Incommensurate icosahedral density waves in rapidly cooled metals

There are three simple phenomenological descriptions of incommensurate icosahedral density waves in rapidly cooled metals, based on stars of the reciprocal-lattice vectors which point to the vertices, edges, or faces of an icosahedron. The icosahedral phase of ${\mathrm{Al}}_{0.86}$ ${\mathrm{Mn}}_{0.14}$ is described by the vertex model. We also discuss extended icosahedral correlations present in a hierarchical Frank-Kasper phase proposed by Mosseri and Sadoc.