content_image/367967.jpg content_image/367968.jpg content_image/367966.jpg
<img src='content_image/367967.jpg'>

<img src='content_image/367968.jpg'>

φ$_{max}$ ( R$_{min}$ ( N )) is clearly greater than or equal to φ $^{∞}$, using Eq. (7) we have 

for N$_{∗}$ ∈$_{N}$ in$_{R}$ $^{d}$. 

<img src='content_image/367966.jpg'>

Equation (8) is true for all feasible φ ∞ including the maximal infinite-volume packing 

