Thomson problem

{{Short description|Max distance configuration for N points on a sphere}}

The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of {{mvar|N}} electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist J. J. Thomson posed the problem in 1904{{cite journal|last=Thomson| first=Joseph John|author-link=Joseph John Thomson| title=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|journal=Philosophical Magazine | doi=10.1080/14786440409463107 | series=Series 6 | volume=7 |number=39|pages=237–265|date= March 1904|url=http://www.cond-mat.physik.uni-mainz.de/~oettel/ws10/thomson_PhilMag_7_237_1904.pdf|url-status=dead|archive-url=https://web.archive.org/web/20131213172104/http://www.cond-mat.physik.uni-mainz.de/~oettel/ws10/thomson_PhilMag_7_237_1904.pdf|archive-date=13 December 2013}} 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 {{mvar|N}} 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 $r_{ij}=|\mathbf{r}_i - \mathbf{r}_j|$ is the distance between each pair of electrons located at points on the sphere defined by vectors $\mathbf{r}_i$ and $\mathbf{r}_j$ , 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 energies

: $U(N) = \sum_{i < j} \frac{1}{r_{ij}}.$

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".

{{cite journal

| first = S. | last = Smale

|title = Mathematical Problems for the Next Century

|journal = Mathematical Intelligencer

|date =1998

|volume=20

|number=2

|pages=7–15

|citeseerx = 10.1.1.35.4101

|doi=10.1007/bf03025291

| s2cid = 1331144

}}

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, $r_{ij} = 2r = 2$ , or

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

Known exact solutions

File:N 2 to 5 ThomsonSolutions.png

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.{{cite journal|first1=L.|last1=Föppl |title=Stabile Anordnungen von Elektronen im Atom | journal = J. Reine Angew. Math. |number=141 |year=1912 |volume=141 |pages=251–301|doi=10.1515/crll.1912.141.251 |s2cid=120309200 |url=http://eudml.org/doc/149380}}. 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 2018 with electrons residing at vertices of a triangular dipyramid.{{cite arXiv|last=Schwartz|first=Richard|eprint=1001.3702|title=The 5 electron case of Thomson's Problem|date=2018-10-30|class=math.MG}} 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.{{cite journal|first=V.A. |last=Yudin |title=The minimum of potential energy of a system of point charges |journal=Discretnaya Matematika |volume=4 |number=2|year=1992|pages= 115–121 (in Russian)}}; {{cite journal|first=V. A. |last=Yudin|journal= Discrete Math. Appl.|volume= 3 |number=1| year=1993|pages=75–81 |doi=10.1515/dma.1993.3.1.75 |title=The minimum of potential energy of a system of point charges|s2cid=117117450}} 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.{{cite journal|first=N.N. |last= Andreev|title=An extremal property of the icosahedron| journal=East J. Approximation|volume=2 |number=4 |year=1996|pages=459–462}} {{MR|1426716}}, {{Zbl|0877.51021}}

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.{{cite arXiv |title=Polyhedra in physics, chemistry and geometry |date=2003 |eprint=math-ph/0303071 |last1=Atiyah |first1=Michael |last2=Sutcliffe |first2=Paul }}

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

$\sum_{i < j} f(|x_i-x_j|).$

Traditionally, one considers $f(x)=x^{-\alpha}$ also known as Riesz $\alpha$ -kernels. For integrable Riesz kernels see the 1972 work of Landkof.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. For non-integrable Riesz kernels, the poppy-seed bagel theorem holds, see the 2004 work of Hardin and Saff.Hardin, D. P.; Saff, E. B. Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc. 51 (2004), no. 10, 1186–1194 Notable cases include:{{cite news |last1=Batagelj |first1=Vladimir |last2=Plestenjak |first2=Bor |title=Optimal arrangements of n points on a sphere and in a circle |url=https://www.fmf.uni-lj.si/~plestenjak/Talks/preddvor.pdf |archive-date=25 June 2018 |archive-url=https://web.archive.org/web/20180625050324/https://www.fmf.uni-lj.si/~plestenjak/Talks/preddvor.pdf |publisher=IMFM/TCS}}

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:

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=

File:ThomsonProblem Related Energies LaFave.png

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

: $U_{\text{shell}}(N)=\frac{N^2}{2}$

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

: $U_{\text{rand}}(N)=\frac{N(N-1)}{2}$

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, $U_{\text{rand}}(N) < U_{\text{shell}}(N)$ .

=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,{{cite journal|last=LaFave Jr| first=Tim| title=Discrete transformations in the Thomson Problem|journal=Journal of Electrostatics | doi=10.1016/j.elstat.2013.11.007| volume=72 |number=1|pages=39–43|date= February 2014| arxiv=1403.2592| s2cid=119309183|url=https://www.sciencedirect.com/science/article/abs/pii/S0304388613001460}}

: $U_0(N+1)=U_{\text{Thom}}(N)+N.$

Thus, if $U_{\text{Thom}}(N)$ is known exactly, then $U_0(N+1)$ is known exactly.

In general, $U_0(N+1)$ is greater than $U_{\text{Thom}}(N+1)$ , but is remarkably closer to each $(N+1)$ th Thomson solution than $U_{\text{shell}}(N+1)$ and $U_{\text{rand}}(N+1)$ . 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.{{cite journal|first1=Y.|last1=Levin|first2=J. J. |last2= Arenzon |title=Why charges go to the Surface: A generalized Thomson Problem|journal=Europhys. Lett.|volume=63|issue=3|page=415|year=2003|arxiv=cond-mat/0302524 |doi=10.1209/epl/i2003-00546-1 |bibcode=2003EL.....63..415L|s2cid=18929981}}

{{quote box|width=23em|"No fact discovered about the atom can be trivial, nor fail to accelerate the progress of physical science, for the greater part of natural philosophy is the outcome of the structure and mechanism of the atom."|—Sir J. J. ThomsonSir J.J. Thomson, The Romanes Lecture, 1914 (The Atomic Theory)}}

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.{{cite journal|last=LaFave Jr|first=Tim|title=Correspondences between the classical electrostatic Thomson problem and atomic electronic structure|journal=Journal of Electrostatics|volume=71|issue=6|pages=1029–1035|year=2013|doi=10.1016/j.elstat.2013.10.001|arxiv=1403.2591|s2cid=118480104}}

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{{cn|date=April 2024}} $N$ is the number of points (charges) in a configuration, $U_{\textrm{Thom}}$ 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.

Kevin Brown.

[http://mathpages.com/home/kmath005/kmath005.htm "Min-Energy Configurations of Electrons On A Sphere"].

Retrieved 2014-05-01.

class="wikitable"
N ! $U_{\textrm{Thom}}$ ! Symmetry ! $\left\| \sum \mathbf{r}_i \right\|$ ! $v_3$ ! $v_4$ ! $v_5$ ! $v_6$ ! $v_7$ ! $v_8$ ! $e$ ! $f_3$ ! $f_4$ ! $\theta_1$ ! Equivalent polyhedron
align=right \| 2 \| align=right \| 0.500000000 \| align=center \| $D_{\infty h}$ \| 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 {3,5+}_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 {3,5+}_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 {3,5+}_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 {3,5+}_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 {3,5+}_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 {3,5+}_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 {3,5+}_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 {3,5+}_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 {3,5+}_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 {3,5+}_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°

class="wikitable"

! $U_{\textrm{Thom}}$

! Symmetry

! $\left| \sum \mathbf{r}_i \right|$

! $v_3$

! $v_4$

! $v_5$

! $v_6$

! $v_7$

! $v_8$

! $e$

! $f_3$

! $f_4$

! $\theta_1$

! Equivalent polyhedron

align=right | 2

| align=right | 0.500000000

| align=center | $D_{\infty h}$

| align=center | 0

| align=center | –

| align=right | 1

| align=center | –

| align=right | 180.000°

| digon

align=right | 3

| align=right | 1.732050808

| align=center | $D_{3h}$

| align=center | 0

| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 12

| align=right | 0

| align=right | 30

| align=right | 20

| align=right | 0

| align=right | 63.435°

| icosahedron
(geodesic sphere {3,5+}_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 | 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 | 12

| align=right | 2

| 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 | 12

| align=right | 3

| 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 | 12

| align=right | 4

| 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 | 12

| align=right | 5

| 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 | 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 | 14

| align=right | 5

| 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 | 12

| align=right | 8

| 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 | 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 | 12

| align=right | 10

| 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 | 12

| align=right | 11

| 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 | 24

| 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 | 14

| align=right | 11

| 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 | 12

| align=right | 14

| 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 | 12

| align=right | 15

| 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 | 12

| align=right | 16

| 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 | 12

| align=right | 17

| 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 | 12

| align=right | 18

| 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 | 12

| align=right | 19

| 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 | 12

| align=right | 20

| align=right | 0

| align=right | 90

| align=right | 60

| align=right | 0

| align=right | 37.377°

| pentakis dodecahedron
(geodesic sphere {3,5+}_1,1)

align=right | 33

| align=right | 440.204057448

| align=center | $C_s$

| align=center | 0.004356481

| 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 | 12

| align=right | 22

| 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 | 12

| align=right | 23

| 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 | 12

| align=right | 24

| 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 | 12

| align=right | 25

| 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 | 12

| align=right | 26

| 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 | 12

| align=right | 27

| 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 | 12

| align=right | 28

| 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 | 12

| align=right | 29

| 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 | 12

| align=right | 30

| 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 | 12

| align=right | 31

| 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 | 24

| align=right | 20

| 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 | 12

| align=right | 33

| 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 | 12

| align=right | 34

| 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 | 14

| align=right | 33

| 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 | 24

| 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 | 12

| align=right | 37

| 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 | 12

| align=right | 38

| 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 | 12

| align=right | 39

| 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 | 12

| align=right | 40

| 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 | 18

| align=right | 35

| 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 | 12

| align=right | 42

| 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 | 12

| align=right | 43

| 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 | 12

| align=right | 44

| 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 | 12

| align=right | 45

| 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 | 12

| align=right | 46

| 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 | 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 | 12

| align=right | 48

| 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 | 12

| align=right | 49

| 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 | 12

| align=right | 50

| 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 | 12

| align=right | 51

| 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 | 12

| align=right | 52

| 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 | 12

| align=right | 53

| 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 | 12

| align=right | 54

| 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 | 12

| align=right | 55

| 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 | 12

| align=right | 56

| 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 | 12

| align=right | 57

| 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 | 12

| align=right | 50

| 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 | 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 | 12

| align=right | 60

| align=right | 0

| align=right | 210

| align=right | 140

| align=right | 0

| align=right | 24.492°

| geodesic sphere {3,5+}_2,1

align=right | 73

| align=right | 2320.633883745

| align=center | $C_{2}$

| align=center | 0.001572959

| align=right | 0

| align=right | 12

| align=right | 61

| 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 | 12

| align=right | 62

| 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 | 12

| align=right | 63

| 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 | 12

| align=right | 64

| 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 | 12

| align=right | 65

| 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 | 12

| align=right | 66

| 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 | 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 | 16

| align=right | 64

| 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 | 12

| align=right | 69

| 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 | 12

| align=right | 70

| 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 | 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 | 12

| align=right | 72

| 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 | 12

| align=right | 73

| 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 | 12

| align=right | 74

| 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 | 12

| align=right | 75

| 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 | 12

| align=right | 76

| 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 | 12

| align=right | 77

| 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 | 12

| align=right | 78

| 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 | 12

| align=right | 79

| 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 | 12

| align=right | 80

| 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 | 12

| align=right | 81

| 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 | 12

| align=right | 82

| 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 | 12

| align=right | 83

| 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 | 12

| align=right | 84

| 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 | 12

| align=right | 85

| 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 | 12

| align=right | 86

| 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 | 12

| align=right | 87

| 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 | 12

| align=right | 88

| 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 | 12

| align=right | 89

| 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 | 12

| align=right | 90

| 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 | 12

| align=right | 91

| 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 | 12

| align=right | 92

| 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 | 12

| align=right | 93

| 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 | 12

| align=right | 94

| 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 | 12

| align=right | 95

| 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 | 12

| align=right | 96

| 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 | 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 | 12

| align=right | 98

| 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 | 12

| align=right | 99

| 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 | 12

| align=right | 100

| 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 | 12

| align=right | 101

| 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 | 12

| align=right | 102

| 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 | 12

| align=right | 103

| 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 | 12

| align=right | 104

| 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 | 12

| align=right | 105

| 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 | 12

| align=right | 106

| 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 | 12

| align=right | 107

| 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 | 12

| align=right | 108

| 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 | 12

| align=right | 109

| 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 | 12

| align=right | 110

| align=right | 0

| align=right | 360

| align=right | 240

| align=right | 0

| align=right | 18.612°

| geodesic sphere {3,5+}_2,2

align=right | 123

| align=right | 6811.827228174

| align=center | $C_{2v}$

| align=center | 0.001939754

| 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 | 12

| align=right | 112

| 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 | 12

| align=right | 113

| 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 | 12

| align=right | 115

| 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 | 12

| align=right | 116

| 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 | 12

| align=right | 117

| 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 | 12

| align=right | 118

| 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 | 12

| align=right | 119

| 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 | 12

| align=right | 120

| align=right | 0

| align=right | 390

| align=right | 260

| align=right | 0

| align=right | 17.958°

| geodesic sphere {3,5+}_3,1

align=right | 133

| align=right | 7998.179212898

| align=center | $C_{3}$

| align=center | 0.000591438

| align=right | 0

| align=right | 12

| align=right | 121

| 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 | 12

| align=right | 122

| 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 | 12

| align=right | 123

| 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 | 12

| align=right | 124

| 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 | 12

| align=right | 125

| 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 | 12

| align=right | 126

| 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 | 12

| align=right | 127

| 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 | 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 | 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 | 12

| align=right | 130

| 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 | 12

| align=right | 131

| 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 | 12

| align=right | 132

| 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 | 12

| align=right | 133

| 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 | 12

| align=right | 134

| 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 | 12

| align=right | 135

| 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 | 12

| align=right | 136

| 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 | 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 | 12

| align=right | 138

| 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 | 12

| align=right | 139

| 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 | 12

| align=right | 140

| 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 | 12

| align=right | 141

| 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 | 12

| align=right | 142

| 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 | 12

| align=right | 143

| 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 | 12

| align=right | 144

| 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 | 12

| align=right | 145

| 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 | 12

| align=right | 146

| 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 | 12

| align=right | 147

| 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 | 12

| align=right | 148

| 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 | 12

| align=right | 149

| 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 | 12

| align=right | 150

| 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 | 12

| align=right | 151

| 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 | 12

| align=right | 152

| 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 | 12

| align=right | 153

| 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 | 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 | 12

| align=right | 155

| 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 | 12

| align=right | 156

| 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 | 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 | 12

| align=right | 158

| 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 | 12

| align=right | 159

| 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 | 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 | 12

| align=right | 161

| 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 | 12

| align=right | 162

| 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 | 12

| align=right | 163

| 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 | 12

| align=right | 164

| 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 | 12

| align=right | 165

| 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 | 12

| align=right | 166

| 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 | 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 | 12

| align=right | 168

| 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 | 12

| align=right | 169

| 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 | 12

| align=right | 170

| 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 | 12

| align=right | 171

| 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 | 12

| align=right | 172

| 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 | 12

| align=right | 173

| 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 | 12

| align=right | 174

| 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 | 12

| align=right | 175

| 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 | 12

| align=right | 176

| 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 | 12

| align=right | 177

| 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 | 12

| align=right | 178

| 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 | 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 | 12

| align=right | 180

| align=right | 0

| align=right | 570

| align=right | 380

| align=right | 0

| align=right | 14.819°

| geodesic sphere {3,5+}_3,2

align=right | 193

| align=right | 17144.564740880

| align=center | $C_{2}$

| align=center | 0.000985192

| align=right | 0

| align=right | 12

| align=right | 181

| 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 | 12

| align=right | 182

| 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 | 12

| align=right | 183

| 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 | 12

| align=right | 184

| 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 | 12

| align=right | 185

| 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 | 12

| align=right | 186

| 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 | 12

| align=right | 187

| 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 | 12

| align=right | 188

| 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 | 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 | 12

| align=right | 190

| 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 | 12

| align=right | 191

| 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 | 12

| align=right | 192

| 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 | 12

| align=right | 200

| align=right | 0

| align=right | 630

| align=right | 420

| align=right | 0

| align=right | 14.118°

| geodesic sphere {3,5+}_4,1

align=right | 214

| align=right | 21169.910410375

| align=center | $T$

| align=center | 0

| align=right | 0

| align=right | 12

| align=right | 202

| 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 | 12

| align=right | 204

| 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 | 12

| align=right | 205

| 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 | 12

| align=right | 220

| 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 | 12

| align=right | 243

| 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 | 12

| align=right | 244

| 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 | 12

| align=right | 245

| 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 | 12

| align=right | 260

| align=right | 0

| align=right | 810

| align=right | 540

| align=right | 0

| align=right | 12.335°

| geodesic sphere {3,5+}_3,3

align=right | 282

| align=right | 37147.294418462

| align=center | $I$

| align=center | 0

| align=right | 0

| align=right | 12

| align=right | 270

| align=right | 0

| align=right | 840

| align=right | 560

| align=right | 0

| align=right | 12.166°

| geodesic sphere {3,5+}_4,2

align=right | 292

| align=right | 39877.008012909

| align=center | $D_{5}$

| align=center | 0

| align=right | 0

| align=right | 12

| align=right | 280

| 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 | 12

| align=right | 294

| 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 | 12

| align=right | 300

| 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 | 12

| align=right | 303

| 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 | 12

| align=right | 305

| 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 | 12

| align=right | 306

| 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 | 12

| align=right | 322

| 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 | 12

| align=right | 336

| 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 | 12

| align=right | 345

| 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 | 12

| align=right | 346

| 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 | 12

| align=right | 360

| align=right | 0

| align=right | 1110

| align=right | 740

| align=right | 0

| align=right | 10.531°

| geodesic sphere {3,5+}_4,3

align=right | 382

| align=right | 68839.426839215

| align=center | $D_{5}$

| align=center | 0

| align=right | 0

| align=right | 12

| align=right | 370

| 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 | 12

| align=right | 378

| 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 | 12

| align=right | 380

| 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 | 12

| align=right | 388

| 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 | 12

| align=right | 390

| 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 | 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 | 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 | 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 | 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 | 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.{{Cite web|url=https://oeis.org/A008486|title=Sloane's A008486 (see the comment from Feb 03 2017)|last=|first=|date=|website=The On-Line Encyclopedia of Integer Sequences|publisher=OEIS Foundation|access-date=2017-02-08}}{{clarify|date=October 2021}}

References

Notes

{{cite journal|first1=L.L. |last1=Whyte|title=Unique arrangements of points on a sphere | journal = Amer. Math. Monthly |year=1952 |volume=59 | number=9 | pages=606–611|doi=10.2307/2306764|jstor=2306764}}
{{cite journal|first1=Harvey |last1=Cohn|year=1956|title= Stability configurations of electrons on a sphere|journal=Math. Comput. |volume=10|issue=55| pages=117–120 | doi=10.1090/S0025-5718-1956-0081133-0 |doi-access=free}}
{{cite journal|first1=Michael | last1=Goldberg | title=Stability configurations of electrons on a sphere|journal=Math. Comp. | year=1969 | volume=23 | issue=108 | pages=785–786 | doi=10.1090/S0025-5718-69-99642-2 | doi-access=free }}
{{cite journal|first1=T. |last1=Erber|first2=G. M. |last2=Hockney | title= equilibrium configurations of N equal charges on a sphere|year=1991|journal=J. Phys. A: Math. Gen. |volume=24 | pages=L1369|doi=10.1088/0305-4470/24/23/008|number=23|bibcode=1991JPhA...24L1369E|s2cid=122561279 }}
{{cite journal|first1=J. R. |last1=Morris|first2=D. M. |last2=Deaven|first3=K. M. | last3=Ho|title=Genetic-algorithm energy minimization for point charges on a sphere | journal = Phys. Rev. B |volume=53 |issue=4| year=1996|pages=R1740–R1743 |doi=10.1103/PhysRevB.53.R1740|pmid=9983695|bibcode=1996PhRvB..53.1740M|citeseerx=10.1.1.28.93}}
{{cite book|first1=T.|last1= Erber |first2= G. M.|last2= Hockney|chapter=Complex Systems: Equilibrium Configurations of $N$ Equal Charges on a Sphere $(2\leq N\leq 112)$ |title= Advances in Chemical Physics|volume= 98|pages=495–594|year= 1997 |doi=10.1002/9780470141571.ch5|isbn= 9780470141571 }}.
{{cite journal|first1=E. L. |last1=Altschuler | first2=T. J. |last2=Williams|first3=E. R. |last3=Ratner |first4=R. |last4=Tipton |first5=R. |last5=Stong |first6=F. |last6=Dowla | first7=F. | last7=Wooten|title=Possible global minimum lattice configurations for Thomson's problem of charges on a sphere|year=1997|journal = Phys. Rev. Lett. |volume=78 |issue=14 | pages=2681–2685 |doi=10.1103/PhysRevLett.78.2681|bibcode=1997PhRvL..78.2681A|url=https://www.mcs.anl.gov/~zippy/publications/thomson/thomsonPRL.html }}
{{cite journal|first1=M. |last1=Bowick|first2=A. | last2=Cacciuto|first3=D. R. | last3=Nelson|first4=A. | last4=Travesset |title=Crystalline order on a sphere and the generalized Thomson Problem|journal=Phys. Rev. Lett. | volume= 89 |year=2002 |pages=249902 | doi=10.1103/PhysRevLett.89.185502|pmid=12398614| issue=18|arxiv=cond-mat/0206144|bibcode=2002PhRvL..89r5502B|s2cid=20362989}}
{{cite journal|first1=P. D.|last1= Dragnev|first2= D. A.|last2= Legg|first3= D. W. |last3= Townsend |title=Discrete logarithmic energy on the sphere|journal=Pacific J. Math. |volume=207 |year=2002|number= 2|pages= 345–358|doi=10.2140/pjm.2002.207.345|doi-access=free}}.
{{cite journal|first1=A. | last1=Katanforoush |first2=M. |last2=Shahshahani | title=Distributing points on the sphere. I|year=2003 | journal=Exper. Math. |volume=12|number=2| pages=199–209 |doi=10.1080/10586458.2003.10504492| s2cid=7306812 | url=http://projecteuclid.org/euclid.em/1067634731 }}
{{cite journal|first1=David J. | last1=Wales |first2=Sidika|last2=Ulker| title=Structure and dynamics of spherical crystals characterized for the Thomson problem | journal=Phys. Rev. B | volume=74 | year=2006|number=21|pages=212101 |doi=10.1103/PhysRevB.74.212101|bibcode=2006PhRvB..74u2101W| s2cid=119932997 }} Configurations reprinted in {{cite web | first1=D. J. | last1=Wales | first2=S. | last2=Ulker | url=http://www-wales.ch.cam.ac.uk/~wales/CCD/Thomson/table.html|title=The Cambridge cluster database}}
{{cite journal|first1=A. |last1=Slosar|first2=R. | last2=Podgornik|title= On the connected-charges Thomson problem|journal=Europhys. Lett.|year=2006|volume=75|number=4|pages=631|doi=10.1209/epl/i2006-10146-1|arxiv=cond-mat/0606765|bibcode=2006EL.....75..631S|s2cid=119005054}}
{{cite journal|first1=Henry |last1= Cohn |first2= Abhinav|last2= Kumar |title=Universally optimal distribution of points on spheres|journal=J. Amer. Math. Soc. |volume= 20 |year=2007|number= 1|pages=99–148 |doi=10.1090/S0894-0347-06-00546-7|arxiv=math/0607446|bibcode=2007JAMS...20...99C|s2cid= 26614691 }}
{{cite journal|first1=D. J. | last1=Wales|first2=H. | last2=McKay | first3=E. L. | last3=Altschuler| title=Defect motifs for spherical topologies| journal=Phys. Rev. B|year=2009 | volume=79 | pages=224115 | issue=22 | doi=10.1103/PhysRevB.79.224115| bibcode=2009PhRvB..79v4115W}}. Configurations reproduced in {{cite web|first1=D. J. | last1=Wales | first2=S. | last2=Ulker | url=http://www-wales.ch.cam.ac.uk/~wales/CCD/Thomson2/table.html|title=The Cambridge cluster database}}
{{cite journal|first1=W. J. M. |last1=Ridgway |first2=A. F. | last2=Cheviakov|doi=10.1016/j.cpc.2018.03.029| year=2018|journal=Comput. Phys. Commun. | title=An iterative procedure for finding locally and globally optimal arrangements of particles on the unit sphere|volume=233|pages=84–109|bibcode=2018CoPhC.233...84R |s2cid=52097788 }}
{{cite web|first1= Cris|last1= Cecka|first2= Mark J.|last2= Bowick|first3= Alan A.|last3= Middleton|url= http://thomson.phy.syr.edu/|title= Thomson Problem @ S.U.|access-date= 2009-11-24|archive-date= 2018-04-09|archive-url= https://web.archive.org/web/20180409064239/http://thomson.phy.syr.edu/|url-status= dead}}
This webpage contains many more electron configurations with the lowest known energy: https://www.hars.us.

Category:Electrostatics

Category:Electron

Category:Circle packing

Category:Unsolved problems in mathematics