Incommensurate structure and stability of mullite

A new complete scheme of symmetry analysis for incommensurate modulated structures is applied for the first time to the important ceramic material mullite ${\mathrm{Al}}_2$(${\mathrm{Al}}_{2+2\mathrm{x}}$${\mathrm{Si}}_{2\mathrm{-}2\mathrm{x}}$)O ${}_{10\mathrm{-}x}{}^{}{}_{}{}^{}\square _x^{}{}_{}{}^{}$. Together with what is known of the average structure, the symmetry suffices to determine the complete Al-Si and oxygen-vacancy ordering patterns. The reason why the incommensurate structure is so stable becomes immediately apparent.