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.