Alternating-radius square circle packing ($\eta =\pi (r_1^2+r_2^2)/(2d^2)$)

Place equal numbers of radii $r_1,r_2$ on alternating sites of a square nearest-neighbour lattice of spacing $d$. Nonoverlap requires $d\ge \max (r_1+r_2,\sqrt{2}{r}_1,\sqrt{2}{r}_2)$. A two-circle cell has area $2d^2$, giving the displayed packing fraction. Its maximum within this geometry occurs at large-to-small radius ratio $1+\sqrt{2}$ and is $\pi (1-1/\sqrt{2})$; smaller interstitial circles leave unused space, while larger ones force the large-circle lattice apart.