Cut and Project

A fundamental result in the theory of nonperiodic tilings was the discovery of the fact that some substitution tilings can be obtained by projecting certain points from higher dimensional point lattices. This was first carried out by deBruijn for the Penrose Rhomb tilings [de81] . In the following years it was developed further by a lot of authors (we cannot list all of them; for a start, see the references in [Moo00] and [Fog02] ). This work culminated in the development of the algebraic theory of model sets. Quickly it was realized that this theory is a reformulation of the work of Meyer [Lag96] , [Moo97] . The ingredients for a cut and project scheme are the ‘direct space’ G, where the model set (or the tiling) lives, the internal space $H$, a lattice $L$ in $G \times H$ and a compact set $W$ - the window - in $H$. The lattice $L$ is embedded in $G \times H$ such that the projection $p_H(L)$ of $L$ to $H$ is dense, and the projection $p_G$ to $G$ is invertible on $p_G(L)$. In general, $G$ and $H$ can be choosen as locally compact Abelian groups. In many cases it suffices to choose $G$ and $H$ as Euclidean vector spaces $R^d$ (but see $p$-adic window). Then the set $W$ tells us which points of $L$ will be projected by $p_G$ to $G$: Those which are projected to $W$ by $p_H$. Under these conditions, $V = \{ p_G(x) \mid x \in L, p_H(x) \in W \}$ is a uniformly discrete and relatively dense point set in $G$, called model set. There are many variations of this construction, some of them yielding tilings rather than point sets (e.g., the Klotz construction in [KS89] ). But essentially, the construction above is the main idea behind all the variations of the theme. A simple example is visualized in the description of the Fibonacci tiling. In this example, $G=H=R$, and $W$ is an interval: the strip in the image is $W \times R$.

References

[KS89]
Kramer, P, and Schlottmann, M
Dualisation of Voronoi domains and Klotz construction: a general method for the generation of proper space fillings
J. Phys. A 1989, 22, 23, pp. L1097--L1102, MR1026579

[Moo97]
Moody, R V
Meyer Sets and Their Duals.
The Mathematics of Aperiodic Order 1997, Moody, R V, NATO ASI Series C489, pp. 403-441,

[Lag96]
Lagarias, J C
Meyer's concept of quasicrystal and quasiregular sets.
Comm. Math. Phys. 1996, 179, 2, pp. 365-376, MR1400744

[Moo00]
Moody, R V
Model sets: A Survey
From Quasicrystals to More Complex Systems 2000, Axel F, DÃ©noyer F, and Gazeau J P, Centre de physique Les Houches, Springer,

[Fog02]
N. Pytheas Fogg
Substitutions in Dynamics, Arithmetics and Combinatorics
Lecture Notes in Mathematics,Springer, Berlin 2002, 1794,

[de81]
de Bruijn, N G
Algebraic theory of Penrose's nonperiodic tilings of the plane. I, II
Nederl. Akad. Wetensch. Indag. Math. 1981, 43, no. 1, 39-52, 53-66, 82e:05055