(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 The energy of a continuous spherical shell of charge distributed across its surface is given by 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 represents the uniform distribution of a single electron's charge, across the entire shell. Randomly distributed point charges The global energy of a system of electrons distributed in a purely random manner across the surface of the sphere is given by 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, Charge-centered distribution For every Nth solution of the Thomson problem there is an 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, Thus, if is known exactly, then is known exactly. In general, is greater than , but is remarkably closer to each th Thomson solution than and . 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. 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. 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