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Sphere Packing — Definition, Formula & Examples

Sphere packing is the study of how to arrange identical spheres in space so they occupy the largest possible fraction of the total volume. The classic problem asks for the densest arrangement, which in three dimensions turns out to be the face-centered cubic (FCC) lattice, filling about 74% of space.

A sphere packing in Rn\mathbb{R}^n is a collection of non-overlapping spheres of equal radius. The packing density η\eta is the fraction of space covered by the spheres. In R3\mathbb{R}^3, the Kepler conjecture (proved by Hales in 1998) establishes that the maximum density is η=π32≈0.7405\eta = \frac{\pi}{3\sqrt{2}} \approx 0.7405, achieved by the FCC and hexagonal close-packed (HCP) arrangements.

Key Formula

η=π32≈0.7405\eta = \frac{\pi}{3\sqrt{2}} \approx 0.7405
Where:
  • η\eta = Packing density — the fraction of total space occupied by spheres in the densest 3D arrangement

How It Works

Imagine stacking oranges in a crate. The first layer arranges circles in a hexagonal pattern, each touching six neighbors. The second layer nests into the gaps of the first. For the third layer, you can place spheres directly above the first-layer gaps (FCC) or directly above the first-layer spheres (HCP). Both achieve the same maximum density. The packing density is computed by dividing the volume of spheres inside a repeating unit cell by the total volume of that cell.

Worked Example

Problem: Compute the packing density of spheres of radius rr arranged in a face-centered cubic lattice with cube edge length a=22 ra = 2\sqrt{2}\,r.
Count spheres per unit cell: An FCC unit cell has 8 corner spheres (each shared among 8 cells) and 6 face-center spheres (each shared between 2 cells).
N=8×18+6×12=4N = 8 \times \tfrac{1}{8} + 6 \times \tfrac{1}{2} = 4
Total sphere volume in the cell: Four complete spheres contribute:
Vspheres=4×43πr3=163πr3V_{\text{spheres}} = 4 \times \tfrac{4}{3}\pi r^3 = \tfrac{16}{3}\pi r^3
Unit cell volume: The cube has edge a=22 ra = 2\sqrt{2}\,r, so:
Vcell=a3=(22 r)3=162 r3V_{\text{cell}} = a^3 = (2\sqrt{2}\,r)^3 = 16\sqrt{2}\,r^3
Compute density: Divide sphere volume by cell volume:
η=163πr3162 r3=π32≈0.7405\eta = \frac{\tfrac{16}{3}\pi r^3}{16\sqrt{2}\,r^3} = \frac{\pi}{3\sqrt{2}} \approx 0.7405
Answer: The FCC packing density is π32≈74.05%\dfrac{\pi}{3\sqrt{2}} \approx 74.05\%.

Why It Matters

Sphere packing appears in crystallography, where atoms in metals like copper and aluminum adopt FCC structures. It also underlies error-correcting codes in information theory, where "spheres" in high-dimensional spaces represent codeword neighborhoods, and denser packings yield more efficient codes.

Common Mistakes

Mistake: Assuming simple cubic packing is the densest arrangement
Correction: Simple cubic packing has density π/6≈52.4%\pi/6 \approx 52.4\%, far below the FCC/HCP maximum of about 74%. Always distinguish between different lattice types when comparing densities.