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 $W \times R$.


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