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.

A triangular array with nearest-centre spacing

2 r

has primitive cell area

23 r^{2}

. Dividing the disc area

π r^{2}

by that area gives packing fraction

π / (23) ≃ 0.9069

Sphere packing

 1  0

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

A model assigning nonoverlapping spheres to particles; contacts express geometric constraints. A crystallographic sublattice need not be physically close-packed even when its Bravais lattice is face-centred cubic.

Simple-cubic sphere packing ( $η = π /6$ )

 0  0

Equal spheres of radius

r

occupy the sites of a cubic lattice of spacing

2 r

. A cell contains one sphere, so its packing fraction is

(4 π r^{3} /3) / (2 r)^{3} = π /6

. This is an ideal arranged deposit rather than a claim about all settling suspensions.

Circle packing

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

π /4

Descartes' circle theorem

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Four mutually tangent circles with signed curvatures

k_{i}

satisfy

(k_{1} + k_{2} + k_{3} + k_{4})^{2} = 2 (k_{1}^{2} + k_{2}^{2} + k_{3}^{2} + k_{4}^{2})

. Curvature is the reciprocal of the radius, with an enclosing circle assigned negative curvature. Given three curvatures, solving the quadratic equation gives

k_{4} = k_{1} + k_{2} + k_{3} \pm 2 k_{1} k_{2} + k_{2} k_{3} + k_{3} k_{1}

, providing the two completions of the configuration.

Alternating-radius square circle packing ( $η = π (r_{1}^{2} + r_{2}^{2}) / (2 d^{2})$ )

 0  0

Place equal numbers of radii

r_{1}, r_{2}

on alternating sites of a square nearest-neighbour lattice of spacing

d

. Nonoverlap requires

d \geq max (r_{1} + r_{2}, 2 r_{1}, 2 r_{2})

. A two-circle cell has area

2 d^{2}

, giving the displayed packing fraction. Its maximum within this geometry occurs at large-to-small radius ratio

1 + 2

and is

π (1 - 1/ 2)

; smaller interstitial circles leave unused space, while larger ones force the large-circle lattice apart.

Packing fraction ( $η$ )

 0  0

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.