The densest packing of equal circles on a sphere

Conjectured solutions of the packing problem are found by using `exact' equations to refine approximations generated by the repulsion-energy method. Initial positions are selected randomly, and no symmetry constraints are imposed. Improved and new results are presented for many cases in the range of 15–90 circles. Several properties of the resulting structures are discussed with emphasis on symmetry and other characteristics useful for identification and comparison. A preliminary set of rules for producing a standard orientation of such structures is presented.