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 is the number of points (charges) in a configuration, is the energy, the symmetry type is given in Schönflies notation (see Point groups in three dimensions), and 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, is the number of vertices where the given number of edges meet, is the total number of edges, is the number of triangular faces, is the number of quadrilateral faces, and is the smallest angle subtended by vectors associated with the nearest charge pair. Note that the edge lengths are generally not equal. Thus, 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. According to a conjecture, if , p is the polyhedron formed by the convex hull of m points, q is the number of quadrilateral faces of p, then the solution for m electrons is f(m): . (comment: table above must be incorrect since an increase in n must produce a decrease in θ, e.g. θ16 < θ17. There are several other examples.) References Notes . . Configurations reprinted in . Configurations reproduced in This webpage contains many more electron configurations with the lowest known energy: https://www.hars.us. Electrostatics Electron Circle packing Unsolved problems in mathematics