Description of hexagonal Frank-Kasper phases by a projection method

The structure of hexagonal Frank-Kasper (FK) phases can be described by the projection of the seven-dimensional (7D) Cr₃Si and Zr₄Al₃ cubes in a suitable projection subspace onto a 3D hyperplane. The close relationship between hexagonal FK phases and the dodecagonal quasicrystal with 12-fold rotational symmetry is discussed.