each fundamental cell is arbitrary subject only to the overall nonoverlap condition of the periodic packing of spheres. As used here, the term “lattice” is the same as “Bravais lattice” conventionally used in the physics literature. Consider an infinite periodic packing (with lattice basis vectors { u$_{j}$ } ) of nonoverlapping spheres of unit diameter in R $^{d}$, d > 1, with infinite-volume packing fraction φ $^{∞}$. Place an infinite periodic packing (with lattice basis vectors { v$_{j}$ } ) of identical nonoverlapping windows of radius R$_{min}$ ( N ) over the infinite periodic packing of spheres of unit diameter in the manner of the construction discussed previously. For the radius R$_{min}$ ( N ) of the windows, any positive integer N$_{∗}$ ∈$_{N}$ can be considered, where in$_{R}$ d the set$_{N}$ is defined such that each N$_{∗}$ is the greatest number N of spheres for which R$_{min}$ ( N$_{∗}$ ) = R$_{min}$ ( N ). For example, with N = 1 in two dimensions, R$_{min}$ (1) = 1, and the greatest N for which R$_{min}$ ( N ) = 1 is N = N$_{∗}$ = 6 [24]. It is intuitively clear that for any greatest such N = N$_{∗}$ and R = R$_{min}$ ( N$_{∗}$ ) that where φ$_{max}$ ( R ) is defined as the maximal fraction of space that identical nonoverlapping spheres of unit diameter may cover in a spherical window of radius R [25]. Now rigidly rotate the packing of windows around the center of the original window as in Fig. 3. The component of a window-packing basis vector v$_{n}$ that is in the direction of a given unit-diameter sphere packing basis vector u$_{m}$ is v$_{n}$ · u$_{m}$ / | u$_{m}$ | . A rotation may be found such that the ratios { v$_{n}$ · u$_{j}$ / | u$_{j}$ | $^{2}$} of the components of (at least) one of the window basis vectors v$_{n}$ in the directions of each of the sphere packing basis vectors { u$_{j}$ } to the respective magnitudes {| u$_{j}$ |} of the sphere packing basis vectors are all irrational. This concept is illustrated in Fig. 4 in$_{R}$ 2 for parallelogram fundamental cells. After the rotation, due to the irrationality of the ratios of lattice vector components, the average fraction of space ¯ φ ( R$_{min}$ ( N )) covered by the spheres of unit diameter in each window of the window packing is equal to the infinite-volume packing fraction φ ∞ of the unit-diameter spheres. With this construction, the packing fraction φ ∞ of the spheres of unit diameter, equal to the average local packing fraction ¯ φ ( R$_{min}$ ( N )) of the unit-diameter spheres within a window, can be compared to the maximal local packing fraction ˆ φ$_{∗}$ ( N ) of that same window. As 11 ˆ φ$_{∗}$ ( N$_{∗}$ ) ≥ φ$_{max}$ ( R$_{min}$ ( N$_{∗}$ )) , (7)