Symmetry,Landau theory and polytope models of glass

Order in supercooled liquids and metallic glasses is related to a regular icosahedral “crystal” consisting of 120 particles inscribed on the surface of a sphere in four dimensions. Hyperspherical harmonics and the discrete symmetry group of this four-dimensional platonic solid can be used to construct an order parameter for glasses in three-dimensional flat space. A uniformly frustrated Landau expansion in this order parameter suggests a ground state with a regular array of wedge disclination lines. Homotopy theory is used to classify all topologically stable defects. A generalization of Bloch's theorem for electronic states in flat space solids allows explicit diagonalization of tight binding models defined on the curved-space icosahedral crystal.