Icosahedral order in glass: Acoustic properties

An icosahedral crystal in the non-Euclidean three-dimensional space $S^3$ models domains of icosahedral order in metallic glass. Crystal symmetries allow explicit construction and classification of vibrational eigenstates. We evaluate phonon frequencies numerically. Long-wavelength, low-frequency vibrations follow the predictions of continuum elastic theory in $S^3$, and of a hypothetical perfect icosahedral crystal in $R^3$. Isotropy of elastic moduli distinguishes icosahedral order from face-centered-cubic order.