Local rules for pentagonal quasi-crystals (Q1894718)

This is an excellent survey which summarizes the most important methods in the theory of quasi-crystals, and contains some new initiatives of the author as well. The author collects the ideas of different schools from east and west, and illustrates these with the pentagonal quasi-crystals, i.e. generalized Penrose two-dimensional tilings. Let us cite the author's abstract. ``The existence of different kinds of local rules is established for many sets of pentagonal quasi-crystal tilings. For each real \(t\in \mathbb{R}\) there is a set \(\overline{\mathbb{T}}_t\) of pentagonal tilings of the same local isomorphism class; the case \(t = 0\) corresponds to the Penrose tilings. It is proved that the set \(\overline {\mathbb{T}}_t\) admits a local rule which does not involve any colorings (or markings, decorations) if and only if \(t = m + n(1 + \sqrt{5})/2\). In other words, this set of tilings is totally characterized by patches of some finite radius, or \(r\)-maps. When \(t = (m + n\sqrt{5})/q\) \((m\), \(n\), \(q\) are integers) the set \(\overline {\mathbb{T}}_t\) admits a local rule which involves colorings. For the set of Penrose tilings the construction here leads exactly to the Penrose mathing rules. Local rules for the case \(t = 1/2\) are presented''. In Section 1 the basic definitions and facts are introduced, then (Section 2) the cut method as projection of an \(\mathbb{R}^5\)-tiling onto 2-subplanes is described. In Section 3-5 the local rules in the abstract are discussed. The used technical result is proved in Section 6. Section 7 contains the example \(t = 1/2\), some generalizations and concluding remarks. In the Appendix the author proves a generalization of a Levitov's result concerning the weak local rule.