Why perfect icosahedral order cannot tile 3D space — the geometric origin of glass
An icosahedron — 12 spheres around a central one — is the locally optimal packing. Yet icosahedral order cannot tile 3D space: the solid angle deficit causes frustration.
Sadoc & Mosseri (1982): map frustrated packing to a curved manifold (4D sphere S³). Defects in the flat-space arrangement correspond to disclinations in the spherical crystal.
Particles colored by bond-orientational order:
Blue = icosahedral-like (q₆ high)
Red = defect / crystalline
Green = intermediate
Fast quench → many frustrated regions. Slow anneal → local icosahedral clusters grow but cannot percolate → glass.