Which symmetry will an ideal quasicrystal admit

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