The icosahedron holds — but only while small. Grow it and its strain outpaces its smooth surface, until the FCC lattice wins and difference takes over. There is a size. Drag to find it.
The icosahedron is rounder: more compact, lower-energy facets, fewer exposed atoms per atom. Surface cost scales as N2/3, and at small N the surface is almost the whole cluster — so the smoother shape wins. Symmetry is affordable while you're small.
The icosahedron isn't a true lattice — it's twenty slightly-compressed wedges (the ~5% core squeeze from the cell drawing), and that internal strain scales as N, the whole volume. FCC carries no bulk strain. As N grows the strain term overtakes the surface term — and the lattice wins.
The magic numbers don't decide — energy does. Both shapes share the exact same closed-shell counts (1, 13, 55, 147, 309, 561, 923 …); the icosahedron is just the cuboctahedron under a strain-storing distortion, same atoms per shell. So size alone never tells you which you have. The crossing of two energies does.
And that's the whole arc in one line: a perfect mirror is only stable below a critical size. To grow past it you have no choice but to introduce difference — a lattice, a gradient, a defect, a which-way. Uniformity holds; only delta scales. The icosahedron isn't beaten by a better cluster — it's beaten by its own success, the moment it tries to get big.
Honest tags: the functional form is real — surface ~N2/3 against strain ~N is the actual competition that sets cluster shape. The constants and the crossover N* drawn here are illustrative, not a measured gold value; real gold also passes through decahedral (Marks decahedron) structures between icosahedral and FCC, which this two-curve picture omits for clarity. The crossover for real metals lands somewhere in the hundreds-to-thousands of atoms and is model-dependent. Treat the shape of the story as physics; the exact number as a schematic.