Geometric packing

Placement of geometric objects without interior overlap. A packing fraction measures how much area or volume they occupy. Circle packing and sphere packing are examples.

Arrangement of spheres without interior overlap. The occupied volume fraction is its packing fraction; it depends on the spatial arrangement and on the radii.

Equal spheres of radius $r$ occupy the sites of a cubic lattice of spacing $2r$. A cell contains one sphere, so its packing fraction is $(4\pi r^3/3)/(2r{)}^3=\pi /6$. This is an ideal arranged deposit rather than a claim about all settling suspensions.

Arrangement of circles in a plane without interior overlap. The covered area fraction is its packing fraction. Equal-radius square packing has fraction $\pi /4$.

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.

Occupied area or volume divided by the corresponding total area or volume of a geometric packing. For a periodic packing, it is the total object area or volume within a repeat cell divided by the cell measure. It is distinct from the number fraction of objects of a given size.