FIG. 4: Illustration of a rigid rotation of a window packing with parallelogram fundamental cells such that the ratios of the components of v$_{1}$ in the directions of each of u$_{1}$ and u$_{2}$ to the magnitudes of u$_{1}$ and u$_{2}$ , respectively, are both irrational. FIG. 5: maximal packing fraction φ ∞ ∗ = 0 . 906900 . . . compared to putative upper bounds on φ ∞ ∗ for d = 2 as determined by putative optimal solutions to the DLP problem for selected N$_{∗}$ ∈$_{N}$ up to N$_{∗}$ = 348. The minimum upper bound determined here is at N$_{∗}$ = 336, ˆ φ$_{∗}$ (336) = 0 . 928114. φ$_{max}$ ( R$_{min}$ ( N )) is clearly greater than or equal to φ $^{∞}$, using Eq. (7) we have for N$_{∗}$ ∈$_{N}$ in$_{R}$ $^{d}$. Equation (8) is true for all feasible φ ∞ including the maximal infinite-volume packing 12 ˆ φ$_{∗}$ ( N$_{∗}$ ) ≥ φ$_{max}$ ( R$_{min}$ ( N$_{∗}$ )) ≥ φ $^{∞}$, (8) ( v$_{1 }$● u $_{1}$) / | u$_{1}$ | ( v$_{1 }$● u $_{2}$) / | u$_{2}$ | u$_{1}$ u$_{2}$ v$_{1}$ v$_{2}$ 0 50 100 150 200 250 300 350 N 0.2 0.4 0.6 0.8 1 1.2 1.4 Upper bound on  2D maximal packing fraction  /(12) (1/2)