Which symmetry will an ideal quasicrystal admit?

The crystallographic nature of a quasicrystal structure is expressed in terms of the possibility of labeling `translationally' equivalent atomic positions by a set of n integers. The corresponding position vectors are integral linear combinations of n basic ones generating a vector module M of rank n and dimension m. Because of the aperiodic nature of the quasicrystal, n is larger than m. Typical values observed in nature are m = 3 and n = 5 or n = 6. Lattice symmetry is recovered by embedding the quasicrystal in an n-dimensional space (the superspace) in such a way that M is the projection of a lattice 2. The rotational symmetries of the quasicrystal are included in those of the vector module M and, after embedding, appear as n-dimensional rotations leaving Σ invariant and a corresponding Euclidean metric. Scaling symmetries are also possible in the atomic point-like approximation of a quasicrystal. In that case, enlarging by a given constant factor all the distances between `translationally' equivalent atoms, the `inflated' pattern still belongs to the original one: the occupied atomic positions in space are transformed into other ones also occupied in the original structure. This is called inflation procedure (of a scaling invariant pattern), the reverse transformation being a deflation. The module M is then invariant with respect to such discrete dilatations. In the superspace these correspond to crystallographic point-group transformations leaving the lattice Σ and an indefinite metric invariant. Scaling symmetries in space appear as hyperbolic rotations in the superspace. In these non-Euclidean rotations the improper ones are included. The compatibility between the two types of n-dimensional point-group symmetries (Euclidean and non-Euclidean rotations) is discussed both at the level of the quasicrystal structure and of that of the double metrical nature of the translational lattice 2. For a characterization of the symmetry of the quasicrystal, one eventually arrives at the concept of the scale-space group, which includes as its Euclidean subgroup an n-dimensional space group (the super-space group). Examples are taken from aperiodic tilings admitting inflation–deflation symmetry. The vertices of these tilings are supposed to represent `translationally' equivalent atomic positions. A number of basic concepts not expected to be familiar to crystallographers, even if explained in the text, are also listed and defined in an Appendix.