vertices of an equilateral triangle about any great circle. The great circle is often considered to define an equator about the sphere and the two points perpendicular to the plane are often considered poles to aid in discussions about the electrostatic configurations of many-N electron solutions. Also, this represents the first two-dimensional solution. For N = 4, electrons reside at the vertices of a regular tetrahedron. Of interest, this represents the first three-dimensional solution. For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid. 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. 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. 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 whose faces are square and pentagonal, respectively. Generalizations One can also ask for ground states of particles interacting with arbitrary potentials. To be mathematically precise, let f be a decreasing real-valued function, and define the energy functional Traditionally, one considers also known as Riesz -kernels. For integrable Riesz kernels see the 1972 work of Landkof. For non-integrable Riesz kernels, the Poppy-seed bagel theorem holds, see the 2004 work of Hardin and Saff. Notable cases include: 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