Matching Rules

Many interesting substitution tilings can alternatively be generated by a matching rule. Examples are (again) the Penrose Rhomb tilings: Note the red arcs on each tile. The matching rule is given by the condition that tiles have to meet in a way such that the arcs of each tile are connected with arcs on the neighboured tiles. The set of all tilings fulfilling this local condition is exactly the set of all tilings generated by the Penrose Rhomb substitution. In particular, the condition forces nonperiodic tilings. (Of course there are periodic tilings made of the Penrose rhomb tiles, but these do not obey the matching rule.) If one reads ‘this substitution tilings can be generated by a matching rule’ or so, two cases has to be distincted: Usually a matching rule is formulated by using a decoration of the prototiles (like the red arcs on the Penrose Rhomb). Either the original tilings are mld to the decorated ones, or they are just locally derivable from the decorated ones (but not vice versa). In the latter case, there is no way to derive the decorated tiling from the undecorated in a unique way locally. In the former case, the matching rule can be given by an ‘atlas’ of undecorated clusters. For instance, Danzer’s 7-fold tilings can be defined by the atlas of its vertex stars. Those tilings are classified here under ‘known matching rules without decoration’, in contrast to ‘… with decoration’. Goodman-Strauss showed in [Goo98] that every ‘nice’ family of substitution tilings can be generated by a matching rule with decoration.

Preview Ammann-Beenker
Ammann-Beenker

In 1977 R. Ammann found several sets of aperiodic tiles. This one (his set A5) is certainly the best-known of those. It allows tilings with perfect 8fold symmetry. The substitution factor is $1+\sqrt{2}$ - sometimes called the ‘silver mean’ - which was the first irrational inflation factor known which is not related to the golden mean. In 1982 F. Beenker described their algebraic properties, essentially how to obtain it by the projection method, following the lines of N.

With Decoration Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling Rhomb Tiles Mld Class Ammann Matching Rules

Preview Danzer's 7-fold
Danzer's 7-fold

A substitution tiling with three triangles as prototiles, based on 7-fold symmetry. The four different edge lengths occurring are $\sin(\frac{\pi}{7})$, $\sin(\frac{2\pi}{7})$, $\sin(\frac{3\pi}{7})$, $\sin(\frac{2\pi}{7}) + \sin(\frac{3\pi}{7})$, The inflation factor is $1+{\sin(\frac{2\pi}{7})}/{\sin(\frac{\pi}{7})}$ , which is not a PV number. There are simple matching rules for the tiling. In fact, the list of all vertex stars occurring in the substitution tiling serves as one. This is stated in [ND96], but never really published, up to my knowledge.

Without Decoration Finite Rotations Polytopal Tiles Self Similar Substitution Matching Rules

Preview Penrose Rhomb
Penrose Rhomb

Certainly the most popular substitution tilings. Discovered in 1973 and 1974 by R. Penrose in - at least - three versions (Rhomb, Penrose kite-dart and Penrose Pentagon boat star), all of them forcing nonperiodic tilings by matching rules. It turns out that the three versions are strongly related: All three generate the same mld-class. These tiles, their matching rules and the corresponding substitution was studied thoroughly in [GS87] . A lot of information can be found there.

Without Decoration Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling Rhomb Tiles Mld Class Penrose Matching Rules

Preview Shield
Shield

In connection with physical quasicrystals, the most interesting 2dim tilings are based on 5-, 8-, 10- and 12-fold rotational symmetry. This 12-fold tiling was studied by F. Gähler, in particular its cut and project scheme, the local matching rules and diffraction properties [Gah88]. The window of the vertex set of the shield It is mld to the Socolar tiling, thus they share many interesting properties. One is that they possess a local matching rules.

With Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Mld Class Shield And Socolar Matching Rules

Preview Socolar
Socolar

In connection with physical quasicrystals, the most interesting 2dim tilings are based on 5-, 8-, 10- and 12-fold rotational symmetry. This 12-fold tiling was studied thoroughly in [Soc89], where J. Socolar described the generating substitution as well as the local matching rules and the cut and project scheme, As well as the Penrose Rhomb tilings (5- resp. 10-fold) and the Ammann-Beenker tilings (8-fold), it allows a decoration by Ammann bars (see [GS87]).

Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Parallelogram Tiles Canonical Substitution Tiling Mld Class Shield And Socolar Matching Rules

References

[Goo98]
Goodman-Strauss, Chaim
Matching rules and substitution tilings
Ann. of Math. (2) 1998, 147, 1, pp. 181--223, MR1609510