

align=center 0.000776245align=right 0align=right 0align=right 12align=right 54align=right 0align=right 0align=right 192align=right 128align=right 0align=right 24.765°
align=right 67align=right 1942.122700406align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 55align=right 0align=right 0align=right 195align=right 130align=right 0align=right 24.727°
align=right 68align=right 2002.874701749align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 56align=right 0align=right 0align=right 198align=right 132align=right 0align=right 24.433°
align=right 69align=right 2064.533483235align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 57align=right 0align=right 0align=right 201align=right 134align=right 0align=right 24.137°
align=right 70align=right 2127.100901551align=center

D2d

align=center 0align=right 0align=right 0align=right 12align=right 50align=right 0align=right 0align=right 200align=right 128align=right 4align=right 24.291°
align=right 71align=right 2190.649906425align=center

C2

align=center 0.001256769align=right 0align=right 0align=right 14align=right 55align=right 2align=right 0align=right 207align=right 138align=right 0align=right 23.803°
align=right 72align=right 2255.001190975align=center

I

align=center 0align=right 0align=right 0align=right 12align=right 60align=right 0align=right 0align=right 210align=right 140align=right 0align=right 24.492°geodesic sphere 2,1
align=right 73align=right 2320.633883745align=center

C2

align=center 0.001572959align=right 0align=right 0align=right 12align=right 61align=right 0align=right 0align=right 213align=right 142align=right 0align=right 22.810°
align=right 74align=right 2387.072981838align=center

C2

align=center 0.000641539align=right 0align=right 0align=right 12align=right 62align=right 0align=right 0align=right 216align=right 144align=right 0align=right 22.966°
align=right 75align=right 2454.369689040align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 63align=right 0align=right 0align=right 219align=right 146align=right 0align=right 22.736°
align=right 76align=right 2522.674871841align=center

C2

align=center 0.000943474align=right 0align=right 0align=right 12align=right 64align=right 0align=right 0align=right 222align=right 148align=right 0align=right 22.886°
align=right 77align=right 2591.850152354align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 65align=right 0align=right 0align=right 225align=right 150align=right 0align=right 23.286°
align=right 78align=right 2662.046474566align=center

Th

align=center 0align=right 0align=right 0align=right 12align=right 66align=right 0align=right 0align=right 228align=right 152align=right 0align=right 23.426°
align=right 79align=right 2733.248357479align=center

Cs

align=center 0.000702921align=right 0align=right 0align=right 12align=right 63align=right 1align=right 0align=right 230align=right 152align=right 1align=right 22.636°
align=right 80align=right 2805.355875981align=center

D4d

align=center 0align=right 0align=right 0align=right 16align=right 64align=right 0align=right 0align=right 232align=right 152align=right 2align=right 22.778°
align=right 81align=right 2878.522829664align=center

C2

align=center 0.000194289align=right 0align=right 0align=right 12align=right 69align=right 0align=right 0align=right 237align=right 158align=right 0align=right 21.892°
align=right 82align=right 2952.569675286align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 70align=right 0align=right 0align=right 240align=right 160align=right 0align=right 22.206°
align=right 83align=right 3027.528488921align=center

C2

align=center 0.000339815align=right 0align=right 0align=right 14align=right 67align=right 2align=right 0align=right 243align=right 162align=right 0align=right 21.646°
align=right 84align=right 3103.465124431align=center

C2

align=center 0.000401973align=right 0align=right 0align=right 12align=right 72align=right 0align=right 0align=right 246align=right 164align=right 0align=right 21.513°
align=right 85align=right 3180.361442939align=center

C2

align=center 0.000416581align=right 0align=right 0align=right 12align=right 73align=right 0align=right 0align=right 249align=right 166align=right 0align=right 21.498°
align=right 86align=right 3258.211605713align=center

C2

align=center 0.001378932align=right 0align=right 0align=right 12align=right 74align=right 0align=right 0align=right 252align=right 168align=right 0align=right 21.522°
align=right 87align=right 3337.000750014align=center

C2

align=center 0.000754863align=right 0align=right 0align=right 12align=right 75align=right 0align=right 0align=right 255align=right 170align=right 0align=right 21.456°
align=right 88align=right 3416.720196758align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 76align=right 0align=right 0align=right 258align=right 172align=right 0align=right 21.486°
align=right 89align=right 3497.439018625align=center

C2

align=center 0.000070891align=right 0align=right 0align=right 12align=right 77align=right 0align=right 0align=right 261align=right 174align=right 0align=right 21.182°
align=right 90align=right 3579.091222723align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 78align=right 0align=right 0align=right 264align=right 176align=right 0align=right 21.230°
align=right 91align=right 3661.713699320align=center

C2

align=center 0.000033221align=right 0align=right 0align=right 12align=right 79align=right 0align=right 0align=right 267align=right 178align=right 0align=right 21.105°
align=right 92align=right 3745.291636241align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 80align=right 0align=right 0align=right 270align=right 180align=right 0align=right 21.026°
align=right 93align=right 3829.844338421align=center

C2

align=center 0.000213246align=right 0align=right 0align=right 12align=right 81align=right 0align=right 0align=right 273align=right 182align=right 0align=right 20.751°
align=right 94align=right 3915.309269620align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 82align=right 0align=right 0align=right 276align=right 184align=right 0align=right 20.952°
align=right 95align=right 4001.771675565align=center

C2

align=center 0.000116638align=right 0align=right 0align=right 12align=right 83align=right 0align=right 0align=right 279align=right 186align=right 0align=right 20.711°
align=right 96align=right 4089.154010060align=center

C2

align=center 0.000036310align=right 0align=right 0align=right 12align=right 84align=right 0align=right 0align=right 282align=right 188align=right 0align=right 20.687°
align=right 97align=right 4177.533599622align=center

C2

align=center 0.000096437align=right 0align=right 0align=right 12align=right 85align=right 0align=right 0align=right 285align=right 190align=right 0align=right 20.450°
align=right 98align=right 4266.822464156align=center

C2

align=center 0.000112916align=right 0align=right 0align=right 12align=right 86align=right 0align=right 0align=right 288align=right 192align=right 0align=right 20.422°
align=right 99align=right 4357.139163132align=center

C2

align=center 0.000156508align=right 0align=right 0align=right 12align=right 87align=right 0align=right 0align=right 291align=right 194align=right 0align=right 20.284°
align=right 100align=right 4448.350634331align=center

T

align=center 0align=right 0align=right 0align=right 12align=right 88align=right 0align=right 0align=right 294align=right 196align=right 0align=right 20.297°
align=right 101align=right 4540.590051694align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 89align=right 0align=right 0align=right 297align=right 198align=right 0align=right 20.011°
align=right 102align=right 4633.736565899align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 90align=right 0align=right 0align=right 300align=right 200align=right 0align=right 20.040°
align=right 103align=right 4727.836616833align=center

C2

align=center 0.000201245align=right 0align=right 0align=right 12align=right 91align=right 0align=right 0align=right 303align=right 202align=right 0align=right 19.907°
align=right 104align=right 4822.876522746align=center

D6

align=center 0align=right 0align=right 0align=right 12align=right 92align=right 0align=right 0align=right 306align=right 204align=right 0align=right 19.957°
align=right 105align=right 4919.000637616align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 93align=right 0align=right 0align=right 309align=right 206align=right 0align=right 19.842°
align=right 106align=right 5015.984595705align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 94align=right 0align=right 0align=right 312align=right 208align=right 0align=right 19.658°
align=right 107align=right 5113.953547724align=center

C2

align=center 0.000064137align=right 0align=right 0align=right 12align=right 95align=right 0align=right 0align=right 315align=right 210align=right 0align=right 19.327°
align=right 108align=right 5212.813507831align=center

C2

align=center 0.000432525align=right 0align=right 0align=right 12align=right 96align=right 0align=right 0align=right 318align=right 212align=right 0align=right 19.327°
align=right 109align=right 5312.735079920align=center

C2

align=center 0.000647299align=right 0align=right 0align=right 14align=right 93align=right 2align=right 0align=right 321align=right 214align=right 0align=right 19.103°
align=right 110align=right 5413.549294192align=center

D6

align=center 0align=right 0align=right 0align=right 12align=right 98align=right 0align=right 0align=right 324align=right 216align=right 0align=right 19.476°
align=right 111align=right 5515.293214587align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 99align=right 0align=right 0align=right 327align=right 218align=right 0align=right 19.255°
align=right 112align=right 5618.044882327align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 100align=right 0align=right 0align=right 330align=right 220align=right 0align=right 19.351°
align=right 113align=right 5721.824978027align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 101align=right 0align=right 0align=right 333align=right 222align=right 0align=right 18.978°
align=right 114align=right 5826.521572163align=center

C2

align=center 0.000149772align=right 0align=right 0align=right 12align=right 102align=right 0align=right 0align=right 336align=right 224align=right 0align=right 18.836°
align=right 115align=right 5932.181285777align=center

C3

align=center 0.000049972align=right 0align=right 0align=right 12align=right 103align=right 0align=right 0align=right 339align=right 226align=right 0align=right 18.458°
align=right 116align=right 6038.815593579align=center

C2

align=center 0.000259726align=right 0align=right 0align=right 12align=right 104align=right 0align=right 0align=right 342align=right 228align=right 0align=right 18.386°
align=right 117align=right 6146.342446579align=center

C2

align=center 0.000127609align=right 0align=right 0align=right 12align=right 105align=right 0align=right 0align=right 345align=right 230align=right 0align=right 18.566°
align=right 118align=right 6254.877027790align=center

C2

align=center 0.000332475align=right 0align=right 0align=right 12align=right 106align=right 0align=right 0align=right 348align=right 232align=right 0align=right 18.455°
align=right 119align=right 6364.347317479align=center

C2

align=center 0.000685590align=right 0align=right 0align=right 12align=right 107align=right 0align=right 0align=right 351align=right 234align=right 0align=right 18.336°
align=right 120align=right 6474.756324980align=center

Cs

align=center 0.001373062align=right 0align=right 0align=right 12align=right 108align=right 0align=right 0align=right 354align=right 236align=right 0align=right 18.418°
align=right 121align=right 6586.121949584align=center

C3

align=center 0.000838863align=right 0align=right 0align=right 12align=right 109align=right 0align=right 0align=right 357align=right 238align=right 0align=right 18.199°
align=right 122align=right 6698.374499261align=center

Ih

align=center 0align=right 0align=right 0align=right 12align=right 110align=right 0align=right 0align=right 360align=right 240align=right 0align=right 18.612°geodesic sphere 2,2
align=right 123align=right 6811.827228174align=center

C2v

align=center 0.001939754align=right 0align=right 0align=right 14align=right 107align=right 2align=right 0align=right 363align=right 242align=right 0align=right 17.840°
align=right 124align=right 6926.169974193align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 112align=right 0align=right 0align=right 366align=right 244align=right 0align=right 18.111°
align=right 125align=right 7041.473264023align=center

C2

align=center 0.000088274align=right 0align=right 0align=right 12align=right 113align=right 0align=right 0align=right 369align=right 246align=right 0align=right 17.867°
align=right 126align=right 7157.669224867align=center

D4

align=center 0align=right 0align=right 2align=right 16align=right 100align=right 8align=right 0align=right 372align=right 248align=right 0align=right 17.920°
align=right 127align=right 7274.819504675align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 115align=right 0align=right 0align=right 375align=right 250align=right 0align=right 17.877°
align=right 128align=right 7393.007443068align=center

C2

align=center 0.000054132align=right 0align=right 0align=right 12align=right 116align=right 0align=right 0align=right 378align=right 252align=right 0align=right 17.814°
align=right 129align=right 7512.107319268align=center

C2

align=center 0.000030099align=right 0align=right 0align=right 12align=right 117align=right 0align=right 0align=right 381align=right 254align=right 0align=right 17.743°
align=right 130align=right 7632.167378912align=center

C2

align=center 0.000025622align=right 0align=right 0align=right 12align=right 118align=right 0align=right 0align=right 384align=right 256align=right 0align=right 17.683°
align=right 131align=right 7753.205166941align=center

C2

align=center 0.000305133align=right 0align=right 0align=right 12align=right 119align=right 0align=right 0align=right 387align=right 258align=right 0align=right 17.511°
align=right 132align=right 7875.045342797align=center

I

align=center 0align=right 0align=right 0align=right 12align=right 120align=right 0align=right 0align=right 390align=right 260align=right 0align=right 17.958°geodesic sphere 3,1
align=right 133align=right 7998.179212898align=center

C3

align=center 0.000591438align=right 0align=right 0align=right 12align=right 121align=right 0align=right 0align=right 393align=right 262align=right 0align=right 17.133°
align=right 134align=right 8122.089721194align=center

C2

align=center 0.000470268align=right 0align=right 0align=right 12align=right 122align=right 0align=right 0align=right 396align=right 264align=right 0align=right 17.214°
align=right 135align=right 8246.909486992align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 123align=right 0align=right 0align=right 399align=right 266align=right 0align=right 17.431°
align=right 136align=right 8372.743302539align=center

T

align=center 0align=right 0align=right 0align=right 12align=right 124align=right 0align=right 0align=right 402align=right 268align=right 0align=right 17.485°
align=right 137align=right 8499.534494782align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 125align=right 0align=right 0align=right 405align=right 270align=right 0align=right 17.560°
align=right 138align=right 8627.406389880align=center

C2

align=center 0.000473576align=right 0align=right 0align=right 12align=right 126align=right 0align=right 0align=right 408align=right 272align=right 0align=right 16.924°
align=right 139align=right 8756.227056057align=center

C2

align=center 0.000404228align=right 0align=right 0align=right 12align=right 127align=right 0align=right 0align=right 411align=right 274align=right 0align=right 16.673°
align=right 140align=right 8885.980609041align=center

C1

align=center 0.000630351align=right 0align=right 0align=right 13align=right 126align=right 1align=right 0align=right 414align=right 276align=right 0align=right 16.773°
align=right 141align=right 9016.615349190align=center

C2v

align=center 0.000376365align=right 0align=right 0align=right 14align=right 126align=right 0align=right 1align=right 417align=right 278align=right 0align=right 16.962°
align=right 142align=right 9148.271579993align=center

C2

align=center 0.000550138align=right 0align=right 0align=right 12align=right 130align=right 0align=right 0align=right 420align=right 280align=right 0align=right 16.840°
align=right 143align=right 9280.839851192align=center

C2

align=center 0.000255449align=right 0align=right 0align=right 12align=right 131align=right 0align=right 0align=right 423align=right 282align=right 0align=right 16.782°
align=right 144align=right 9414.371794460align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 132align=right 0align=right 0align=right 426align=right 284align=right 0align=right 16.953°
align=right 145align=right 9548.928837232align=center

Cs

align=center 0.000094938align=right 0align=right 0align=right 12align=right 133align=right 0align=right 0align=right 429align=right 286align=right 0align=right 16.841°
align=right 146align=right 9684.381825575align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 134align=right 0align=right 0align=right 432align=right 288align=right 0align=right 16.905°
align=right 147align=right 9820.932378373align=center

C2

align=center 0.000636651align=right 0align=right 0align=right 12align=right 135align=right 0align=right 0align=right 435align=right 290align=right 0align=right 16.458°
align=right 148align=right 9958.406004270align=center

C2

align=center 0.000203701align=right 0align=right 0align=right 12align=right 136align=right 0align=right 0align=right 438align=right 292align=right 0align=right 16.627°
align=right 149align=right 10096.859907397align=center

C1

align=center 0.000638186align=right 0align=right 0align=right 14align=right 133align=right 2align=right 0align=right 441align=right 294align=right 0align=right 16.344°
align=right 150align=right 10236.196436701align=center

T

align=center 0align=right 0align=right 0align=right 12align=right 138align=right 0align=right 0align=right 444align=right 296align=right 0align=right 16.405°
align=right 151align=right 10376.571469275align=center

C2

align=center 0.000153836align=right 0align=right 0align=right 12align=right 139align=right 0align=right 0align=right 447align=right 298align=right 0align=right 16.163°
align=right 152align=right 10517.867592878align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 140align=right 0align=right 0align=right 450align=right 300align=right 0align=right 16.117°
align=right 153align=right 10660.082748237align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 141align=right 0align=right 0align=right 453align=right 302align=right 0align=right 16.390°
align=right 154align=right 10803.372421141align=center

C2

align=center 0.000735800align=right 0align=right 0align=right 12align=right 142align=right 0align=right 0align=right 456align=right 304align=right 0align=right 16.078°
align=right 155align=right 10947.574692279align=center

C2

align=center 0.000603670align=right 0align=right 0align=right 12align=right 143align=right 0align=right 0align=right 459align=right 306align=right 0align=right 15.990°
align=right 156align=right 11092.798311456align=center

C2

align=center 0.000508534align=right 0align=right 0align=right 12align=right 144align=right 0align=right 0align=right 462align=right 308align=right 0align=right 15.822°
align=right 157align=right 11238.903041156align=center

C2

align=center 0.000357679align=right 0align=right 0align=right 12align=right 145align=right 0align=right 0align=right 465align=right 310align=right 0align=right 15.948°
align=right 158align=right 11385.990186197align=center

C2

align=center 0.000921918align=right 0align=right 0align=right 12align=right 146align=right 0align=right 0align=right 468align=right 312align=right 0align=right 15.987°
align=right 159align=right 11534.023960956align=center

C2

align=center 0.000381457align=right 0align=right 0align=right 12align=right 147align=right 0align=right 0align=right 471align=right 314align=right 0align=right 15.960°
align=right 160align=right 11683.054805549align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 148align=right 0align=right 0align=right 474align=right 316align=right 0align=right 15.961°
align=right 161align=right 11833.084739465align=center

C2

align=center 0.000056447align=right 0align=right 0align=right 12align=right 149align=right 0align=right 0align=right 477align=right 318align=right 0align=right 15.810°
align=right 162align=right 11984.050335814align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 150align=right 0align=right 0align=right 480align=right 320align=right 0align=right 15.813°
align=right 163align=right 12136.013053220align=center

C2

align=center 0.000120798align=right 0align=right 0align=right 12align=right 151align=right 0align=right 0align=right 483align=right 322align=right 0align=right 15.675°
align=right 164align=right 12288.930105320align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 152align=right 0align=right 0align=right 486align=right 324align=right 0align=right 15.655°
align=right 165align=right 12442.804451373align=center

C2

align=center 0.000091119align=right 0align=right 0align=right 12align=right 153align=right 0align=right 0align=right 489align=right 326align=right 0align=right 15.651°
align=right 166align=right 12597.649071323align=center

D2d

align=center 0align=right 0align=right 0align=right 16align=right 146align=right 4align=right 0align=right 492align=right 328align=right 0align=right 15.607°
align=right 167align=right 12753.469429750align=center

C2

align=center 0.000097382align=right 0align=right 0align=right 12align=right 155align=right 0align=right 0align=right 495align=right 330align=right 0align=right 15.600°
align=right 168align=right 12910.212672268align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 156align=right 0align=right 0align=right 498align=right 332align=right 0align=right 15.655°
align=right 169align=right 13068.006451127align=center

Cs

align=center 0.000068102align=right 0align=right 0align=right 13align=right 155align=right 1align=right 0align=right 501align=right 334align=right 0align=right 15.537°
align=right 170align=right 13226.681078541align=center

D2d

align=center 0align=right 0align=right 0align=right 12align=right 158align=right 0align=right 0align=right 504align=right 336align=right 0align=right 15.569°
align=right 171align=right 13386.355930717align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 159align=right 0align=right 0align=right 507align=right 338align=right 0align=right 15.497°
align=right 172align=right 13547.018108787align=center

C2v

align=center 0.000547291align=right 0align=right 0align=right 14align=right 156align=right 2align=right 0align=right 510align=right 340align=right 0align=right 15.292°
align=right 173align=right 13708.635243034align=center

Cs

align=center 0.000286544align=right 0align=right 0align=right 12align=right 161align=right 0align=right 0align=right 513align=right 342align=right 0align=right 15.225°
align=right 174align=right 13871.187092292align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 162align=right 0align=right 0align=right 516align=right 344align=right 0align=right 15.366°
align=right 175align=right 14034.781306929align=center

C2

align=center 0.000026686align=right 0align=right 0align=right 12align=right 163align=right 0align=right 0align=right 519align=right 346align=right 0align=right 15.252°
align=right 176align=right 14199.354775632align=center

C1

align=center 0.000283978align=right 0align=right 0align=right 12align=right 164align=right 0align=right 0align=right 522align=right 348align=right 0align=right 15.101°
align=right 177align=right 14364.837545298align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 165align=right 0align=right 0align=right 525align=right 350align=right 0align=right 15.269°
align=right 178align=right 14531.309552587align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 166align=right 0align=right 0align=right 528align=right 352align=right 0align=right 15.145°
align=right 179align=right 14698.754594220align=center

C1

align=center 0.000125113align=right 0align=right 0align=right 13align=right 165align=right 1align=right 0align=right 531align=right 354align=right 0align=right 14.968°
align=right 180align=right 14867.099927525align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 168align=right 0align=right 0align=right 534align=right 356align=right 0align=right 15.067°
align=right 181align=right 15036.467239769align=center

C2

align=center 0.000304193align=right 0align=right 0align=right 12align=right 169align=right 0align=right 0align=right 537align=right 358align=right 0align=right 15.002°
align=right 182align=right 15206.730610906align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 170align=right 0align=right 0align=right 540align=right 360align=right 0align=right 15.155°
align=right 183align=right 15378.166571028align=center

C1

align=center 0.000467899align=right 0align=right 0align=right 12align=right 171align=right 0align=right 0align=right 543align=right 362align=right 0align=right 14.747°
align=right 184align=right 15550.421450311align=center

T

align=center 0align=right 0align=right 0align=right 12align=right 172align=right 0align=right 0align=right 546align=right 364align=right 0align=right 14.932°
align=right 185align=right 15723.720074072align=center

C2

align=center 0.000389762align=right 0align=right 0align=right 12align=right 173align=right 0align=right 0align=right 549align=right 366align=right 0align=right 14.775°
align=right 186align=right 15897.897437048align=center

C1

align=center 0.000389762align=right 0align=right 0align=right 12align=right 174align=right 0align=right 0align=right 552align=right 368align=right 0align=right 14.739°
align=right 187align=right 16072.975186320align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 175align=right 0align=right 0align=right 555align=right 370align=right 0align=right 14.848°
align=right 188align=right 16249.222678879align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 176align=right 0align=right 0align=right 558align=right 372align=right 0align=right 14.740°
align=right 189align=right 16426.371938862align=center

C2

align=center 0.000020732align=right 0align=right 0align=right 12align=right 177align=right 0align=right 0align=right 561align=right 374align=right 0align=right 14.671°
align=right 190align=right 16604.428338501align=center

C3

align=center 0.000586804align=right 0align=right 0align=right 12align=right 178align=right 0align=right 0align=right 564align=right 376align=right 0align=right 14.501°
align=right 191align=right 16783.452219362align=center

C1

align=center 0.001129202align=right 0align=right 0align=right 13align=right 177align=right 1align=right 0align=right 567align=right 378align=right 0align=right 14.195°
align=right 192align=right 16963.338386460align=center

I

align=center 0align=right 0align=right 0align=right 12align=right 180align=right 0align=right 0align=right 570align=right 380align=right 0align=right 14.819°geodesic sphere 3,2
align=right 193align=right 17144.564740880align=center

C2

align=center 0.000985192align=right 0align=right 0align=right 12align=right 181align=right 0align=right 0align=right 573align=right 382align=right 0align=right 14.144°
align=right 194align=right 17326.616136471align=center

C1

align=center 0.000322358align=right 0align=right 0align=right 12align=right 182align=right 0align=right 0align=right 576align=right 384align=right 0align=right 14.350°
align=right 195align=right 17509.489303930align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 183align=right 0align=right 0align=right 579align=right 386align=right 0align=right 14.375°
align=right 196align=right 17693.460548082align=center

C2

align=center 0.000315907align=right 0align=right 0align=right 12align=right 184align=right 0align=right 0align=right 582align=right 388align=right 0align=right 14.251°
align=right 197align=right 17878.340162571align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 185align=right 0align=right 0align=right 585align=right 390align=right 0align=right 14.147°
align=right 198align=right 18064.262177195align=center

C2

align=center 0.000011149align=right 0align=right 0align=right 12align=right 186align=right 0align=right 0align=right 588align=right 392align=right 0align=right 14.237°
align=right 199align=right 18251.082495640align=center

C1

align=center 0.000534779align=right 0align=right 0align=right 12align=right 187align=right 0align=right 0align=right 591align=right 394align=right 0align=right 14.153°
align=right 200align=right 18438.842717530align=center

D2

align=center 0align=right 0align=right 0align=right 12align=right 188align=right 0align=right 0align=right 594align=right 396align=right 0align=right 14.222°
align=right 201align=right 18627.591226244align=center

C1

align=center 0.001048859align=right 0align=right 0align=right 13align=right 187align=right 1align=right 0align=right 597align=right 398align=right 0align=right 13.830°
align=right 202align=right 18817.204718262align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 190align=right 0align=right 0align=right 600align=right 400align=right 0align=right 14.189°
align=right 203align=right 19007.981204580align=center

Cs

align=center 0.000600343align=right 0align=right 0align=right 12align=right 191align=right 0align=right 0align=right 603align=right 402align=right 0align=right 13.977°
align=right 204align=right 19199.540775603align=center

Th

align=center 0align=right 0align=right 0align=right 12align=right 192align=right 0align=right 0align=right 606align=right 404align=right 0align=right 14.291°
align=right 212align=right 20768.053085964align=center

I

align=center 0align=right 0align=right 0align=right 12align=right 200align=right 0align=right 0align=right 630align=right 420align=right 0align=right 14.118°geodesic sphere 4,1
align=right 214align=right 21169.910410375align=center

T

align=center 0align=right 0align=right 0align=right 12align=right 202align=right 0align=right 0align=right 636align=right 424align=right 0align=right 13.771°
align=right 216align=right 21575.596377869align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 204align=right 0align=right 0align=right 642align=right 428align=right 0align=right 13.735°
align=right 217align=right 21779.856080418align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 205align=right 0align=right 0align=right 645align=right 430align=right 0align=right 13.902°
align=right 232align=right 24961.252318934align=center

T

align=center 0align=right 0align=right 0align=right 12align=right 220align=right 0align=right 0align=right 690align=right 460align=right 0align=right 13.260°
align=right 255align=right 30264.424251281align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 243align=right 0align=right 0align=right 759align=right 506align=right 0align=right 12.565°
align=right 256align=right 30506.687515847align=center

T

align=center 0align=right 0align=right 0align=right 12align=right 244align=right 0align=right 0align=right 762align=right 508align=right 0align=right 12.572°
align=right 257align=right 30749.941417346align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 245align=right 0align=right 0align=right 765align=right 510align=right 0align=right 12.672°
align=right 272align=right 34515.193292681align=center

Ih

align=center 0align=right 0align=right 0align=right 12align=right 260align=right 0align=right 0align=right 810align=right 540align=right 0align=right 12.335°geodesic sphere 3,3
align=right 282align=right 37147.294418462align=center

I

align=center 0align=right 0align=right 0align=right 12align=right 270align=right 0align=right 0align=right 840align=right 560align=right 0align=right 12.166°geodesic sphere 4,2
align=right 292align=right 39877.008012909align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 280align=right 0align=right 0align=right 870align=right 580align=right 0align=right 11.857°
align=right 306align=right 43862.569780797align=center

Th

align=center 0align=right 0align=right 0align=right 12align=right 294align=right 0align=right 0align=right 912align=right 608align=right 0align=right 11.628°
align=right 312align=right 45629.313804002align=center

C2

align=center 0.000306163align=right 0align=right 0align=right 12align=right 300align=right 0align=right 0align=right 930align=right 620align=right 0align=right 11.299°
align=right 315align=right 46525.825643432align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 303align=right 0align=right 0align=right 939align=right 626align=right 0align=right 11.337°
align=right 317align=right 47128.310344520align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 305align=right 0align=right 0align=right 945align=right 630align=right 0align=right 11.423°
align=right 318align=right 47431.056020043align=center

D3

align=center 0align=right 0align=right 0align=right 12align=right 306align=right 0align=right 0align=right 948align=right 632align=right 0align=right 11.219°
align=right 334align=right 52407.728127822align=center

T

align=center 0align=right 0align=right 0align=right 12align=right 322align=right 0align=right 0align=right 996align=right 664align=right 0align=right 11.058°
align=right 348align=right 56967.472454334align=center

Th

align=center 0align=right 0align=right 0align=right 12align=right 336align=right 0align=right 0align=right 1038align=right 692align=right 0align=right 10.721°
align=right 357align=right 59999.922939598align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 345align=right 0align=right 0align=right 1065align=right 710align=right 0align=right 10.728°
align=right 358align=right 60341.830924588align=center

T

align=center 0align=right 0align=right 0align=right 12align=right 346align=right 0align=right 0align=right 1068align=right 712align=right 0align=right 10.647°
align=right 372align=right 65230.027122557align=center

I

align=center 0align=right 0align=right 0align=right 12align=right 360align=right 0align=right 0align=right 1110align=right 740align=right 0align=right 10.531°geodesic sphere 4,3
align=right 382align=right 68839.426839215align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 370align=right 0align=right 0align=right 1140align=right 760align=right 0align=right 10.379°
align=right 390align=right 71797.035335953align=center

Th

align=center 0align=right 0align=right 0align=right 12align=right 378align=right 0align=right 0align=right 1164align=right 776align=right 0align=right 10.222°
align=right 392align=right 72546.258370889align=center

C1

align=center 0align=right 0align=right 0align=right 12align=right 380align=right 0align=right 0align=right 1170align=right 780align=right 0align=right 10.278°
align=right 400align=right 75582.448512213align=center

T

align=center 0align=right 0align=right 0align=right 12align=right 388align=right 0align=right 0align=right 1194align=right 796align=right 0align=right 10.068°
align=right 402align=right 76351.192432673align=center

D5

align=center 0align=right 0align=right 0align=right 12align=right 390align=right 0align=right 0align=right 1200align=right 800align=right 0align=right 10.099°
align=right 432align=right 88353.709681956align=center

D3

align=center 0align=right 0align=right 0align=right 24align=right 396align=right 12align=right 0align=right 1290align=right 860align=right 0align=right 9.556°
align=right 448align=right 95115.546986209align=center

T

align=center 0align=right 0align=right 0align=right 24align=right 412align=right 12align=right 0align=right 1338align=right 892align=right 0align=right 9.322°
align=right 460align=right 100351.763108673align=center

T

align=center 0align=right 0align=right 0align=right 24align=right 424align=right 12align=right 0align=right 1374align=right 916align=right 0align=right 9.297°
align=right 468align=right 103920.871715127align=center

S6

align=center 0align=right 0align=right 0align=right 24align=right 432align=right 12align=right 0align=right 1398align=right 932align=right 0align=right 9.120°
align=right 470align=right 104822.886324279align=center

S6

align=center 0align=right 0align=right 0align=right 24align=right 434align=right 12align=right 0align=right 1404align=right 936align=right 0align=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)=\delta0,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