Abstract: The problem of finding the minimum energy configurations of n electrons on the surface of the unit sphere, which is known as the Thomson problem, is extremely challenging for large n. The solutions to the Thomson problem have been found useful in a variety of problems from modeling spherical viral structures to designing acquisitions in Magnetic Resonance Imaging (MRI). In this talk, the focus will be on several variants of the Thomson problem, involving the minimum energy configurations that are endowed with antipodal symmetry and mirror reflection symmetry. It will be argued that the antipodally symmetric configurations are appropriate for MR acquisition design because the raw MR data (in spatial frequency domain) and its image data (in real spatial domain) are related through the Fourier relationship and the image data, to a good approximation, is real-valued. There is an interesting link between the minimum energy configurations subject to a mirror reflection symmetry constraint to those of two-dimensional system of charged particles in a quadratic potential well.