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. 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 10 FIG. 3: Illustration of a rotation of an infinite packing of identical nonoverlapping windows of radius R , arranged on the sites of a triangular lattice, overlayed upon an infinite packing of smaller iden tical nonoverlapping disks. A rotation is selected such that irrational ratios are achiev ed between the components of at least one of the lattice vectors of the packing of windows in the directions of the lattice vectors of the packing of smaller disks. As a result, at large distances from the axis of rotation, any window can be thought of as being placed at random onto the packing of smaller disks. It follows that the average local packing fraction ¯ φ ( R ) of the smaller disks within the windows is equal to the infinite-volume packing fraction φ ∞ of the smaller disks.