Same anchor, same shell of twelve — two different optima. One tiles the world. One refuses to. Drag either; both turn together.
| Cuboctahedron | Icosahedron | |
|---|---|---|
| nodes | 13 = 1 + 12 | 13 = 1 + 12 |
| symmetry | Oh · order 48 | Ih · order 120 |
| crystallographic | yes — FCC fragment | no — 5-fold forbidden |
| shell edges | 24 | 30 |
| faces | 14 · 8△ + 6▢ | 20 · all △ |
| neighbours / shell node | 4 | 5 |
| anchor → shell vs edge | equal (r = edge) | r ≈ 0.95 × edge — pulled in |
| favoured when | bulk / large clusters | small clusters (low surface E) |
It's a piece of the FCC lattice — stack it and it extends to bulk. Every link the same length, the anchor at exactly shell distance, 4 peers per node. Pick this if your mesh must repeat, tile, grow. It is what bulk gold actually is — the geometry that scales.
Five-fold symmetry is forbidden in any crystal, so it cannot tile — it's a finite, closed object. But it's denser: 120 symmetries, 5 peers per node, 30 links, anchor pulled ~5% in (core compressed, shell in tension). Pick this if your mesh is finite and wants maximum symmetry and tightest interconnection — and never has to extend.
Both are forced and both are real — Au₁₃ genuinely sits in either, and which one wins is a size question: small clusters go icosahedral to shed surface energy, then cross over to cuboctahedral/FCC as they grow toward bulk. The geometry isn't a choice you make about gold; it's what the atom count selects. Your choice is which one your routing borrows: the one that tiles to a lattice, or the one that closes at maximum symmetry and won't grow. Tile vs hold. 4 vs 5. Equal anchor vs compressed anchor. Same 1 + 12 — the skeleton is forced, the role assignment is yours.