A crystal minimises its total surface energy for a fixed volume. The Wulff theorem says the equilibrium distance from the crystal's centre to each facet plane is directly proportional to that facet's surface energy:
r_hkl = C · γ_hkl (Wulff's theorem, 1901)
For a face-centred-cubic metal the two lowest-energy facet families are the six {100} planes (cube faces) and the eight {111} planes (octahedron faces). Placing each plane at a distance proportional to its own γ and intersecting all fourteen half-spaces gives the equilibrium polyhedron:
- γ{100} ≫ γ{111} (large ratio) → the {111} planes cut deep into the corners → octahedron.
- γ{100} ≈ γ{111} → a mix of squares and hexagons → truncated octahedron (the real equilibrium shape of gold and platinum nanocrystals, since γ{111}/γ{100} ≈ 0.87 for FCC metals).
- γ{100} ≪ γ{111} (small ratio) → the {100} planes dominate → cube, passing through truncated-cube and the exact cuboctahedron on the way.
The Temperature slider mimics thermal facet roughening: as T rises toward the roughening transition, the surface-energy anisotropy between facet families shrinks and the shape relaxes toward a rounder, more isotropic polyhedron.
Real-world relevance: this exact geometric argument explains why colloidal gold nanoparticles used in catalysis, plasmonics and drug delivery are grown as truncated octahedra and cuboctahedra rather than perfect spheres or cubes — shape controls which facet is exposed, and facet type controls catalytic activity and optical response.