Cubic Unit Cell Calculator
Face-centred cubic reaches 74.05 percent — Kepler conjectured it in 1611 and it was proved in 1998.
The formula
packing = atoms * (4/3) pi r^3 / a^3 ; density = atoms * M / (NA * a^3)
Three ways to stack spheres in a cube
A simple cubic cell has an atom at each corner and fills 52 % of space. Adding one in the middle gives body-centred cubic at 68 %. Putting them on the faces instead gives face-centred cubic at 74 %, which is the densest packing of equal spheres that exists in three dimensions.
Kepler was right, and it took 387 years
Kepler conjectured in 1611 that no arrangement beats 74 %. It was not proved until Thomas Hales did so in 1998, with a proof so dependent on computer verification that the referees eventually certified it only as "99 % certain" — and a fully formal machine-checked proof was not completed until 2014.
Counting atoms in a cell
A corner atom is shared between eight neighbouring cells and so contributes an eighth; a face atom is shared between two and contributes a half; one in the body belongs entirely to its own cell. That bookkeeping is what gives 1, 2 and 4 atoms per cell for the three structures.
Where these models stop
Michaelis-Menten assumes a single substrate and a steady state; Langmuir assumes one layer on identical sites with no interaction between them; Stokes-Einstein assumes a hard sphere much larger than the solvent molecules around it. Each is an idealisation that happens to describe real systems well over a useful range, and each fails predictably outside it. Knowing which assumption a number rests on is usually more valuable than the number.