The optimal design of periodic structures under the macro scale and that of periodic materials under the micro scale are treated differently by the current topology optimization techniques. Nevertheless, a material point in theory could be considered as a unit cell in a periodic structure if the number of unit cells approaches to infinity. In this work, we investigate the equivalence between optimal solutions of periodic structures obtained from the macro scale approach on the structure level and those of material microstructures obtained from the micro scale approach using the homogenization techniques. The minimization of the mean compliance of the macrostructure with a volume constraint is taken as the optimization problem for both structural and material designs. On the macro scale, we solve the optimization problem by gradually increasing the number of unit cells until the solution converges, in terms of both the topology and the objective function. On the micro scale, the optimal microstructure of the material is obtained for the macrostructure under prescribed loading and support conditions. The microstructure of the material compares very well with the corresponding optimal topology from the periodic macrostructural design. This work reveals the equivalence of the solutions from the macro and micro approaches, and proves that an optimal finite periodic solution remains valid through cell refinement to infinite periodicity.