A new criterion for the classification of quasicrystalline structures is proposed. A quasicrystal with composition close to a Frank-Kasper phase, which consists exclusively of tetrahedra, should belong to the i((Al, Zn)49Mg32) class whereas one with composition close to that of the crystal, which consists of both tetrahedra and octahedra, should belong to the i(A1MnSi) class. The Ti2(Ni, V) building block consists of four icosahedra and a truncated tetrahedron. Replacing the eight carbon atoms of a diamond unit cell with this building block produces the Ti2(Ni, V) crystal. Placing a distorted Ti2(Ni, V) building block on a threefold axis of an icosahedrally symmetric point group and operating on it by this point group gives a double Mackay icosahedron. It can be concluded that the Ti2(Ni, V) and α-AlMnSi quasicrystals are isomorphic structures since Ti2(Ni, V) and α-A1MnSi crystals consist of both tetrahedra and octahedra.