Tilings Encyclopedia

Model Set

A model set is a discrete point set (more precisely, a Delone set) arising from a cut and project scheme. A theorem of Hof, generalized by Schlottmann, states that each model set is pure point diffractive [Hof95] , [Sch00] . In connection with quasicrystals, it is of interest if a substitution tiling can serve as a model for a physical quasicrystal. Since physical quasicrystals are detected via their diffraction properties, this leads to the question whether a substitution gives rise to a model set. This is true for many well-known substitution tilings, in particular for the Penrose Rhomb tilings and the Ammann-Beenker tilings. These are mld with model sets with Euclidean internal spaces and with polytopal windows. Other model sets may arise from Euclidean internal spaces and fractally shaped windows, like the conch or the tribonacci tilings. There are also tilings corresponding to model sets with p-adic internal spaces - like the chair and the sphinx tilings - as well as tilings corresponding to model sets with mixed p-adic and Euclidean internal spaces, like the Watanabe Ito Soma 8-fold tilings. Another interesting property of model sets is that each Meyer set is a subset of a model set. A Meyer set is a Delone set L, such that L-L is contained in L+F, where F is some finite set [Mey95] , [Mey72] . A central question in the theory of nonperiodic substitution tilings is the PV-conjecture: Essentially, it asks whether every primitive substitution tiling which inflation factor is an irrational PV number is mld to a model set (possibly under further conditions). Currently, the conjecture is proven for one-dimensional tilings with two prototiles [Hol03] , [Sir02] . The general case (more prototiles, higher dimensions) remains open.

Bowtie-Hexagon

One of several substitution tilings found and submitted in February 2016 by L. Andritz with fivefold symmetry using tiles inspired by Islamic Girih patterns. Despite the similarities the tiling is different to the “Tie and Navette” tiling as discussed in [Lue1990] and [LL1994] .

Finite Rotations Model Set Polytopal Tiles Self Similar Substitution Finite Local Complexity

Watanabe Ito Soma 8-fold

This tiling was originally introduced in [WSI87] , however the description given there admits several substitution rules. This is the version given explicitly in [WSI95] . This is an example of a cut and project with a mixed internal space, a product of Euclidean and $p$-adic spaces, namely $\mathbb{R}^2 \times \mathbb{Q}_2$.

Finite Rotations Model Set Rhomb Tiles Polytopal Tiles Self Similar Substitution Finite Local Complexity

References

[Mey72]
Meyer, Y
Algebraic numbers and harmonic analysis
North-Holland Publishing Co. 1972, Amsterdam, North-Holland Mathematical Library, Vol. 2,

[Sch00]
Schlottmann, M
Generalized model sets and dynamical systems
Directions in mathematical quasicrystals 2000, CRM Monogr. Ser., 13, Amer. Math. Soc, Providence, RI, 2000., 143–159, MR1798991http://www.ams.org/mathscinet-getitem?mr=MR1798991

[Hof95]
Hof, A
On diffraction by aperiodic structures
Comm. Math. Phys. 1995, 169, 1, pp. 25--43, MR1328260

[Hol03]
Hollander, M, Solomyak, B
Two-symbol Pisot substitutions have pure discrete spectrum
Ergodic Theory Dynam. Systems 2003, 23, 2, pp. 533--540, MR1972237

[Mey95]
Meyer, Y
Quasicrystals, Diophantine approximation and algebraic numbers
Beyond quasicrystals (Les Houches, 1994), Springer, Berlin 1995, MR1420415

[Sir02]
Sirvent, V F, Solomyak, B
Pure discrete spectrum for one-dimensional substitution systems of Pisot type
Canad. Math. Bull. 2002, 45, 4, pp. 697--710, MR1941235