A problem of determining the macroscopic or effective thermal conductivity of an N-dimensional composite medium containing N-dimensional nonoverlapping hyperspherical inclusions is considered. Since the macroscopic conductivity is expected to become less sensitive to the detailed spatial distribution of the inclusions for N ≥ 4, only the special case of periodic arrangement of the inclusions is considered. An expression for the macroscopic conductivity correct to O(χ3N + 8), χ being the ratio of "diameter" of the inclusions to the spacing between them, is derived and the numerical results for the conductivity are presented as a function of χ and N for the two special cases of perfectly conducting and insulating inclusions. The effective conductivity of the composite is found to approach that of the continuous matrix in higher dimensions.