Decagrammal symmetry of decagonal Al₇₈Mn₂₂ quasicrystal

General ideas about symmetries of quasicrystals based on simple self-similar tiling models and their mathematical formulation in terms of higher-dimensional multimetrical space groups find extensive confirmation in the structure of the decagonal Al₇₈Mn₂₂ quasicrystal phase. There is an incredible richness and variety of symmetries involving, in addition to mirror, rotation, translation and screw-rotation symmetries, planar and linear scalings as well, together with involutions generating those scalings, with and without associated nonprimitive, translations. The linear parts of these symmetries generate a point group of infinite order, not yet fully investigated but, up to now, consistent with the symmetry of a self-similar decagram. The applicability of these symmetries to the atomic structure of the quasicrystal Al₇₈Mn₂₂ observed in nature requires the concept of higher-dimensional crystal forms and their projections in the physical space and in the internal space, respectively.