Liquids and Glasses in Spaces of Incommensurate Curvature

Supercooled liquids and metallic glasses can be viewed as defected states of bond orientational order. Surfaces of constant negative curvature contain an irreducible density of point disclinations in a hexatic order parameter. Analogous defect lines in an icosahedral order parameter appear in three-dimensional flat space. The Frank-Kasper phases are ordered networks of these lines, which, when disordered, provide an appealing model for structure in metallic glasses.