Closest Packing of Disks and Spheres; Kepler's Conjecture
Closest Packing of Disks and Spheres; Kepler's Conjecture
Disks (like pennies) can be arranged in the plane in a close-packed rhombic lattice, with lattice angle . The fraction of the plane occupied by the disks, known as the packing fraction, is given by
α
f(α)=
π
4sinα
For a square lattice with , . When , the array becomes a hexagonal lattice, with the maximum possible value of the packing fraction: . Choosing "disks" shows the lattice and packing fraction for selected values of .
α=90°
f=≈0.7854
π
4
α=60°
f=≈0.9069
π
2
3
α
The packing of spheres in 3D space is a far more complex problem, as discussed in the Details. The lattice is defined by two angles and , with the second angle determining the relative displacement of successive planar layers. The spheres in each layer are placed directly over the cavities in the preceding layer. Since there are two possible sets of cavities, a regular arrangement of layers can be either a sequence like a b c a b c … or an alternative sequence like a b a b a b …. In either case, the packing fraction is given by
α
β
f(α,β)=
πsinβ
6sinα(1-cosβ)
1+2cosβ
Choosing gives a simple cubic lattice with packing fraction . Choosing gives either the cubic closest packing (for a b c …) or hexagonal closest packing (for a b a b …). In either case, . Kepler's conjecture claims that this is the maximum possible packing fraction for spheres in 3D. Most practicing physicists would be quite satisfied that this result is correct, but mathematicians are another story, as described in Details.
α=β=90°
f=≈0.5236
π
6
α=β=60°
f=≈0.7405
π
3
2
In the graphics for "spheres", you can move the layers apart for a closer view of their structure.