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found such that the average local packing fraction ¯ φ ( R$_{min}$ ( N )) of spheres of unit diameter within the windows is equal to the infinite-volume packing fraction φ ∞ of the spheres of unit diameter in R $^{d}$. The concepts of overlay and rotation are illustrated in Fig. 3 for a triangular lattice of disks of unit diameter. 

<img src='content_image/344618.jpg'>

It suffices to apply the aforementioned construction to periodic packings, as it has been shown that periodic packings in R d can obtain an infinite-volume packing fraction φ ∞ ar- bitrarily close to the maximal infinite-volume packing fraction φ ∞ ∗ (for example, see [22]). A periodic packing can be defined in terms of a lattice Λ, where Λ in R d is a subgroup consisting of the integer linear combinations of a set of vectors that constitute a basis for R $^{d}$. For identical nonoverlapping spheres, a lattice packing is a packing where the centers of the spheres are located at the points of Λ. In such a lattice packing, the space R d can be divided into finite-size identical nonoverlapping regions called fundamental cells, each containing the center of only one sphere. 

A periodic packing is a more general formulation of a lattice packing. For identical nonoverlapping spheres, a periodic packing is obtained by placing a fixed configuration of a number M of spheres in a fundamental cell that is then periodically replicated (without overlap between cells or spheres) to cover$_{R}$ $^{d}$. The fixed configuration of M spheres within 

