A 2D crystal minimises its total edge energy for a fixed cross-sectional area. The Wulff theorem says the equilibrium distance from the crystal's centre to each facet line is directly proportional to that facet's edge energy:
r_hk = C · γ_hk (Wulff's theorem, 1901 — 2D case)
For a square-lattice crystal cross-section the two lowest-energy edge families are the four {10} edges (axis-aligned, like cube faces) and the four {11} edges (diagonal, 45°, like octahedron faces). Placing each line at a distance proportional to its own γ and intersecting all eight half-planes gives the equilibrium polygon:
- γ{11} ≤ γ{10} (diagonal cheap) → the {11} lines cut deep → pure diamond (rotated square).
- γ{10} < γ{11} < 2·γ{10} → both families expose an edge → octagon, continuously reshaping as the ratio changes (the real equilibrium cross-section of small gold and platinum nanoislands, since γ{11}/γ{10} ≈ 0.87·√2 ≈ 1.23 for these metals).
- γ{11} ≥ 2·γ{10} (diagonal expensive) → the {11} lines never reach the corners → pure square.
Unlike the 3D Wulff polyhedron (which passes through five distinct combinatorial shapes — cube, truncated cube, cuboctahedron, truncated octahedron, octahedron), the 2D construction only has three: a 2D crystal has one fewer degree of geometric freedom, so the "in-between" region is a single continuously-deforming octagon rather than several distinct polytopes.
The Temperature slider mimics thermal edge roughening: as T rises toward the roughening transition, the edge-energy anisotropy between facet families shrinks and the shape relaxes toward a rounder, more isotropic polygon.
Real-world relevance: this same 2D Wulff argument sets the equilibrium island shape of atomic terraces and 2D epitaxial islands grown on a crystal surface (e.g. Pt or Au adatom islands on a (100) face) — scanning-tunnelling-microscope images of such islands show exactly this square/octagon/diamond family of shapes as growth conditions change the relative step-edge energies.