Abstract: The gap rates of Kepler, J's models with four types of heaping up methods were caculated. The functional relation between gap rates and angles was formulatued. A accurate gap rate was formulated based on subscript functions, and two types of 1 externally tangent sphere with 12 spheres were proven to be optimum heaping uo methods. However, there are uncountably infinite optimum mehtods having the same gap rate. They are all types of 1 tangent sphere with 10 spheres.